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= exp−1 φ(q) = exp−1 φ(q) expφ(p) R dφ(p)v and so dφ(q) ◦ FR,q,p = FR,φ(q),φ(p) ◦ dφ(p) for all p, q ∈ M and all R > 0. Divide by the norm and take the limit R → ∞ to obtain (7.5.11). This proves Lemma 7.5.3. 372 CHAPTER 7. TOPICS IN GEOMETRY Deﬁnition 7.5.4. Let G ⊂ GL(n, R) be a Lie group. A G-action on M by isometries is a Lie group homomorphism G → I(M ) : g → φg, i.e. the map G × M → M : (g, p) → φg(p) is a smooth group action (Deﬁnition 2.5.40) and the map φg : M → M is an isometry for each g ∈ G. In this situation we say that the G action has a ﬁxed point iﬀ there exists an element p ∈ M such that φg(p) = p for all g ∈ G. We say that the G-action has a ﬁxed point at inﬁnity iﬀ the induced G-action on the sphere at inﬁnity has a ﬁxed point, i.e. there exist elements p ∈ M and v ∈ Sp such that dφg(p)v = Fφg(p),p(v) for all g ∈ G. If such a pair (p, v) ∈ S(M ) does not exist, we say that the Gaction has no ﬁxed point at inﬁnity. Deﬁnition 7.5.5. A smooth function f : M → R is called convex iﬀ the function f ◦ γ : R → R is convex for every geodesic γ : R → M. With these preparations in place we are ready to state the following existence theorem for critical points of a convex function (see [17, Theorem
4]). Theorem 7.5.6 (Donaldson). Let M be a Hadamard manifold equipped with a smooth action G → I(M ) : g → φg of a Lie group G by isometries, let K be a compact subgroup of G, and let f : M → R be a convex function such that f ◦ φg = f for all g ∈ G. Assume that the G-action has no ﬁxed point at inﬁnity. Then there exists an element p0 ∈ M such that f (p0) = inf p∈M f (p), φu(p0) = p0 for all u ∈ K. (7.5.12) As pointed out in [17], similar results can be found in the works of Bishop–O’Neill [7] and Bridson–Haeﬂiger [11, Lemma 8.26]. We remark that the compactness of the subgroup K ⊂ G is only needed for an appeal to Cartan’s Fixed Point Theorem 6.5.6. If we assume instead that the action of K on M has a ﬁxed point, compactness is not required. The proof of Theorem 7.5.6 is based in the following lemma (see [17, Lemma 5]). 7.5. CONVEX FUNCTIONS ON HADAMARD MANIFOLDS* 373 Lemma 7.5.7. Let pi be a sequence in M such that limi→∞ d(p, pi) = ∞ for some (and hence every) p ∈ M. Let I ⊂ I(M ) be a collection of isometries of M such that supi d(pi, φ(pi)) < ∞ for all φ ∈ I. Then the isometries in I have a common ﬁxed point at inﬁnity, i.e. there exists an element [p, v] ∈ S∞(M ) such that φ∗[p, v] = [p, v] for all φ ∈ I. Proof. Fix an element p ∈ M and deﬁne vi := exp−1 p (pi) Ri, Ri := \|exp−1 p (pi)\| = d(p,
pi), (7.5.13) for each i ∈ N such that pi = p. Passing to a subsequence, if necessary, we may assume that pi = p for all i ∈ N and that the limit v := limi→∞ vi ∈ Sp exists. We will prove that dφ(p)v = Fφ(p),p(v) for all φ ∈ I. To see this, let φ ∈ I, choose c > 0 such that d(pi, φ(pi)) ≤ c for all i, and deﬁne v i := exp−1 p (φ(pi)) R i, R i := \|exp−1 p (φ(pi))\| = d(p, φ(pi)). Since expp(R iv i) = φ(pi) and expp(Rivi) = pi, Theorem 6.5.2 asserts that \|Rivi − R iv i\| ≤ d(pi, φ(pi)) ≤ c for all i. Since \|Ri − R i\| ≤ c by the triangle inequality, it follows that \|vi − v i\| ≤ 2c Ri and hence lim i→∞ R i = ∞, lim i→∞ v i = lim i→∞ vi = v. Since expp(R iv i) = φ(pi), this implies Fq,p(v) = lim i→∞ exp−1 \|exp−1 q (expp(R q (expp(R iv iv i)) i))\| = lim i→∞ exp−1 \|exp−1 q (φ(pi)) q (φ(pi))\| for all q ∈ M. Take q = φ(p) and use the identity exp−1 and equation (7.5.13) to obtain φ(p) ◦φ = dφ(p) ◦ exp−1 p Fφ(p),p(v) = lim i→∞ dφ(p) exp−1 \|dφ(p) exp−1 p (pi) p (pi)\| = dφ(p) lim i→∞ vi = dφ(p)v. Hence φ∗[p, v] = [p, v] for all φ ∈
I and this proves Lemma 7.5.7. 374 CHAPTER 7. TOPICS IN GEOMETRY The next step in the proof of Theorem 7.5.6 is to examine the gradient ﬂow of the convex function f : M → R. The ﬂow equation has the form ˙γ(t) = −∇f (γ(t)), (7.5.14) where the gradient vector ﬁeld is deﬁned by ∇f (p), v := df (p)v for p ∈ M and v ∈ TpM. An important consequence of convexity is that the distance between any two solutions of the gradient ﬂow equation is nonincreasing. This is the content of the next lemma (see [17, Lemma 6]). Lemma 7.5.8. Let f : M → R be a smooth function. Then f is convex if and only if it satisﬁes the condition ∇v∇f (p), v ≥ 0 (7.5.15) for all p ∈ M and all v ∈ TpM. If f is convex, then the following holds. (i) Equation (7.5.14) has a solution γ : [0, ∞) → M on the entire positive real axis for every initial condition γ(0) = p0. (ii) Let γ0 : [0, ∞) → M and γ1 : [0, ∞) → M be two solutions of (7.5.14). Then the function [0, ∞) → R : t → d(t) := d(γ0(t), γ1(t)) is nonincreasing. Proof. Let γ : R → M be a geodesic. Then f ◦ γ : R → R is convex if and only if 0 ≤ d2 dt2 f (γ(t)) = d dt ∇f (γ(t)), ˙γ(t) = ∇˙γ(t)∇f (γ(t)), ˙γ(t) for all t. This holds for all geodesics if and only if f satisﬁes (7.5.15). In the remainder of the proof we assume that f is convex
. Let p0 ∈ M and for T > 0 deﬁne KT := {p ∈ M \| d(p0, p) ≤ T \|∇f (p0)\|}. This set is compact because M is complete. Now let γ : [0, T ) → M be the solution of (7.5.14) with γ(0) = p0 on some time interval [0, T ). Then d dt \|∇f (γ)\|2 = 2 ∇˙γ∇f (γ), ∇f (γ) = −2 ∇˙γ∇f (γ), ˙γ ≤ 0 by (7.5.15). Thus the function t → \|∇f (γ(t))\| = \| ˙γ(t)\| is nonincreasing and so γ(t) ∈ KT for 0 ≤ t < T. Since KT is a compact subset of M, the solution γ extends to a longer time interval [0, T + δ) for some δ > 0 by Corollary 2.4.15. Since T > 0 was chosen arbitrary, this proves (i). 7.5. CONVEX FUNCTIONS ON HADAMARD MANIFOLDS* 375 We prove part (ii). Assume without loss of generality that γ0(0) = γ1(0) and so γ0(t) = γ1(t) for all t. For t ≥ 0 let [0, 1] → M : s → γ(s, t) be the unique geodesic that satisﬁes γ(0, t) = γ0(t) and γ(1, t) = γ1(t). Then d(t) = d(γ0(t), γ1(t)) = L(γ(·, t)) = \|∂sγ(s, t)\| for all s, t and hence ˙d(t) = 1 0 ∂sγ(s, t), ∇t∂sγ(s, t) \|∂sγ(s, t)\| ds = 1 d(t) 1 0 ∂ ∂s ∂sγ(s, t), ∂tγ(s, t) ds = − = − 1
d(t) 1 d(t) ∂sγ(1, t), ∇f (γ(1, t)) − ∂sγ(0, t), ∇f (γ(0, t)) 1 0 ∂sγ(s, t), ∇∂sγ(s,t)f (γ(s, t)) ds ≤ 0 by (7.5.15). This proves (ii) and Lemma 7.5.8. Lemma 7.5.9. Let f : M → R be a convex function that has a critical point p∞. Then f (p) ≥ f (p∞) =: c for all p ∈ M, and the set Cf := f −1(c) of minima of f is geodesically convex. Proof. Let p ∈ M and let γ : [0, 1] → M be the unique geodesic with the endpoints γ(0) = p∞ and γ(1) = p. Then β := f ◦ γ : [0, 1] → R is a convex function satisfying β(0) = f (p∞) = c and ˙β(0) = 0, hence β(t) ≥ c for all t, and so f (p) = β(1) ≥ c. Thus f attains its minimum at p∞. Now let p0, p1 ∈ Cf and let γ : [0, 1] → M be the unique geodesic with the endpoints γ(0) = p0 and γ(1) = p1. Then the function β := f ◦ γ : [0, 1] → R is convex, satisﬁes β(0) = β(1) = c, and takes values in the interval [c, ∞). Hence β ≡ c and so γ(t) ∈ Cf for all t. This proves Lemma 7.5.9. Proof of Theorem 7.5.6. Choose an element p0 ∈ M and let γ : [0, ∞) → M be the unique solution of equation (7.5.14) that satisﬁes the initial condition γ(0) = p0. Assume ﬁrst that d(p0
, γ(t)) = ∞ sup t≥0 (7.5.16) and choose a sequence ti → ∞ such that limi→∞ d(p0, γ(ti)) = ∞. Now let g ∈ G. Since φg : M → M is an isometry and f ◦ φg = f, it follows that dφg(p)∇f (p) = ∇f (φg(p)) for all p ∈ M (Exercise: Prove this.) Hence the curve [0, ∞) → M : t → φg(γ(t)) is another solution of equation (7.5.14) and hence d(γ(ti), φg(γ(ti))) ≤ d(p0, φg(p0)) for all i and all g ∈ G by part (ii) of Lemma 7.5.8. Hence Lemma 7.5.7 asserts that there exists a (p, v) ∈ SM such that dφg(p)v = Fφg(p),p(v) for all g ∈ G, in contracdiction to our assumption that the G-action has no ﬁxed point at inﬁnity. 376 CHAPTER 7. TOPICS IN GEOMETRY This shows that our assumption (7.5.16) must have been wrong. Thus d(p0, γ(t)) =: R < ∞, sup t≥0 and so our solution γ : [0, ∞) → M of (7.5.14) takes values in the compact set B := {p ∈ M \| d(p0, p) ≤ R}. Since the function t → f (γ(t)) is nonincreasing, this implies that the limit c := lim t→∞ f (γ(t)) ≥ min p∈B f (p) (7.5.17) exists and is a real number (and not −∞). Since d dt f (γ(t)) = −\|∇f (γ(t))\|2 and the function t → f (γ(t)) is bounded below by c, there must exist a sequence ti → ∞ such that lim i→∞ ∇f (γ(ti
)) = 0. Since γ(ti) ∈ B for all i, we may also assume that the limit p∞ := lim i→∞ γ(ti) (7.5.18) (7.5.19) exists (after passing to a subsequence, if necessary). This limit is a critical point of f by (7.5.18), and f (p∞) = c by (7.5.17). Hence, by Lemma 7.5.9, f attains its minimum at p∞ and the set Cf := {p ∈ M \| f (p) = c} of minima of f is geodesically convex. We must ﬁnd an element of Cf that is ﬁxed under the action of K. By Cartan’s Fixed Point Theorem 6.5.6 there exists a q ∈ M such that φu(q) = q for all u ∈ K. Since M is complete and Cf is a nonempty closed subset of M, there exists an element p0 ∈ Cf such that d(q, p0) = inf p∈Cf d(q, p) =: δ. (7.5.20) We claim that φu(p0) = p0 for all u ∈ K. To see this, ﬁx an element u ∈ K, let γ : [0, 1] → M be the geodesic joining γ(0) = p0 to γ(1) = φu(p0), and denote by m := γ(1/2) the midpoint of this geodesic. Then Lemma 6.5.7 asserts that 2d(q, m)2 + d(p0, φu(p0))2 2 ≤ d(q, p0)2 + d(q, φu(p0))2. (7.5.21) Since φu(q) = q and φu is an isometry, we have d(q, φu(p0)) = d(q, p0) = δ, and since Cf is geodesically convex, we have m ∈ Cf and so d(q, m) ≥ δ. Hence it follows from (7.5.21) that p0 =
φu(p0). This shows that p0 ∈ Cf is a ﬁxed point for the action of K on M and proves Theoren 7.5.6. 7.5. CONVEX FUNCTIONS ON HADAMARD MANIFOLDS* 377 Example 7.5.10. To illustrate the argument in the proof of Theorem 7.5.6, take M = C with the standard ﬂat metric. Then the equivalence relation on the sphere bundle SM = C × S1 is given by translation in C and so the sphere at inﬁnity is S∞(M ) = S1. The orthogonal group G = O(2) acts by isometries on M and has no ﬁxed point at inﬁnity. The subgroup K = Z/2Z acts by complex conjugation and its ﬁxed point set is the real axis. Choose a smooth convex function h : R → R that vanishes on the interval [−1, 1] and is positive elsewhere. Then the function f (z) := h(\|z\|) is convex and Ginvariant and the set Cf of minima of f is the closed unit disc in C. If q = 2 is the ﬁxed point of the K-action chosen in the proof, then p0 = 1 ∈ Cf. If G = K = Z/2Z, then ±1 are the ﬁxed points at inﬁnity and f (z) := eRe(z) is convex and G-invariant, but does not take on its inﬁmum. Example 7.5.11 ([17]). Consider the case where M = Dm is the Poincar´e model of hyperbolic space (Exercise 6.4.22). Then the sphere at inﬁnity is the boundary ∂Dm = Sm−1 (Exercise 6.4.24). If the convex function f extends continuously to the closed ball and does not take on its minimum in M, then it attains its minimum at a unique point on the boundary, because any two boundary points are the asymptotic limits of a geodesic in M. Hence the minimum on the boundary is ﬁxed under the action of any Lie group on M by
isometries, that leave f invariant. This is reminiscent of the Kempf Uniqueness Theorem in GIT (see [37] and [20, Theorem 10.2]), where M = G/K is associated to the complexiﬁcation G of a compact Lie group K and f is the Kempf–Ness function (see [38, 53] and [20, §4]). 7.5.2 Inner Products and Weighted Flags We will now turn to a speciﬁc example, where the Hadamard manifold is the space of positive deﬁnite symmetric matrices with determinant one (§6.5.3). Following [17], we choose a ﬁnite-dimensional real vector space V equipped with a ﬁxed inner product ·, ·. Then every inner product on V has the form v, vP := v, P −1v for some self-adjoint positive deﬁnite automorphism P. Denote the set of such automorphisms with determinant one by P0(V ) := P ∈ End(V ) P ∗ = P > 0, det(P ) = 1. (7.5.22) Here P ∗ ∈ End(V ) is deﬁned by v, P ∗v := P v, v for v, v ∈ V, and the notation “P > 0” means v, P v > 0 for all v ∈ V \ {0}. Thus P0(V ) is a codimension-1 submanifold of the space of self-adjoint endomorphisms of V. Its tangent space at P ∈ P0(V ) is given by TP P0(V ) := P ∈ End(V ) P = P ∗, trace P P −1 = 0. (7.5.23) 378 CHAPTER 7. TOPICS IN GEOMETRY The Riemannian metric on P0(V ) is deﬁned by \| P \|P := trace P P −1 P P −1 (7.5.24) for P ∈ P0(V ) and P ∈ TP P0(V ) as in §6.5.3, and so P0(V ) is a Hadamard manifold by Theorem 6.5.10. For
P ∈ TP P0(V ) the endomorphism P P −1 is self-adjoint with respect to the inner product ·, P −1· on V. The group SL(V ) ⊂ GL(V ) of automorphisms of V with determinant one acts on P0(V ) by the isometries φg(P ) := gP g∗ for g ∈ SL(V ). The isotropy subgroup of 1l ∈ P0(V ) is the special orthogonal group SO(V ). The action of a subgroup G ⊂ SL(V ) on V is called irreducible iﬀ there does not exist a linear subspace E ⊂ V, other than E = {0} and E = V, such that gE = E for all g ∈ G. This notion can be used to carry over the general existence theorem in §7.5.1 for critical points of a convex function to the present setting (see [17, Theorem 3]). Theorem 7.5.12 (Donaldson). Let G ⊂ SL(V ) be a Lie subgroup such that the action of G on V is irreducible. Let K ⊂ G be a compact subgroup and let f : P0(V ) → R be a convex function such that f (gP g∗) = f (P ) for all g ∈ G and all P ∈ P0(V ). Then there exists a P0 ∈ P0(V ) such that f (P0) = inf P ∈P0(V ) f (P ), uP0u∗ = P0 for all u ∈ K. (7.5.25) The goal will be to deduce Theorem 7.5.12 from Theorem 7.5.6. Thus we must understand the sphere at inﬁnity of the space P0(V ). This will be accomplished with the help of the following deﬁnition. Deﬁnition 7.5.13. A weighted ﬂag in V is a pair (F, µ), where F is a ﬁnite sequence of linear suspaces {0} = F0 ⊂ F1 ⊂ F2 ⊂ · · · ⊂ Fr = V such that ni
:= dim(Fi)/ dim(Fi−1) > 0 for i = 1,..., r, and µ is a ﬁnite sequence of real numbers µ1 > µ2 > · · · > µr satisfying the conditions r i=1 niµi = 0, r i=1 niµ2 i = 1. (7.5.26) Let F = F (V ) be the set of weighted ﬂags. The group SL(V ) acts on F (V ) by g · (F, µ) := (gFi, µi)i for (F, µ) = (Fi, µi)i ∈ F (V ) and g ∈ SL(V ). 7.5. CONVEX FUNCTIONS ON HADAMARD MANIFOLDS* 379 For P ∈ P0(V ) the unit sphere in the tangent space TP P0(V ) is the set P P −1 P P −1 = 1. P = P ∗ ∈ End(V ) P P −1 = 0, trace trace SP := Let P ∈ P0(V ) and P ∈ SP. Then the endomorphism P P −1 is self-adjoint with respect to the inner product ·, P −1· and hence has only real eigenvalues µ1 > µ2 > · · · > µr. For each i let Ei ⊂ V be the eigenspace for µi and deﬁne ni := dim(Ei). Since trace P P −1 P P −1 = 1, the µi, ni satisfy (7.5.26). The weighted ﬂag of (P, P ) is deﬁned by ιP ( P ) := (F, µ) = (F1, µ1),..., (Fr, µr) ∈ F (V ), P P −1 = 0 and trace (7.5.27) where Fi := E1 ⊕ E2 ⊕ · · · ⊕ Ei for i = 1,..., r. For each P ∈ P0(V ) the map ιP : SP → F (V ) deﬁned by (7.5.27) is bijective, and thus induces a (compact, metrizable) top
ology on F (V ). This topology is independent of P as the next lemma shows. The lemma also shows that the space of weighted ﬂags is the sphere at inﬁnity (see [17, Lemma 4]). Lemma 7.5.14. The equivalence relation in Deﬁnition 7.5.1 on the unit sphere bundle SP0(V ) is given by (P, P ) ∼ (Q, Q) ⇐⇒ ιP ( P ) = ιQ( Q) (7.5.28) for P, Q ∈ P0(V ), P ∈ SP, and Q ∈ SQ. Thus the map FQ,P : SP → SQ as deﬁned in Lemma 7.5.2 is given by FQ,P = ιQ ◦ ι−1 P. Moreover, the map SP0(V ) → F (V ) : (P, P ) → ιp( P ) is SL(V )-equivariant, i.e. ιgP g∗(g P g∗) = g · ιP ( P ) (7.5.29) for all P ∈ P0(V ), all P ∈ SP, and all g ∈ SL(V ). Proof. Since (g P g∗)(gP g∗)−1 = g( P P −1)g−1 for all P ∈ SP and g ∈ SL(V ), equation (7.5.29) follows directly from the deﬁnitions. The proof that the equivalence relation satisﬁes (7.5.28) rests on the following claims. Claim 1. Let S ∈ S1l, let (Fi, µi)r i=1 = ι1l(S) be the ﬂag associated to S, and let h ∈ SL(V ) be an automorphism of V with determinant one such that hFi = Fi for i = 1,..., r. (7.5.30) Then (1l, S) ∼ (hh∗, hSh∗). Claim 2. Let g ∈ SL(V ) and ﬁx any ﬂag {0} = F0 F1 · · · Fr
= V. Then there exist elements h ∈ SL(V ) and u ∈ SO(V ) such that h satisﬁes (7.5.30) and g = hu. 380 CHAPTER 7. TOPICS IN GEOMETRY We prove, that these two claims imply (7.5.28). Fix any element S ∈ S1l, i=1 = ι1l(S) be the weighted ﬂag of S, and let P ∈ P0(V ). Choose let (Fi, µi)r an element g ∈ SL(V ) such that gg∗ = P, and choose u and h as in Claim 2. Then hh∗ = P, and the pairs (P, P ) := (hh∗, hSh∗) and (1l, S) have the same ﬂag by (7.5.30) and are equivalent by Claim 1. Hence any pair (P, P ) is equivalent to (1l, S) if and only if it has the same ﬂag. By transitivity of the equivalence relation we deduce that (7.5.28) holds for all P, Q ∈ P0(V ). i := g−1Fi and mi := dim(Fi) for i = 1,..., r. 1,..., e m of V such that for, mi i := ei i = u−1Fi and hence gu−1Fi = Fi for all i. Then choose orthonormal bases e1,..., em and e each i, the vectors e1,..., emi form a basis of Fi and the vectors e form a basis of F for i = 1,..., m. Thus h := gu−1 satisﬁes the requirements of Claim 2. i. Deﬁne the orthogonal transformation u by ue It satisﬁes F We prove Claim 2. Let F 1,..., e We prove Claim 1, following [17, Lemma 4]. Deﬁne the geodesics γ0, γ1 in P0(V ) by γ0(t) = exp(tS) and γ1(t) = h exp(tS)h∗ (see Lemma 6.5
.18). By equation (6.5.12) the square of their distance is given by ρ(t) := d(exp(tS), h exp(tS)h∗)2 = trace M (t) := exp−tS/2h exptS/2. log (M (t)M (t)∗)2, (7.5.31) In the eigenspace decomposition V = E1 ⊕ · · · ⊕ Er of the self-adjoint endomorphism S the automorphisms h and exp(tS/2) have the form h =       h11 h12 0... 0 h22... · · · h1r... · · ·...... hr−1,r hrr 0      , exp(tS/2) = diagetµ1/21lE1, etµ2/21lE2,..., etµr/21lEr Here the upper triangular form of h follows from (7.5.30). Hence. M (t) =       h11 h12(t) 0... 0 h22... · · · h1r(t)... · · ·...... hr−1,r(t) 0 hrr      , (7.5.32) (7.5.33) where hij(t) := e−t(µi−µj )/2hij for 1 ≤ i < j ≤ r. Since µi > µj for i < j, it follows that the limit M∞ := limt→∞ M (t) = diag(h11,..., hrr) exists and is an invertible endomorphism of V. Hence the function ρ : [0, ∞) → R in (7.5.31) is bounded. This proves Claim 1 and Lemma 7.5.14. 7.5. CONVEX FUNCTIONS ON HADAMARD
MANIFOLDS* 381 Proof of Theorem 7.5.12. Let G ⊂ SL(V ) be a Lie subgroup which acts irreducibly on V. Then G acts on P0(V ) by isometries. By Lemma 7.5.14 the induced action on the sphere at inﬁnity S∞(P0(V )) ∼= F (V ) is given by G × F (V ) → F (V ) : (g, (Fi, µi)i=1,...,r) → (gFi, µi)i=1,...,r. This action has no ﬁxed points because the action of G on V is irreducible and r ≥ 2 for each weighted ﬂag (Fi, µi)i=1,...,r ∈ F (V ). Hence all the assertions of Theorem 7.5.12 follow directly from Theorem 7.5.6 with M = P0(V ). 7.5.3 Lengths of Vectors The material in this section goes back to ideas in geometric invariant theory developed by Kempf–Ness [38], Ness [53], and Kirwan [40] in the complex setting and by Richardson–Slodowy [59] and Marian [48] in the real setting. We assume throughout that V, W are ﬁnite-dimensional real vector spaces and ρ : SL(V ) → SL(W ) is a Lie group homomorphism. Note that every Lie group homomorphism from GL(V ) to GL(W ) restricts to a Lie group homomorphism from SL(V ) to SL(W ), because every Lie group homomorphism from GL(V ) to the multiplicative group of nonzero real numbers is some power of the determinant. Fix a nonzero vector w ∈ W and denote by Gw ⊂ SL(V ) the connected component of the identity in the isotropy subgroup of w, i.e. Gw :=   g ∈ SL(V )  ∃ a smooth path γ : [0, 1] → SL(V ) such that γ(0) = 1l, γ(1) = g, and ρ(γ(t))w = w for 7.5.34) By Theorem 2.5.
27 this is a Lie subgroup of SL(V ) with the Lie algebra gw := ξ ∈ sl(V ) ˙ρ(ξ)w = 0. There are many examples of this setup that are related to interesting questions in geometry. The vector space W can be the space of all symmetric bilinear forms on V and w can be an inner product, in which case Gw is the special orthogonal group associated to the inner product, or w can be the quadratic form (6.4.9), in which case Gw is the identity component of the isometry group of hyperbolic space. Or W can be the space of skewsymmetric bilinear forms on V and w a symplectic form, in which case Gw is the symplectic linear group. Or W can be the space of skew-symmetric bilinear maps on V with values in V. Then w can be a cross product in dimension three or seven, or w can be the Lie bracket of a Lie algebra g = V and then Gw is the identity component in the group of automorphisms of g. The latter example will be examined in detail in §7.6.2. Of particular interest are the cases where the group Gw is noncompact. 382 CHAPTER 7. TOPICS IN GEOMETRY Deﬁnition 7.5.15. An inner product ·, · on V is called (ρ, w)-symmetric iﬀ the Lie subalgebra gw ⊂ sl(V ) is invariant under the involution A → A∗, deﬁned by v, A∗v := Av, v for v, v ∈ V. Exercise 7.5.16. Let ·, · be a (ρ, w)-symmetric inner product on V. Prove that g ∈ Gw implies g∗ ∈ Gw. Hint: Choose a smooth path g : [0, 1] → Gw with the endpoints g(0) = 1l and g(1) = g, and deﬁne ξ(t) := g(t)−1 ˙g(t). Show that the initial value problem ˙h(t) = ξ(t)∗h(t), h(0) = 1, has a unique solution h : [0, 1] → Gw (
Exercise 2.5.36) and that h(t) = g(t)∗ for all t. The following theorem asserts the existence of a (ρ, w)-symmetric inner product on V under an irreducibility assumption (see [17, Theorem 2]). Theorem 7.5.17. Assume that the group Gw in (7.5.34) acts irreducibly on V. Then there exists a (ρ, w)-symmetric inner product on V with the following properties. The subgroup Kw := Gw ∩ SO(V ) is connected and is a maximal compact subgroup of Gw. Moreover, every compact subgroup of Gw is conjugate in Gw to a Lie subgroup of Kw. Thus, if K is any maximal compact subgroup of Gw, there exists an element h ∈ Gw such that K = hKwh−1. Proof. See Lemma 7.5.23. Example 7.5.18. The hypothesis that the group Gw acts irreducibly on V cannot be removed in Theorem 7.5.17. Consider the case where W = V has dimension at least two, the homomorphism ρ : SL(V ) → SL(V ) is the identity, and w ∈ V is any nonzero vector. Then the one-dimensional linear subspace Rw ⊂ V is evidently invariant under the action of Gw, and there does not exist any (ρ, w)-symmetric inner product on V. To begin with, we will ﬁx any inner product ·, ·V on V and deﬁne the space P0(V ) of self-adjoint positive deﬁnite automorphisms of V with determinant one in terms of this ﬁxed inner product. We will then use Theorem 7.5.12 to ﬁnd an element P ∈ P0(V ) such that the inner product v, v V,P := v, P −1v V (7.5.35) on V satisﬁes the requirements of Theorem 7.5.17. The proof is based on three lemmas. The fourth lemma restates the theorem in a modiﬁed form. 7.5. CONVEX FUNCTIONS ON HADAMARD MANIFOLDS* 383 Lemma 7.5.19.
There exists an inner product ·, ·W on W such that ρ(SO(V )) ⊂ SO(W ) and ˙ρ(A∗) = ˙ρ(A)∗ for all A ∈ sl(V ). Proof. The proof follows the argument in [17, §3]. Assume without loss of generality that V = Rm is equipped with the standard inner product, and that W = Rn and that ρ : SL(m, R) → SL(n, R) is a Lie group homomorphism. We prove ﬁrst that there exists a unique Lie group homomorphism ρc : SL(m, C) → SL(n, C) (the complexiﬁcation of ρ) such that ρc\|SL(m,R) = ρ, ˙ρc(A + iB) = ˙ρ(A) + i ˙ρ(B) (7.5.36) for all A, B ∈ sl(m, R). Since SL(m, C) is connected, we can deﬁne ρc(g) for g ∈ GL(m, C) by choosing a smoth path α : [0, 1] → SL(m, C) with the endpoints α(0) = 1lm and α(1) = g, and taking ρc(g) := β(1), β(s)−1 ˙β(s) = ˙ρcα(s)−1 ˙α(s), β(0) = 1ln. (7.5.37) To verify that β(1) is independent of the choice of α, one can choose a smooth map [0, 1]2 → SL(m, C) : (s, t) → α(s, t) satisfying α(0, t) = 1lm, α(1, t) = g, deﬁne S := α−1∂sα and T := α−1∂tα, and deﬁne β : [0, 1]2 → SL(n, C) as the solution of the initial value problem β−1∂sβ = ˙ρc(S), β(0, t) = 1
ln. Since ˙ρc is a Lie algebra homomorphism and ∂tS − ∂sT = [S, T ], it follows that β−1∂tβ = ˙ρc(T ) and so β(1, t) is independent of t. Moreover, SL(m, C) retracts onto SU(m) by polar decomposition and so is simply connected by a standard homotopy argument. That the map ρc : SL(m, C) → SL(n, C) thus deﬁned is smooth follows from the smooth dependence of solutions on the parameter in a smooth family of diﬀerential equations. That it is a group homomorphism follows by catenation of paths, and that it satisﬁes (7.5.36) follows directly from the deﬁnition. Now consider the action of the compact subgroup SU(m) ⊂ SL(m, C) on the Hadamard manifold Q0 of positive deﬁnite Hermitian n × n-matrices Q of determinant one (Remark 6.5.21) by (g, Q) → ρc(g)Qρc(g)∗. This action is by isometries and hence, by Cartan’s Fixed Point Theorem 6.5.6, there exists an element Q0 ∈ Q0 such that ρc(g)Q0ρc(g)∗ = Q0 for all g ∈ SU(m). Diﬀerentiate this equation at g = 1lm to obtain ˙ρc(B)Q0 + Q0 ˙ρc(B)∗ = 0 for every skew-Hermitian matrix B = −B∗ ∈ sl(m, C) with trace zero. Now let A ∈ sl(m, R), deﬁne R := 1 2 (A − AT), S := 1 2 (A + AT), and take B = R and B = iS. Then ˙ρ(R)Q0 + Q0 ˙ρ(R)T = 0 and ˙ρ(S)Q0 = Q0 ˙ρ(S)T. Hence the positive deﬁnite symmetric n × n-matrix Q
:= Re(Q0) satisﬁes ˙ρAT = ˙ρ−R + S = Q ˙ρ(R + S)TQ−1 = Q ˙ρ(A)TQ−1 for all A ∈ sl(m, R). This shows that the inner product w, w := wTQ−1w on Rn satisﬁes the requirements of Lemma 7.5.19. 384 CHAPTER 7. TOPICS IN GEOMETRY In the remainder of this subsection we will ﬁx an inner product on W as in Lemma 7.5.19. Recall also that we have already chosen a nonzero vector w ∈ W. The norm squared of this vector determines a function on the space of inner products on V (see [17, Lemma 1]). Lemma 7.5.20. Deﬁne the function fw : P0(V ) → R by fw(P ) = w, ρ(P −1)w for P ∈ P0(V ). Then an element P ∈ P0(V ) is a critical point of fw if and only if ˙ρ(A)w, ρ(P −1)wW = 0 for all A ∈ sl(V ). Moreover, if P is a critical point of fw, then the inner product (7.5.35) on V is (ρ, w)-symmetric. Proof. Fix an element P ∈ P0(V ) and a tangent vector P ∈ TpP0(V ). Then it follows from Lemma 7.5.19 that (7.5.38) W dfw(P ) P = − = − w, ˙ρ(P −1 P )ρ(P −1)w ˙ρ( P P −1)w, ρ(P −1)w W W. Now let A ∈ sl(V ) and take P := AP + P A∗ to obtain dfw(P )(AP + P A∗) = − ˙ρ(A + P A∗P −1)w, ρ(P −1)w W (7.5.39) = −2 ˙ρ(A)w, ρ(P −1)w W. Thus
P is a critical point of fw if and only if the right hand side of (7.5.39) vanishes for all A ∈ sl(V ). To prove the last assertion, deﬁne the norm \|w\|W,P := w, ρ(P −1)wW for w ∈ W. Now let P ∈ P0(g) be a critical point of fw, let ξ ∈ sl(V ), and take A := [ξ, P ξ∗P −1] ∈ sl(V ) in (7.5.39). Then 0 = ˙ρ([ξ, P ξ∗P −1])w, ρ(P −1)w W = \| ˙ρ(P ξ∗P −1)w\|2 W,P − \| ˙ρ(ξ)w\|2 W,P. If ξ ∈ gw, then ˙ρ(ξ)w = 0, hence ˙ρ(P ξ∗P −1)w = 0, and hence P ξ∗P −1 ∈ gw. But the endomorphism P ξ∗P −1 is the adjoint of ξ with respect to the inner product (7.5.35) on V and this proves Lemma 7.5.20. Example 7.5.21. This example shows that the (ρ, w)-symmetry of the inner product (7.5.35) does not imply that P is a critical point of fw. Take W = V × V and let ρ : SL(V ) → SL(W ) be the diagonal action. Assume dim(V ) = 2 and choose w = (u, v) ∈ W such that u, v ∈ V are linearly independent. Then Gw = {1l} and so every inner product on V is (ρ, w)symmetric (and the assertions of Theorem 7.5.17 are satisﬁed), however, the function fw(P ) = u, P −1uV + v, P −1vV does not have any critical point. 7.5. CONVEX FUNCTIONS ON HADAMARD MANIFOLDS* 385 Let Mw ⊂ P0(V ) be
the set of minima of the function fw : P0(V ) → R in Lemma 7.5.20 and let Crit(fw) ⊂ P0(V ) be its set of critical points. The next result shows that fw is convex and that Gw acts transitively on Mw whenever this set is nonempty (see [17, Lemma 2]). Lemma 7.5.22. The function fw has the following properties. (i) fw is convex and Gw-invariant, and thus Mw = Crit(fw). (ii) Fix two elements P0 ∈ Mw and P ∈ P0(V ). Then P ∈ Mw if and only if there exists an element η ∈ gw such that η = P0η∗P −1 and exp(η) = P P −1, or equivalently ρ(P P −1 )w = w. 0 (iii) The group Gw acts transitively on Mw. Proof. We prove part (i). Let γ : R → P0(V ) be a geodesic and deﬁne 0 0 P := γ(0), P := ˙γ(0), S := P −1/2 P P −1/2. Then, by Lemma 6.5.18, γ(t) = P 1/2 exp(tS)P 1/2 and hence fw(γ(t)) = ρ(P −1/2)w, exp(−t ˙ρ(S))ρ(P −1/2)wW. This implies d2 dt2 fw(γ(t)) = ˙ρ(S)ρ(P −1/2)w, exp(−t ˙ρ(S)) ˙ρ(S)ρ(P −1/2)wW ≥ 0 for all t and hence fw is convex. Hence Mw = Crit(fw) by Lemma 7.5.9. That fw is Gw-invariant follows directly from the deﬁnition. This proves (i)., exp(η) = P P −1, )w, hence We prove part (ii). If η ∈ gw satisﬁes η = P0�
�∗P −1 0 )w = w, then ρ(P −1)w = ρ(P −1 then ρ(P P −1 0 0 0 0 )w = w. If ρ(P P −1 ˙ρ(A)w, ρ(P −1)wW = ˙ρ(A)w, ρ(P −1 )wW = 0 0 for all A ∈ sl(V ), and hence P ∈ Mw by Lemma 7.5.20 and part (i). Now assume P ∈ Mw and let γ : [0, 1] → P0(V ) be the unique geodesic with the endpoints γ(0) = P0 and γ(1) = P ∈ Mw. Then γ(t) ∈ Mw for all t by Lemma 7.5.9. Hence ˙ρ(η)w, ρ(γ(t)−1)wW = 0 for all t and all η ∈ sl(V ), by Lemma 7.5.20. Diﬀerentiate this equation at t = 0 to obtain ˙ρ(η)w, ρ(P −1 ) ˙ρ( P P −1 )wW = 0, P := ˙γ(0) ∈ TP0 P0(V ). 0 Take η := P P −1 By Lemma 6.5.18 we also have P = γ(1) = exp( P P −1 this proves (ii). 0 to obtain ˙ρ(η)w = 0, and thus η ∈ gw and η = P0η∗P −1. )P0 = exp(η)P0 and 0 0 0 We prove part (iii). Let P0, P ∈ Mw, choose an element η ∈ gw as in (ii) so that ηP0 = P0η∗ and P = exp(η)P0, and deﬁne h := exp(η/2) ∈ Gw to obtain P = hP0h∗. This proves (iii) and Lemma 7.5.22. 386 CHAPTER 7. TOPICS IN GEOMETRY With these
preparations we are ready to prove Theorem 7.5.17. Lemma 7.5.23. (i) If Gw acts irreducibly on V, then Mw = ∅. (ii) If P ∈ Mw, then the inner product ·, P −1·V on V satisﬁes all the requirements of Theorem 7.5.17. Proof. The function fw is convex and Gw-invariant by Lemma 7.5.22. Hence part (i) follows from Theorem 7.5.12. To prove part (ii), assume that P is a critical point of fw. Then, by Lemma 7.5.20, the inner product ·, P −1·V is (ρ, w)-symmetric. We prove the remaining assertions in four steps. Deﬁne KP := Gw ∩ SO(V, ·, P −1·V ) = u ∈ Gw \| uP u∗P −1 = 1l. Step 1. Let K ⊂ Gw be any compact subgroup. Then there exists an element h ∈ Gw such that h−1Kh ⊂ KP. By Theorem 7.5.12, there exists a P0 ∈ P0(V ) such that fw(P0) = inf fw and uP0u∗ = P0 for all u ∈ K. By Lemma 7.5.22, there exists an element h ∈ Gw such that P0 = hP h∗. Hence uhP h∗u∗ = hP h∗ for all u ∈ K, and hence the automorphism h−1uh is orthogonal with respect to the inner product ·, P −1·V for every u ∈ K. Step 2. KP is a compact connected subgroup of Gw. By the Closed Subgroup Theorem 2.5.27 KP is a closed, and hence compact, subgroup of SO(V, ·, P −1·V ) and so is a compact Lie subgroup of Gw. We prove that KP is connected. Let u ∈ KP ⊂ Gw. Since Gw is connected, there exists a smooth path g : [0, 1] → Gw such that g(0) = 1l and g(1) = u. Thus, by Exercise 7.5.16,
we have g(t)P g(t)∗P −1 ∈ Gw for all t. Hence, by part (ii) of Lemma 7.5.22, there exists a smooth path η : [0, 1] → gw such that η(t) = P η(t)∗P −1, exp(η(t)) = g(t)P g(t)∗P −1, and η(0) = η(1) = 0. Hence u(t) := exp(−η(t)/2)g(t) is a path in KP joining u(0) = 1l to u(1) = u. Step 3. KP is a maximal compact subgroup of Gw. Let K ⊂ Gw be a compact subgroup containing KP. Then by Step 1 there exists an h ∈ Gw such that h−1Kh ⊂ KP. Hence KP ⊂ K ⊂ hKP h−1 and so Lie(KP ) ⊂ Lie(K) ⊂ hLie(KP )h−1. Since Lie(KP ) and hLie(KP )h−1 have the same dimension, this implies Lie(KP ) = hLie(KP )h−1. Since KP and hKP h−1 are connected, this implies KP = hKP h−1 and so KP = K. Step 4. Let K be a maximal compact subgroup of Gw. Then there exists an element h ∈ Gw such that K = hKP h−1. By Step 1 there exists an h ∈ Gw such that h−1Kh ⊂ KP, thus K ⊂ hKP h−1 and so K = hKP h−1, because K is a maximal compact subgroup of Gw. This proves Step 4, Lemma 7.5.23, and Theorem 7.5.17. 7.5. CONVEX FUNCTIONS ON HADAMARD MANIFOLDS* 387 Lemma 7.5.23 shows that all minima of the function fw in Lemma 7.5.20 give rise to inner products that satisfy the requirements of Theorem 7.5.17. The next lemma shows that the set of minima of fw is a Hadamard manifold. Lemma 7.
5.24. The set Mw is a geodesically convex and totally geodesic submanifold of P0(V ). Hence, if it is nonempty, it is a Hadamard manifold and a symmetric space. Proof. By Lemma 7.5.22 the function fw is convex, and so Mw is geodesically convex by Lemma 7.5.9. Now assume Mw is nonempty and ﬁx any element P0 ∈ Mw. Then the exponential map TP0 P0(V ) → P0(V ) : P → exp( P P −1 is a diﬀeomorphism (Theorem 6.5.10 and Lemma 6.5.18) and Mw is the 0 ∈ gw} under this diﬀeoimage of the linear subspace { P ∈ TP0 morphism (Lemma 7.5.22). Since P0 can be chosen to be any element of Mw, this shows that Mw is a totally geodesic submanifold of P0(V ). Moreover, the isometry P0(V ) → P0(V ) : P → φ0(P ) = P0P −1P0 in Step 2 of the proof of Theorem 6.5.10 satisﬁes P0(g) \| P P −1 )P0 0 φ0 exp P P −1 0 P0 = exp− P P −1 0 P0 for all P ∈ TP P0(V ) and so restricts to an isometry of Mw. Hence Mw is a symmetric space and this proves Lemma 7.5.24. We emphasize that Lemma 7.5.24 does not require the hypothesis that the group Gw acts irreducibly on V. This hypothesis was only used to prove that the space Mw is nonempty. The next example shows that fw can have critical points in cases where Gw acts reducibly on V. Example 7.5.25. Let V = R2, let W = S ⊂ R2×2 be the space of symmetric matrices, equipped with the standard inner product and the standard action S → gSgT of SL(2, R), and let w = S := diag(1, −1) ∈ S so that The action of
GS on R2 is reducible, because the diagonal in R2 is GSinvariant, however, the function fS(P ) = trace(SP −1SP −1) attains its minimum on the set MS = GS ⊂ P0(R2), corresponding to the symmetric inner products on R2. If one modiﬁes this example by taking W = S × S and w = (1l, S), then fw(P ) = trace(P −2 + SP −1SP −1) has a unique critical point at P = 1l, the group Gw = {1l} acts reducibly on V, and the assertions of Theorem 7.5.17 are trivially satisﬁed for every inner product on V. a b b a a > 0, a2 − b2 = 1 GS =. 388 CHAPTER 7. TOPICS IN GEOMETRY Example 7.5.21 and Example 7.5.25 show that, in general, one cannot expect there to be a one-to-one correspondence between the minima of fw and the (ρ, w)-symmetric inner products on V. However, we shall see below that such a one-to-one correspondence does exist in many cases. Remark 7.5.26. Once it is known, that the function fw : P0(V ) → R has a critical point P0 ∈ P0(V ), we can modify the entire setup as follows. Replace the inner product on V by ·, ·V,0 := ·, P −1 ·V and the inner product on W by ·, ·W,0 := ·, ρ(P −1 )·W. This pair of inner products satisﬁes the requirements of Lemma 7.5.19. Let P00(V ) be the space of selfadjoint positive deﬁnite endomorphisms with respect to the new inner product and deﬁne the function fw,0 : P00(V ) → R by the analogous formula. Then P ∈ P0(V ) if and only if P P −1 ) = fw(P ) for all P ∈ P0(V ). Thus fw,0 attains its minimum at P = 1l. 0 ∈ P00(V ) and fw,0(P P −
1 0 0 0 In the next corollary we do not assume that Gw acts irreducibly on V. Corollary 7.5.27 (Cartan Decomposition). Assume the function fw in Lemma 7.5.20 has a critical point at P0 = 1l and deﬁne Kw := Gw ∩ SO(V ), pw := {η ∈ gw \| η = η∗}. (7.5.40) Then the map Kw × pw → Gw : (u, η) → exp(η)u =: φw(u, η) (7.5.41) is a diﬀeomorphism. Hence the map Gw → P0(V ) : g → a diﬀeomorphism from the quotient space Gw/Kw to Mw = Gw ∩ P0(V ). gg∗ descends to √ Proof. By part (ii) of Lemma 7.5.22 with P0 = 1l the function fw attains its minimum on the set Mw = Gw ∩ P0(g). Now deﬁne the map ψw : Gw → Kw × pw by ψw(g) := (u, η), where η := 1 2 exp−1(gg∗) ∈ pw, u := exp(−η)g ∈ Kw for g ∈ Gw. This map is well deﬁned and smooth because, for every g ∈ Gw, we have gg∗ ∈ Gw ∩ P0(V ) = Mw (see Exercise 7.5.16), and the exponential map descends to a diﬀeomorphism exp : pw → Mw (see Lemma 7.5.22). Since φw ◦ ψw = id and ψw ◦ φw = id, it follows that φw is a diﬀeomorphism. This proves Corollary 7.5.27. 7.5. CONVEX FUNCTIONS ON HADAMARD MANIFOLDS* 389 As a warmup for the main application of Theorem 7.5.17 in §7.6 it may be useful to consider the following two examples. Exercise 7.5.28. (i
) Let V be an m-dimensional real vector space and W := S2V ∗ be the vector space of all symmetric bilinear forms Q : V × V → R. Deﬁne the homomorphism ρ : SL(V ) → SL(W ) by the standard action of the group SL(V ) on W, i.e. ρ(g)Q(v, v) := Q(g−1v, g−1v) for g ∈ SL(V ), Q ∈ S2V ∗, and v, v ∈ V. Assume that V is equipped with an inner product and an orthonormal basis e1,..., em, and deﬁne an inner product on W by Q, Q := i,j Q(ei, ej)Q(ei, ej) for Q, Q ∈ S2V ∗. Show that this inner product is independent of the choice of the orthonormal basis and satisﬁes the requirements of Lemma 7.5.19. (ii) The inner product on V is an element Q0 ∈ W whose isotropy subgroup is the special orthogonal group SO(V ). Show that the function fQ0 in (7.5.38) is given by fQ0(P ) = trace(P 2) for P ∈ P0(V ) and that it has a unique critical point at P = 1l. (iii) Examine the case where V = Rm+1 is equipped with the standard inner product and Q ∈ W is the quadratic form in (6.4.9). Relate this example to the isometry group of hyperbolic space (§6.4.3). Find a maximal compact subgroup of the identity component of O(m, 1). Exercise 7.5.29. (i) Let V be a 2n-dimensional real vector space and W = Λ2V ∗ be the vector space of all skew-symmetric bilinear forms τ : V × V → R. Deﬁne the homomorphism ρ : SL(V ) → SL(W ) by the standard action of the group SL(V ) on W, i.e. ρ(g)τ (v, v) := g∗τ (v, v) := τ (
g−1v, g−1v) for g ∈ SL(V ), τ ∈ Λ2V ∗, and v, v ∈ V. Assume that V is equipped with an inner product and an orthonormal basis e1,..., e2n, and deﬁne an inner product on W by τ, τ := i,j τ (ei, ej)τ (ei, ej) for τ, τ ∈ Λ2V ∗. Show that this inner product is independent of the choice of the orthonormal basis and satisﬁes the requirements of Lemma 7.5.19. 390 CHAPTER 7. TOPICS IN GEOMETRY (ii) Let ω : V × V → R be a nondegenerate skew-symmetric bilinear form. Then the pair (V, ω) is called a symplectic vector space and ω is called a symplectic form on V. The isotropy subgroup of ω in GL(V ) is called the symplectic linear group. Denote this group and its Lie algebra by ω(g·, g·) = ω, Sp(V, ω) := g ∈ GL(V ) sp(V, ω) := Lie(Sp(V, ω)) = A ∈ End(V ) ω(A·, ·) + ω(·, A·) = 0. The group Sp(V, ω) is connected and contained in SL(V ) (see [49]). An automorphism J : V → V is called a linear complex structure iﬀ J 2 = −1l. A linear complex structure J is called compatible with ω iﬀ the bilinear form ω(·, J·) is an inner product on V. An inner product ·, · on V is called compatible with ω iﬀ there exists a linear complex structure J such that ω(·, J·) = ·, ·. Prove that an inner product on V is compatible with ω if and only if it satisﬁes the conditions (for any basis e1,..., e2n of V ) detω(ei, ej) = detei, ej, =⇒ A∗ ∈ sp(V,
ω). A ∈ sp(V, ω) (7.5.42) (7.5.43) If an inner product on V is compatible with ω, prove that g ∈ Sp(V, ω) implies g∗ ∈ Sp(V, ω) (without using the fact that Sp(V, ω) is connected). Hint: Deﬁne J ∈ End(V ) by ω(·, J·) := ·, · and show that J + J ∗ = 0. Show that A ∈ sp(V, ω) if and only if AJ + JA∗ = 0. Use (7.5.43) to prove that J 2 commutes with every self-adjoint endomorphism of V and hence satisﬁes J 2 = λ1l for some λ ∈ R. Use (7.5.42) to conclude that λ = −1. (iii) Fix an inner product on V and a symplectic form ω : V × V → R that satisﬁes (7.5.42). Then the function fω in (7.5.38) is given by fω(P ) = i,j ω(ei, ej)ω(P ei, P ej), P ∈ P0(V ), (7.5.44) where e1,..., e2n is an orthonormal basis of V. Prove that this is the norm squared of ω with respect to the inner product ·, P −1·. Prove that P is a critical point of fω if and only if the inner product ·, P −1· is compatible with ω. Prove that the space J (V, ω) of ω-compatible linear complex structures is a Hadamard manifold (Exercise 6.5.24). Prove that, if J ∈ J (V, ω), then the unitary group U(V, ω, J) := g ∈ Sp(V, ω) gJg−1 = J is a maximal compact subgroup of Sp(V, ω) and every compact subgroup of Sp(V, ω) is conjugate to a subgroup of U(V, ω, J). All this is of course well known
, but this exercise shows how these results can be derived from Theorem 7.5.17. Moreover, it is not necessary to assume that Sp(V, ω) is connected. One can start with the identity component of Sp(V, ω), prove that it is contained in SL(V ), and deduce the connectivity of Sp(V, ω) from that of U(V, ω, J). 7.6. SEMISIMPLE LIE ALGEBRAS* 391 7.6 Semisimple Lie Algebras* This section discusses applications of the results in §7.5.3 to Lie algebra theory, following the work of Donaldson [17]. It examines symmetric inner products on Lie algebras (§7.6.1), establishes their existence on simple Lie algebras (§7.6.2), and derives as consequences several standard results in Lie algebra theory, such as the uniqueness of maximal compact subgroups up to conjugation for semisimple Lie algebras (§7.6.3), and Cartan’s theorem about the compact real form of a semisimple complex Lie algebra (§7.6.4). Here are some basic deﬁnitions that will be used throughout this section. Let g be a ﬁnite-dimensional real Lie algebra (Deﬁnition 2.4.22). A vector subspace h ⊂ g is called an ideal iﬀ [ξ, η] ∈ h for every ξ ∈ g and every η ∈ h. The Lie algebra g is called abelian iﬀ the Lie bracket vanishes. It is called simple iﬀ it is not abelian and does not contain any ideal other than h = {0} and h = g. Examples of ideals in any Lie algebra g are the center Z(g) (Exercise 2.5.34) and the commutant [g, g] (Exercise 5.2.24), deﬁned by Z(g) := {ξ ∈ g \| [ξ, η] = 0 for all η ∈ g}, [g, g] := span {[ξ, η] \| ξ, η ∈ g}. Recall also the deﬁnition of
the adjoint representation ad : g → Der(g) in Example 2.5.23 by ad(ξ) := [ξ, ·] for ξ ∈ g and the deﬁnition of the Killing form κ : g × g → R in Example 5.2.25 by κ(ξ, η) := trace(ad(ξ)ad(η)) for ξ, η ∈ g. Let Aut0(g) be the connected component of the identity in the group Aut(g) of automorphisms of g. This is a Lie subgroup of GL(g) whose Lie algebra is the space Lie(Aut0(g)) = Der(g) of derivations on g. 7.6.1 Symmetric Inner Products In [17] Donaldson introduced the following notion. Deﬁnition 7.6.1. Let g be a ﬁnite-dimensional real Lie algebra. An inner product ·, · on g is called symmetric iﬀ it satisﬁes the condition δ ∈ Der(g) =⇒ (7.6.1) Here δ∗ : g → g denotes the adjoint of the endomorphism δ with respect to the inner product, i.e. it satisﬁes ξ, δ∗η = δξ, η for all ξ, η ∈ g. δ∗ ∈ Der(g). Exercise 7.6.2. Let g be a ﬁnite-dimensional real Lie algebra equipped with a symmetric inner product. Prove that g ∈ Aut0(g) =⇒ g∗ ∈ Aut0(g). Hint: See Exercise 7.5.16. Some consequences of the existence of a symmetric inner product are derived in Lemma 7.6.8 below. To begin with we discuss some examples. 392 CHAPTER 7. TOPICS IN GEOMETRY Every vector space endomorphism of a Lie algebra g that takes values in the center Z(g) and vanishes on the commutant [g, g] is a derivation. Conversely, every derivation that takes values in Z(g) necessarily vanishes on [g, g]. The space of such derivations is an ideal in Der(g),
denoted by im(δ) ⊂ Z(g). DerZ(g) := δ ∈ Der(g) (7.6.2) Example 7.6.3. Consider the abelian Lie algebra g = Rm. The adjoint representation is trivial and Der(g) = gl(m, R) = Rm×m is the Lie algebra of all vector space endomorphisms of g. Thus Der(g) = DerZ(g), the Killing form on g vanishes, and every inner product on g is symmetric. Example 7.6.4. Consider the Lie algebra g = gl(m, R) with [g, g] = sl(m, R) and Z(g) = R1l. It satisﬁes g = [g, g] ⊕ Z(g) and Der(g) = ad(g) ⊕ DerZ(g), where DerZ(g) ⊂ Der(g) is the one-dimensional subspace generated by the derivation δZ(A) = trace(A)1l (whose trace is m). Moreover, the standard inner product A, A = trace(ATA) on g is symmetric and the kernel of the Killing form κ(A, A) = 2mtrace(AA) − 2trace(A)trace(A) is Z(g). Example 7.6.5. Consider the Lie algebra g = gl(m, R) × Rm with the Lie bracket [(A, v), (A, v)] := ([A, A], Av − Av). This Lie algebra can be identiﬁed with the space of all aﬃne vector ﬁelds on Rm. It has a trivial center and the commutant [g, g] = sl(m, R) × Rm has codimension one. Moreover, trace(ad(A, v)) = trace(A) for every (A, v) ∈ g, the kernel of the Killing form κ((A, v), (A, v)) = (2m + 1)trace(AA) − 2trace(A)trace(A) is the abelian ideal {0} × Rm, and the adjoint representation ad : g → Der(g) is a Lie algebra isomorphism. Example
7.6.6. Consider the Heisenberg algebra h = V ×R of a symplectic vector space (V, ω) with the Lie bracket [(v, t), (v, t)] = (0, ω(v, v)) (see Exercise 2.5.15). It satisﬁes Z(h) = [h, h] = {0} × R and the Killing form vanishes. Every derivation on h has the form δ(v, t) = (Av + λv, Λ(v) + 2λt), where λ ∈ R, Λ ∈ V ∗, and A ∈ sp(V, ω). The subspace ad(h) = DerZ(h) consists of all derivations of the form δ(v, t) = (0, Λ(v)). Example 7.6.7. The Heisenberg algebra of a symplectic vector space (V, ω) extends to a Lie algebra g = sp(V, ω) × V × R with the Lie bracket [(A, v, t), (A, v, t)] = ([A, A], Av − Av, ω(v, v)). It satisﬁes [g, g] = g and has a one-dimensional center Z(g) = {0} × {0} × R. The kernel of the Killing form is the Heisenberg algebra and the adjoint representation ad : g → Der(g) is surjective. The next lemma shows that the Lie algebras in Examples 7.6.5, 7.6.6, and 7.6.7 do not admit symmetric inner products. 7.6. SEMISIMPLE LIE ALGEBRAS* 393 Lemma 7.6.8. Let g be a ﬁnite-dimensional real Lie algebra equipped with a symmetric inner product. Then Der(g) = ad(g) ⊕ DerZ(g), (7.6.3) the kernel of the Killing form κ : g × g → R is the center of g, and there exists an involution g → g : ξ → ξ∗ such that, for all ξ, η ∈ g, g = [g, g] ⊕ Z(g), ad(
ξ∗) = ad(ξ)∗, [ξ, η]∗ = [η∗, ξ∗]. (7.6.4) Proof. Consider the orthogonal decomposition Der(g) = A ⊕ B, A := ad(g), B := A⊥, (7.6.5) with respect to the inner product δ, δ = trace(δ∗δ) for δ, δ ∈ Der(g). Since [δ, ad(ξ)] = ad(δξ) for δ ∈ Der(g) and ξ ∈ g, the subspace A is an ideal in Der(g). Moreover, if δ ∈ B and ε ∈ Der(g), then ε∗ ∈ Der(g), hence trace([ε, δ]∗ad(ξ)) = trace(δ∗[ε∗, ad(ξ)]) = trace(δ∗ad(ε∗ξ)) = 0 for all ξ ∈ g, and hence [ε, δ] ∈ B. Thus B is also an ideal in Der(g). Next deﬁne DerZ(g)∗ := {δ∗ \| δ ∈ DerZ(g)}. We prove that (7.6.6) B = DerZ(g) = DerZ(g)∗. Since A and B are ideals we have [δ, δ] = 0 for all δ ∈ B and δ ∈ A. Thus ad(δξ) = [δ, ad(ξ)] = 0 for all δ ∈ B and ξ ∈ g, hence im(δ) ⊂ Z(g) for all δ ∈ B, and so B ⊂ DerZ(g). Now let δ ∈ DerZ(g)∗. Then δ∗ ∈ DerZ(g), hence [g, g] ⊂ ker(δ∗), hence trace(δ∗ad(ξ)) = 0 for all ξ ∈ g, and so δ ∈ B. Thus DerZ(g)∗ ⊂ B ⊂ Der
Z(g) and so (7.6.6) holds for dimensional reasons. The ﬁrst equation in (7.6.3) follows directly from (7.6.5) and (7.6.6). It follows also from (7.6.6) that trace(δ∗ad(ξ)∗) = trace(ad(ξ)δ) = 0 for all ξ ∈ g and all δ ∈ B, and so ad(ξ)∗ ∈ B⊥ = A for all ξ ∈ g. Thus δ ∈ A =⇒ δ∗ ∈ A. (7.6.7) By (7.6.7), an element ζ ∈ g belongs to Z(g) if and only if ad(ξ)∗ζ = 0 for all ξ ∈ g if and only if ζ, ad(ξ)η = 0 for all ξ, η ∈ g, if and only if ζ ∈ [g, g]⊥. Thus [g, g]⊥ = Z(g) and this proves the second equation in (7.6.3). By (7.6.3), the map ad : g → Der(g) restricts to a Lie algebra isomorphism from [g, g] to A. Hence, by (7.6.7) there exists a unique involution [g, g] → [g, g] : ξ → ξ∗ that satisﬁes (7.6.4) for all ξ, η ∈ [g, g]. By (7.6.3) this involution extends uniquely to an involution g → g : ξ → ξ∗ such that ζ ∗ = ζ for all ζ ∈ Z(g), and the extended involution satisﬁes (7.6.4) for all ξ, η ∈ g. Now let ζ ∈ g belong to the kernel of the Killing form, i.e. κ(ζ, ξ) = 0 for all ξ ∈ g. Then \|ad(ζ)\|2 = κ(ζ, ζ ∗) = 0 and hence ζ ∈
Z(g). Conversely, it follows directly from the deﬁnitions that Z(g) is contained in the kernel of the Killing form and this proves Lemma 7.6.8. 394 CHAPTER 7. TOPICS IN GEOMETRY 7.6.2 Simple Lie Algebras The goal of this section is to establish the existence of symmetric inner products on simple Lie algebras. First, the following lemma derives some immediate consequences of the deﬁnition of a simple Lie algebra. Lemma 7.6.9. Let g be a ﬁnite-dimensional simple real Lie algebra. Then the center of g is trivial, the adjoint representation ad : g → Der(g) is injective, the commutant is [g, g] = g, and trace(ad(ξ)) = 0 for all ξ ∈ g. Proof. The center Z(g) is an ideal in g. It is not equal to g because g is not abelian, and hence Z(g) = {0} because g is simple. The subspace [g, g] is also an ideal in g. It is nonzero because g is not abelian, and hence [g, g] = g because g is simple. The adjoint representation ad : g → Der(g) is injective because its kernel is the center of g. The last assertion follows from the fact that [g, g] = g and tracead([ξ, η]) = trace[ad(ξ), ad(η)] = 0 for all ξ, η ∈ g. This proves Lemma 7.6.9. That the Killing form of a simple Lie algebra is nondegenerate is a deeper result that does not follow directly from the deﬁnition. In [17, Theorem 1] Donaldson deduces the existence of symmetric inner products on simple Lie algebras from Theorem 7.5.17 and derives as corollaries various standard results in Lie algebra theory, including nondegeneracy of the Killing form. Theorem 7.6.10. Let g be a ﬁnite-dimensional simple real Lie algebra. Then the Killing form on g is nondegenerate, the adjoint representation ad : g → Der(g) is bijective, every derivation δ : g
→ g has trace zero, and every automorphism in the identity component Aut0(g) has determinant one. In particular, Theorem 7.6.10 establishes for every simple Lie algebra g the existence of a connected Lie group Aut0(g) ⊂ SL(g) whose Lie algebra is isomorphic to g. Theorem 7.6.11 (Donaldson). Every ﬁnite-dimensional simple real Lie algebra admits a symmetric inner product. Moreover, if SO(g) is the special orthogonal group associated to a symmetric inner product on g, then K := Aut0(g) ∩ SO(g) (7.6.8) is connected and is a maximal compact subgroup of Aut0(g), every compact subgroup of Aut0(g) is conjugate to a Lie subgroup of K, and every maximal compact subgroup of Aut0(g) is conjugate to K. 7.6. SEMISIMPLE LIE ALGEBRAS* 395 Given an inner product ·, · on any m-dimensional Lie algebra g and an orthonormal basis e1,..., em of g, deﬁne the function fg : P0(g) → R by fg(P ) := i,j [ei, ej], P −1[P ei, P ej] (7.6.9) for P ∈ P0(g). The right hand side of (7.6.9) is independent of the choice of the orthonormal basis and is the norm squared of the Lie bracket with respect to the inner product ·, P −1· on g. Theorem 7.6.12 (Donaldson). Let g be a ﬁnite-dimensional simple real Lie algebra equipped with a symmetric inner product. Then the set Mg := P ∈ P0(g) dfg(P ) = 0 = P ∈ P0(g) fg(P ) = inf fg = P0(g) ∩ Aut(g) = exp(δ) = P ∈ P0(g) the inner product ·, P −1· is symmetric δ ∈ Der(g), δ = δ∗ (7.6.10) of critical points of fg is a geodesically convex and totally geodesic subman
ifold of P0(g). Hence it is a Hadamard manifold and a symmetric space. The proofs require three preparatory lemmas. We do not assume that every derivation has trace zero. Thus it is necessary as an intermediate step to introduce the subspace Der0(g) := {δ ∈ Der(g) \| trace(δ) = 0}. We will consider inner products on g that satisfy the condition δ ∈ Der0(g) =⇒ δ∗ ∈ Der0(g). (7.6.11) Lemma 7.6.13. Let g be a ﬁnite-dimensional real Lie algebra satisfying the conditions Z(g) = {0} and [g, g] = g, and ﬁx an inner product ·, · on g. Then the inner product is symmetric if and only if it satsﬁes (7.6.11). Moreover, if such an inner product exists, then Der(g) = ad(g) = Der0(g). Proof. Assume (7.6.11) and consider the decomposition Der0(g) = A0 ⊕ B0, where A0 := ad(g) ⊂ Der0(g) (because [g, g] = g) and B0 := A⊥ 0 are ideals in Der0(g) as in the proof of Lemma 7.6.8. Then ad(δξ) = [δ, ad(ξ)] = 0 for all δ ∈ B0 and all ξ ∈ g. Since Z(g) = {0}, this implies B0 = 0, and hence the adjoint representation ad : g → Der0(g) is bijective. By (7.6.11), this implies the existence of an involution ξ → ξ∗ such that ad(ξ∗) = ad(ξ)∗ for all ξ ∈ g. Since the adjoint representation ad : g → Der(g) is injective, this in turn implies that the Killing form is nondegenerate and so Der(g) = ad(g) = Der0(g) by Lemma 7.4.3. Thus the inner product on g is symmetric. Conversely, if the inner product on g is symm
etric, then it satisﬁes (7.6.11) because δ and δ∗ have the same trace. This proves Lemma 7.6.13. 396 CHAPTER 7. TOPICS IN GEOMETRY Lemma 7.6.14. Let g be a ﬁnite-dimensional simple real Lie algebra with a symmetric inner product, and let g → g : ξ → ξ∗ be the unique involution that satisﬁes (7.6.4). Then there exists a constant c > 0 such that κ(ξ∗, η) = cξ, η for all ξ, η ∈ g. (7.6.12) Proof. By Lemma 7.6.9 the adjoint representation ad : g → Der(g) is injective. Hence the map g × g → R : (ξ, η) → κ(ξ∗, η) = trace(ad(ξ)∗ad(η)) is an inner product on g. Thus there exists a self-adjoint positive deﬁnite vector space isomorphism A : g → g such that κ(ξ∗, η) = Aξ, η for all ξ, η ∈ g. Let c > 0 be an eigenvalue of A and deﬁne h := η ∈ g Aη = cη. We prove that h is an ideal in g. Let ξ ∈ g and η ∈ h. Then, for all ζ ∈ g, A[ξ, η], ζ = κ([ξ, η]∗, ζ) = κ([η∗, ξ∗], ζ) = κ(η∗, [ξ∗, ζ]) = Aη, [ξ∗, ζ] = Aη, ad(ξ∗)ζ = ad(ξ)Aη, ζ = [ξ, Aη], ζ = c[ξ, η], ζ. Hence A[ξ, η] = c[ξ, η] and so [ξ, η] ∈ h
. This shows that h is a nonzero ideal and hence h = g. Thus A = c1l and this proves Lemma 7.6.14. Given any inner product on g, call an element P ∈ P0(g) symmetric iﬀ the inner product ·, P −1· on g is symmetric. Thus every symmetric element P ∈ P0(g) determines an involution τP : Der(g) → Der(g) given by τP (δ) := P δ∗P −1. Denote its determinant by εP := det(τP ) ∈ {−1, +1}. Lemma 7.6.15. Let g be a ﬁnite-dimensional simple real Lie algebra, choose any inner product on g, and let P, P0 ∈ P0(g) be symmetric elements. Then P P −1 0 ∈ Aut(g). Proof. The composition of the involutions τP and τP0 is the Lie algebra automorphism τP ◦τP0(δ) = P P0δ∗P −1 0 δP0P −1 for δ ∈ Der(g). By Lemma 7.6.8 and Lemma 7.6.9 the very existence of a symmetric inner product on g implies that the adjoint representation ad : g → Der(g) is a Lie algebra isomorphism. Hence there exists a g ∈ Aut(g) such that ∗P −1 = P P −1 0 ad(gξ) = P P −1 0 ad(ξ)P0P −1 for all ξ ∈ g. (7.6.13) Since ad(gξ) = gad(ξ)g−1, this automorphism satisﬁes the equations g−1P P −1 0 [ξ, η] = [ξ, g−1P P −1 0 η] for all ξ, η ∈ g, (7.6.14) det(g) = εP εP0 ∈ {−1, +1}. (7.6.15) 7.6. SEMISIMPLE LIE ALGEBRAS* 397 Since P0 is symmetric, it follows from Lemma 7.
6.8 and Lemma 7.6.14 that there exists an involution g → g : ξ → ξ∗ and a constant c > 0 such that ad(ξ∗) = P0ad(ξ)∗P −1 0, κ(ξ∗, η) = cξ, P −1 0 η (7.6.16) for all ξ, η ∈ g. By (7.6.13) and (7.6.16) we have κ((gξ)∗, η) = traceP0ad(gξ)∗P −1 0 ad(η) 0 ad(ξ)P0P −1∗P −1 = traceP P −1 = traceP −1P0ad(ξ)∗P −1 = tracead(ξ∗)P P −1 = tracead(ξ∗)ad(gη) = κ(ξ∗, gη). 0 P P −1 0 ad(η)P0P −1 0 ad(η)P0 0 ad(η)P0 0 ·, and so det(g) = 1 by (7.6.15). Since P P −1 This shows that g is self-adjoint and positive deﬁnite with respect to the inner product ·, P −1 is selfadjoint and positive deﬁnite with respect to the same inner product, the vector space isomorphism g−1P P −1 0 g−1/2)g1/2 has only positive real eigenvalues. Let λ > 0 be one such eigenvalue. Then the eigenspace ker(λ1l − g−1P P −1 ) is a nonzero ideal in g by (7.6.14), and so is equal to g, because g is simple. Thus g−1P P −1 0 = λ1l, and since g, P, P0 all have determinant one, it follows that λ = 1 and so P P −1 0 = g ∈ Aut(g). This proves Lemma 7.6.15. 0 = g−1/2(g−1/2P P −1 0 0 Proof of Theorems 7
.6.10, 7.6.11, and 7.6.12. We use the results of §7.5.3 in the situation where V := g is the Lie algebra itself and W := Λ2g∗ ⊗ g is the space of all skew-symmetric bilinear maps τ : g × g → g. The Lie group homomorphism ρ : SL(g) → SL(W ) is given by the standard action of the group SL(g) on W, i.e. (ρ(g)τ )(ξ, η) := gτ (g−1ξ, g−1η) for g ∈ SL(g), τ ∈ W, and ξ, η ∈ g. Fix an inner product ·, · on g and an orthonormal basis e1,..., em of g, and deﬁne σ, τ W := i,j σ(ei, ej), τ (ei, ej) (7.6.17) for σ, τ ∈ W. This inner product satisﬁes the requirements of Lemma 7.5.19, i.e. ρ(A∗) = ρ(A)∗ for all A ∈ sl(g). The vector w := [·, ·] ∈ W is chosen to be the Lie bracket. This vector is nonzero because g is not abelian. The isotropy subgroup of w is the group Aut(g) ∩ SL(g) of all automorphisms of determinant one. Denote its identity component by G ⊂ Aut0(g) ∩ SL(g). This is a Lie subgroup of SL(g) with the Lie algebra Lie(G) = Der0(g). 398 CHAPTER 7. TOPICS IN GEOMETRY Claim 1. G acts irreducibly on g. Let h ⊂ g be a subspace that is invariant under the action of G. Then δh ⊂ h for every δ ∈ Der0(g). By Lemma 7.6.9 this implies ad(ξ)h ⊂ h for all ξ ∈ g, so h is an ideal. Thus h = {0} or h = g because g is simple. Claim 2.
fg is convex and G-invariant and has a critical point. Moreover, every critical point P ∈ P0(g) is symmetric, and Der0(g) = Der(g). In the present setting the function fw in Lemma 7.5.20 agrees with the function fg in (7.6.9) and Gw = G. Hence fg is convex and G-invariant by Lemma 7.5.22, and G acts irreducibly on g by Claim 1. Thus Lemma 7.5.23 asserts that fg has a critical point. Let P ∈ P0(g) be a critical point of fg. Then by Lemma 7.5.23 the inner product ·, P −1· satisﬁes (7.6.11) and so, by Lemma 7.6.13, this inner product is symmetric and Der(g) = Der0(g). Claim 3. Fix a critical point P0 ∈ P0(g) of fg and any element P ∈ P0(g). Then the following are equivalent. (a) P is a critical point of fg. (b) fg(P ) = inf fg. (c) P P −1 0 ∈ Aut(g). (d) There exists a δ ∈ Der(g) such that δ = P0δ∗P −1 (e) The inner product ·, P −1· on g is symmeric. 0, exp(δ) = P P −1 0. Since fg is convex by Claim 2, the equivalence of (a) and (b) follows from Lemma 7.5.9. Since Der(g) = Der0(g) by Claim 2, the equivalence of (b), (c), and (d) follows from part (ii) of Lemma 7.5.22. Moreover, (a) implies (e) by Claim 2, and (e) implies (c) by Lemma 7.6.15. This proves Claim 3. The existence of a symmetric inner product on g was proved in Claim 2. Thus the nondegeneracy of the Killing form follows from Lemma 7.6.8. The remaining assertions of Theorem 7.6.10 are direct consequences of the nondegeneracy of the
Killing form. In particular, the adjoint representation ad : g → Der(g) is bijective and Der(g) ⊂ sl(g) by Lemma 7.4.3. Hence Aut0(g) ⊂ SL(g) and so Gw = G = Aut0(g) in the notation of §7.5.3. Now ﬁx a symmetric inner product on g. Then P0 = 1l is a critical point of fg by “(e) =⇒ (a)” in Claim 3. Hence the assertions about the subgroup K = Aut0(g) ∩ SO(g) in Theorem 7.6.11 follow from Lemma 7.5.23. The equalities in (7.6.10) follow from the equivalence of (a), (b), (c), (d), (e) in Claim 3 with P0 = 1l, and the remaining assertions about the space Mg in Theorem 7.6.12 follow from Lemma 7.5.24. This completes the proof of Theorems 7.6.10, 7.6.11, and 7.6.12. 7.6. SEMISIMPLE LIE ALGEBRAS* 399 7.6.3 Semisimple Lie Algebras The following theorem characterizes semisimple Lie algebras in terms of symmetric inner products. First observe that, if h is an ideal in a ﬁnitedimensional real Lie algebra g, then the subspace h := η ∈ g \| κ(η, η) = 0 for all η ∈ h (7.6.18) is also an ideal, because every η ∈ h satisﬁes κ(η, [ξ, η]) = κ([η, ξ], η) = 0 for all ξ ∈ g and all η ∈ h, and so [ξ, η] ∈ h for all ξ ∈ g. Theorem 7.6.16 (Semisimple Lie algebras). Let g be a ﬁnite-dimensional real Lie algebra. Then the following are equivalent. (i) g has a trivial center and admits a symmetric inner product. (ii) The Killing form �
� : g × g → R is nondegenerate. (iii) If h ⊂ g is an ideal, then g = h ⊕ h. (iv) g is a direct sum of simple ideals. (v) There exists an inner product ·, · on g, an involution g → g : ξ → ξ∗, and a constant c > 0 such that, for all ξ, η ∈ g, ad(ξ∗) = ad(ξ)∗, [ξ, η]∗ = [η∗, ξ∗], κ(ξ∗, η) = cξ, η. (7.6.19) Deﬁnition 7.6.17. A real Lie algebra g is called semisimple iﬀ it is ﬁnitedimensional and satisﬁes the equivalent conditions in Theorem 7.6.16. Proof of Theorem 7.6.16. That (i) implies (ii) was shown in Lemma 7.6.8. We prove that (ii) is equivalent to (iii). Assume ﬁrst that the Killing form is nondegenerate and let h ⊂ g be an ideal. Then η ∈ h, η ∈ h =⇒ [η, η] = 0, (7.6.20) because κ([η, η], ξ) = κ(η, [η, ξ]) = 0 for all ξ ∈ g, all η ∈ h, and all η ∈ h. Now let η ∈ h ∩ h. Then η := ad(ξ)ad(η)ζ = [ξ, [η, ζ]] ∈ h for all ξ, ζ ∈ g. Hence, by (7.6.20) we have (ad(ξ)ad(η))2ζ = [ξ, [η, η]] = 0 for all ξ, ζ ∈ g. Thus (ad(ξ)ad(η))2 = 0 and so κ(ξ, η) = trace(ad(ξ)ad(η)) = 0
for all ξ ∈ g. This implies η = 0 by nondegeneracy of the Killing form. Thus h ∩ h = {0} and so g = h ⊕ h because dim(h) + dim(h) = dim(g). This shows that (ii) implies (iii). To prove the converse, take h = g so that h = {0} is the kernel of the Killing form. It follows from (ii) and (iii) that, for every ideal h ⊂ g, the Killing forms of h and h are both nondegenerate. Hence an induction argument shows that (iii) implies (iv). 400 CHAPTER 7. TOPICS IN GEOMETRY We prove that (iv) implies (v). Assume that g = g1 ⊕ · · · ⊕ gr is a direct sum of simple ideals gj ⊂ g. Fix an index j ∈ {1,..., r} and denote by κj : gj × gj → R the Killing form of gj. By Theorem 7.6.11 the Lie algebra gj admits a symmetric inner product ·, ·j. Hence, by Lemma 7.6.8 there exists an involution gj → gj : ξ → ξ∗ that satisﬁes (7.6.4), and by Lemma 7.6.14 there exists a constant cj > 0 such that κj(ξ∗, η) = cjξ, ηj for all ξ, η ∈ gj. Thus the inner product ξ, η := j cjξj, ηjj for ξj, ηj ∈ gj and ξ = ξ1 + · · · + ξr, η = η1 + · · · + ηr satisﬁes the requirements of part (v) with c = 1 and ξ∗ := ξ∗ 1 + · · · + ξ∗ r. If (v) holds, then the Killing form is nondegenerate, hence Z(g) = 0 and Der(g) = ad(g) by Lemma 7.4.3, and hence the inner product in (v) is symmetric. Thus (v) implies (i)
and this proves Theorem 7.6.16. Corollary 7.6.18 (Cartan Involution). Let g be a nonzero semisimple real Lie algebra equipped with a symmetric inner product and let ξ → ξ∗ be the involution in Lemma 7.6.8. Then the map g → g : ξ → −ξ∗ (7.6.21) is a Lie algebra homomorphism (called a Cartan involution). The Cartan involution (7.6.21) gives rise to a splitting g = k ⊕ p, k := {ξ ∈ g \| ξ + ξ∗ = 0}, p := {η ∈ g \| η = η∗}, (7.6.22) such that [k, k] ⊂ k, [k, p] ⊂ p, (7.6.23) Moreover, k is nontrivial, the Killing form κ : g × g → R is negative deﬁnite on k and positive deﬁnite on p, and κ(ξ, η) = 0 for all ξ ∈ k and all η ∈ p. [p, p] ⊂ k. Proof. That the map (7.6.21) is a Lie algebra homomorphism follows directly from (7.6.4). It follows also from (7.6.4) that the subspaces k, p ⊂ g in (7.6.22) satisfy (7.6.23). That g is the direct sum of these subspaces, follows from the identity ζ ∗∗ = ζ for ζ ∈ g, which implies ζ = ξ + η, ξ := 1 2 ζ − ζ ∗ ∈ k, η := 1 2 ζ + ζ ∗ ∈ p. By (7.6.23) the summand k must be nontrivial, because g is nonzero and hence is not abelian. Next observe that the formula A, B := trace(A∗B) deﬁnes an inner product on End(g) with the norm \|A\| := trace(A∗A), and that κ(ξ, η∗) =
κ(ξ∗, η) = tracead(ξ)∗ad(η) = ad(ξ), ad(η). For ξ ∈ k and η ∈ p this implies κ(ξ, ξ) = −\|ad(ξ)\|2, κ(η, η) = \|ad(η)\|2, and κ(ξ, η) = 0. This proves Corollary 7.6.18. 7.6. SEMISIMPLE LIE ALGEBRAS* 401 Lemma 7.6.19. Let g be a semisimple real Lie algebra equipped with a symmetric inner product, let ξ → ξ∗ be the involution in Lemma 7.6.8, let g = g1 ⊕ · · · ⊕ gr be a decomposition into simple ideals gj, and let k, p ⊂ g be as in Corollary 7.6.18. Then the following holds. (i) The simple ideals gj ⊂ g are pairwise orthogonal. (ii) Each ideal gj is invariant under the involution ξ → ξ∗. (iii) p is the orthogonal complement of k. (iv) For each j the restriction of the inner product to gj is symmetric and gj = kj ⊕ pj, where kj = k ∩ gj, pj = p ∩ gj are as in Corollary 7.6.18. for all ξ ∈ g, η ∈ g⊥ Proof. We prove part (i). For each i the orthogonal complement g⊥ is an i ideal, because [ξ, η], ζ = η, [ξ∗, ζ] = 0 for all ξ ∈ g, η ∈ g⊥ i, ζ ∈ gi, and i. This implies that hj := i=j g⊥ so [ξ, η] ∈ g⊥ is an i i ideal, and so is the subspace hj ∩ gj. So either hj ∩ gj = {0} or hj = gj, If hj ∩ gj = {0},
then [ξ, η] = 0 for all ξ ∈ gj and because gj is simple. all η ∈ hj, hence gj ⊂ Z(g), and this is impossible because the center of g is trivial. Thus hj = gj for all j and this proves (i). We prove part (ii). The subspace g∗ j := {η∗ \| η ∈ gj} is an ideal, because [ξ, η∗] = [η, ξ∗]∗ ∈ g∗ for all ξ ∈ g and all η ∈ gj. By part (i) we j have [ξ, η∗], ζ = −ξ, [η, ζ] = 0 for all ξ ∈ gi and η ∈ gj with i = j, and all ζ ∈ g. Hence [gi, g∗ j ∩ gj cannot be zero, because otherwise [gj, g∗ j = gj, because gj is simple. This proves (ii). j ] = 0 for i = j. Hence the ideal g∗ j ⊂ Z(g). Hence g∗ j ] = 0 and so g∗ We prove part (iii). By (ii) and Lemma 7.6.14 the involution ξ → ξ∗ preserves the inner product on gj, and so by (i) it preserves the inner product on all of g. Hence ξ, η = ξ∗, η∗ = −ξ, η for all ξ ∈ k and all η ∈ p. Thus k and p are orthogonal to each other and this proves (iii). Part (iv) follows directly from (ii) and this proves Lemma 7.6.19. Theorem 7.6.20. Let g be an m-dimensional real Lie algebra that is not abelian and ﬁx an inner product on g and an orthonormal basis e1,..., em of g. Then the following are equivalent. (i) P = 1l is a critical point of fg. (ii) There exists a real number c such that 2ad(ei)∗ad(ei) − ad(ei)ad(
ei)∗ = c1l. (7.6.24) m i=1 (iii) g is semisimple, the inner product is symmetric, and there exists an involution g → g : ξ → ξ∗ and a constant c > 0 such that (7.6.19) holds. 402 CHAPTER 7. TOPICS IN GEOMETRY Proof. By Lemma 7.5.20 the element P = 1l ∈ P0(g) is a critical point of the function fg in (7.6.9) if and only if, for all A ∈ sl(g), m 0 = [ei, ej], A[ei, ej] − [Aei, ej] − [ei, Aej] i,j=1 m trace = i=1 ad(ei)∗Aad(ei) − 2ad(ei)∗ad(ei)A. This holds if and only if there exists a constant c ∈ R that satisﬁes (7.6.24). Thus we have proved that (i) is equivalent to (ii). We prove that (i) and (ii) imply (iii). Since P = 1l is a critical point of fg, Lemma 7.5.20 asserts that the inner product on g is symmetric. Thus by Lemma 7.6.8 there exists an involution g → g : ξ → ξ∗ that satisﬁes (7.6.4). Hence by (ii) there exists a real number c such that Qg := m i=1 2ad(e∗ i )ad(ei) − ad(ei)ad(e∗ i ) = c1l. (7.6.25) Since g is not abelian, the endomorphism Qg has a positive trace, so c > 0. This implies that the center of g is trivial, because Z(g) ⊂ ker(Qg). Hence, by Lemma 7.6.8, the Killing form on g is nondegenerate. By Lemma 7.6.19 this implies that the decomposition g = k ⊕ p in Corollary 7.6.18 is orthogonal. Hence the orthonormal basis e1,..., em of g can be chosen such that
e1,..., ek is a basis of k and ek+1,..., em is a basis of p. Thus e∗ i = −ei for i ≤ k and e∗ i = ei for i > k, and so it follows from (7.6.25) that m i=1 ad(e∗ i )ad(ei) = c1l. (7.6.26) Hence κ(ξ∗, η) = all ξ, η ∈ g. This proves (iii). iei, ad(ξ∗)ad(η)ei = iξ, ad(e∗ i )ad(ei)η = cξ, η for That (iii) implies (ii) follows by reversing this argument. By (iii) the Killing form is nondegenerate and the inner product is symmetric and is preserved by the involution ξ → ξ∗. Thus the splitting g = k ⊕ p is orthogonal, and hence the orthonormal basis e1,..., em of g can be chosen such that e1,..., ek is a basis of k and ek+1,..., em is a basis of p. Moreover, m ξ, ad(e∗ i )ad(ei)η = m ei, ad(ξ∗)ad(η)ei = κ(ξ∗, η) = cξ, η i=1 i=1 for all ξ, η ∈ g by (7.6.19). This implies (7.6.26). Since e∗ equation (7.6.24) follows from (7.6.26). This proves Theorem 7.6.20. i = ±ei for all i, 7.6. SEMISIMPLE LIE ALGEBRAS* 403 Corollary 7.6.21 (Cartan Decomposition). Let g be a semisimple real Lie algebra equipped with a symmetric inner product, let g → g : ξ → ξ∗ be the involution in Lemma 7.6.8, and deﬁne K := Aut0(g) ∩ SO(g), Der+(g) := {�
� ∈ Der(g) \| δ = δ∗}. Then the following holds. (i) K is connected and is a maximal compact subgroup of Aut0(g), every compact subgroup of Aut0(g) is conjugate in Aut0(g) to a Lie subgroup of K, and every maximal compact subgroup of Aut0(g) is conjugate to K. (ii) The map K × Der+(g) → Aut0(g) : (u, δ) → exp(δ)u is a diﬀeomorphism. (iii) If there exists a c > 0 such that κ(ξ∗, η) = cξ, η for all ξ, η ∈ g, then Mg := Crit(fg) = P0(g) ∩ Aut(g) = {exp(δ) \| δ ∈ Der+(g)} is a totally geodesic and geodesically convex submanifold of P0(g) and so is a Hadamard manifold and a symmetric space, and the map Aut0(g) → P0(g) : g → gg∗ descends to a diﬀeomorphism from the quotient space Aut0(g)/K to Mg. Proof. By Theorem 7.6.16 there exists a splitting g = g1 ⊕ · · · ⊕ gr into simple ideals, this splitting is preserved by every derivation of g, and by Lemma 7.6.19 it is also preserved by the involution ξ → ξ∗. Thus Aut0(g) is isomorphic to the product of the groups Aut0(gj), and K = Aut0(g) ∩ SO(g) is isomorphic to the product of the subgroups Kj := Aut0(gj) ∩ SO(gj). Hence part (i) follows from Theorem 7.6.11. Moreover, by Lemma 7.6.19, we have p = p1 ⊕ · · · ⊕ pr, and so part (ii) follows from Corollary 7.5.27. Under the assumptions of (iii) Theorem 7.6.20 asserts that P0 = 1l is a critical point of fg, so (iii)
follows from Lemma 7.5.22, Lemma 7.5.24, and Corollary 7.5.27. This proves Corollary 7.6.21. Remark 7.6.22. The Lie algebra of the group K = Aut0(g) ∩ SO(g) in Corollary 7.6.21 is given by Lie(K) = {ad(ξ) \| ξ ∈ k} (see Corollary 7.6.18). If the summand p in (7.6.22) is trivial, then Aut0(g) = K is a compact Lie group. If p is nontrivial, then the quotient space Aut0(g)/K is a nontrivial Hadamard manifold diﬀeomorphic to p. 404 CHAPTER 7. TOPICS IN GEOMETRY Remark 7.6.23. One can replace the space P0(g) of positive deﬁnite selfadjoint vector space isomorphisms P : g → g of determinant one by the space Hg of inner products on g with a ﬁxed determinant (Remark 6.5.11). This eliminates the dependence on the background inner product and there is then only one function fg : Hg → R whose set of minima is the totally geodesic submanifold Mg ⊂ Hg of all symmetric inner products on g with a ﬁxed determinant that satisfy part (iii) of Theorem 7.6.20. The main result asserts that, when g is not abelian, the space Mg is nonempty if and only if g is semisimple (Theorem 7.6.16 and Theorem 7.6.20). 7.6.4 Complex Lie Algebras A complex Lie algebra is a complex vector space g equipped with a Lie bracket g × g → g : (ξ, η) → [ξ, η] that is complex bilinear, i.e. it is a skewsymmetric bilinear map that satisﬁes the Jacobi identity and [iξ, η] = [ξ, iη] = i[ξ, η] for all ξ, η ∈ g. Thus every complex Lie algebra is also a real Lie algebra.
Let g be a ﬁnite-dimensional complex Lie algebra. A complex ideal in g is a complex linear subspace h ⊂ g that satisﬁes [ξ, η] ∈ h for all ξ ∈ g and all η ∈ h. The complex Lie algebra g is called simple iﬀ it is not abelian and has no complex ideals other than h = {0} and h = g. It is called semisimple iﬀ it is ﬁnite-dimensional and the complex Killing form κc : g × g → C, deﬁned by κc(ξ, η) := tracec(ad(ξ)ad(η)) for ξ, η ∈ g, is nondegenerate. Since κ = 2Reκc, a complex Lie algebra is semisimple if and only if it is semisimple as a real Lie algebra. The next lemma shows that the analogous assertion holds for simple complex Lie algebras (see [17, Lemma 7]). Lemma 7.6.24. A ﬁnite-dimensional complex Lie algebra g is simple if and only if it is simple as a real Lie algebra. Proof. Let g be a simple complex Lie algebra of complex dimension n and let h ⊂ g be a real linear subspace of g that satisﬁes [ξ, η] ∈ h for all ξ ∈ g and all η ∈ h. Then the subspaces h ∩ ih, h + ih are complex ideals in g, and their real dimensions satisfy the equation dimR(h ∩ ih) + dimR(h + ih) = 2 dimR(h). Since both summands on the left are either 0 or 2n, the real dimension of h is either 0, n, or 2n. We claim that the dimension cannot be n. 7.6. SEMISIMPLE LIE ALGEBRAS* 405 Assume, by contradiction, that dimR(h) = n. Then h ∩ ih = {0}, h + ih = g. Hence, for all ζ, ζ ∈ g there exist ξ, η ∈ h such that ζ = ξ + iη, and so
[ζ, ζ ] = [ξ, ζ ] + [η, iζ ] = i[ξ, −iζ ] + [η, ζ ] ∈ h ∩ ih = {0}. This contradicts the fact that g is not abelian. Thus h = {0} or h = g, and this proves Lemma 7.6.24. Lemma 7.6.25. Let g be a semisimple complex Lie algebra. Then g is a direct sum of simple complex ideals. Proof. Let h ⊂ g be a real ideal and let h ⊂ g be as in (7.6.18). Then it follows from “(ii) =⇒ (iii)” in Theorem 7.6.16 that [h + ih, h] = {0}. Hence h + ih ⊂ h = h and so ih = h. Thus every real ideal in g is a complex ideal. Hence the assertion follows from “(ii) =⇒ (iv)” in Theorem 7.6.16. The next result is a theorem of Cartan [13] which asserts that every semisimple complex Lie algebra has a compact real form. The proof given here is due to Donaldson [17, Lemma 8]. Theorem 7.6.26 (Cartan). Let g be a nonzero semisimple complex Lie algebra equipped with a symmetric inner product. Then p = ik, g = k ⊕ ik, (7.6.27) where k, p are as in Corollary 7.6.18. Moreover, the group Aut0(g) is the complexiﬁcation of the maximal compact subgroup K = Aut0(g) ∩ SO(g), i.e. Aut0(g) = {exp(iδ)u \| u ∈ K, δ ∈ Lie(K)} (7.6.28) and the map K × Lie(K) → Aut0(g) : (u, δ) → exp(iδ)u is a diﬀeomorphism. Proof. Assume ﬁrst that g is simple. Then g is simple as a real Lie algebra (Lemma 7.6.24), and so has a
nondegenerate Killing form (Theorem 7.6.10). By the inclusions in (7.6.23) the subspace h := (k ∩ ip) + (p ∩ ik) = (k ∩ ip) + i(k ∩ ip) is a complex ideal in g. Hence it is either {0} or g. If h = g, then it follows from (7.6.22) that p = ik. Assume, by contradiction, that h = {0}. Then k ∩ ip = {0}, g = k ⊕ ip. 406 CHAPTER 7. TOPICS IN GEOMETRY Hence the map σ : g → g, deﬁned by σ(ξ + iη) := ξ − iη for ξ ∈ k and η ∈ p is a Lie algebra homomorphism and an involution. This implies ad(σ(ζ)) = σad(ζ)σ and so κ(σ(ζ), σ(ζ )) = κ(ζ, ζ ) for all ζ, ζ ∈ g. Hence κ(ξ, iη) = 0 for all ξ ∈ k and all η ∈ p. Thus ip is the orthogonal complement of k with respect to the Killing form. Thus, by Corollary 7.6.18, p = ip. However, for all η ∈ p we have κ(iη, iη) = −κ(η, η) = −\|ad(η)\|2. Hence the Killing form is negative deﬁnite on ip and positive deﬁnite on p, and hence p = {0}. This implies k = ik. Since the Killing form is positive deﬁnite on ik and negative deﬁnite on k, this is a contradiction. This contradiction shows that our assumption h = {0} must have been wrong. Thus h = g and hence p = ik. This completes the proof of (7.6.27) in the simple case. For general semisimple complex Lie algebras, the proof of the equations in (7.6.27) reduces to the simple case by Lemma 7.6.19 and Lem
ma 7.6.25. Equation (7.6.28) follows directly from (7.6.27) and Corollary 7.6.21. This proves Theorem 7.6.26. Remark 7.6.27. Let g be a semisimple complex Lie algebra. Then Aut(g) is a complex Lie group, i.e. it admits the structure of a complex manifold such that the structure maps G × G → G : (h, g) → hg, G → G : g → g−1 dt (g−1 g) + [g−1 ˙g, g−1 are holomorphic. The proof uses the fact that Der(g) is isomorphic to g and that the resulting almost complex structure on Aut(g) is integrable (as it is preserved by the torsion-free connection g−1∇g = d g]). Theorem 7.6.26 asserts that the identity component G = Aut0(g) of the group of automorphisms of g is the complexiﬁcation of the maximal compact subgroup K = Aut0(g) ∩ SO(g), i.e. its Lie algebra Der(g) = ad(g) is the complexiﬁcation of the Lie algebra Lie(K) = ad(k) and the quotient space G/K is contractible. These conditions imply the universality property that every Lie group homomorphism ρ : K → G with values in a complex Lie group G extends to a unique holomorphic Lie group homomorphism ρc : G → G such that ρc\|K = ρ. Such a complexiﬁcation exists for every compact Lie group K, whether or not it is semisimple. (For an exposition see [20, Appendix B].) 7.6. SEMISIMPLE LIE ALGEBRAS* 407 The following exercise is inspired by a remark in [17, §5.1] concerning a positive curvature manifold that is dual to Mg. Exercise 7.6.28. Let g be a semisimple real Lie algebra equipped with a symmetric inner product that satisﬁes condition (iii) in Theorem 7.6.20. Consider the complexiﬁed Lie algebra with the Lie bracket gc := g ⊕ ig [�
�, ζ ] := [ξ, ξ] − [η, η] + i[ξ, η] + [η, ξ] (7.6.29) and the Hermitian form ζ, ζ c := ξ, ξ + η, η + iξ, η − η, ξ (7.6.30) for ζ = ξ + iη ∈ gc and ζ = ξ + iη ∈ gc. With this convention the Hermitian form (7.6.30) is complex anti-linear in the ﬁrst variable and complex linear in the second variable. Prove the following. (a) gc is semisimple. Hint: gc has a trivial center and the real part of (7.6.30) is a symmetric inner product on gc. (b) If g is simple, then gc is simple. Hint: If hc is a complex ideal in gc, then the linear subspace h := {Re(ζ) \| ζ ∈ hc} is an ideal in g. (c) Every real linear derivation on gc is complex linear and has complex trace zero. (d) The identity component Aut0(gc) of the group of real linear Lie algebra automorphisms of gc consists of complex linear automorphisms of complex determinant one. (Complex conjugation is a Lie algebra automorphism of gc not in the identity component.) (e) The subgroup Kc := Aut0(gc) ∩ SU(gc) is connected and is a maximal compact subgroup of Aut0(gc). Its Lie algebra Lie(Kc) ∼= k + ip is the compact real form of gc (Corollary 7.6.18 and Theorem 7.6.26). 408 CHAPTER 7. TOPICS IN GEOMETRY (f ) Let Φ + iΨ : g → gc be a Lie algebra homomorphism, i.e. for all ξ, η ∈ g, Φ[ξ, η] = [Φξ, Φη] − [Ψξ, Ψη], Ψ[ξ, η] = [Φξ, Ψ
η] + [Ψξ, Φη]. Assume Φ + iΨ is injective and denote its image by l := {Φξ + iΨξ \| ξ ∈ g}. Then the following are equivalent. (I) The real part of (7.6.30) restricts to a symmetric inner product on l. (II) Φ∗Φ + Ψ∗Ψ ∈ Aut(g). (III) l is a Lagrangian subspace of gc with respect to the imaginary part of (7.6.30), i.e. Φ∗Ψ − Ψ∗Φ = 0. Hint: If Φ∗Φ + Ψ∗Ψ ∈ Aut(g), then there exists a derivation α ∈ Der(g) such that α = α∗ and exp(2α) = Φ∗Φ + Ψ∗Ψ (Corollary 7.6.21). Prove that δ := exp(−α)(Φ∗Ψ − Ψ∗Φ) exp(−α) is a derivation whose image is abelian. Prove that κ(δξ, δη) = 0 for all ξ, η ∈ g and deduce that δ∗δ = −δ2 = 0. (g) The space of oriented Lagrangian Lie subalgebras l ⊂ gc isomorphic to g (that can be joined to g by a path of such subspaces) is diﬀeomorphic to the quotient space Lg := Kc/K, where Kc := Aut0(gc) ∩ SU(gc) and K := Aut0(g) ∩ SO(g). Hint: Choose the embedding Φ + iΨ in (f) such that Φ∗Φ + Ψ∗Ψ = 1l, Φ∗Ψ − Ψ∗Φ = 0, and extend it to a unitary automorphism of gc. (h) The space Lg in part (g) embeds as a totally geodesic submanifold of dimension dim(Lg) = dim(p) into the symmetric space U
(gc)/SO(g) of all oriented Lagrangian subspaces of gc. Hence Lg has nonnegative sectional curvature. (Hint: Example 5.2.18.) One can think of the positive curvature manifold Lg of all Lagrangian Lie subalgebras of gc that are isomorphic to g as dual to the negative curvature manifold Mg of all symmetric inner products on g that satisfy condition (iii) in Theorem 7.6.20. 7.6. SEMISIMPLE LIE ALGEBRAS* 409 Remark 7.6.29. The idea of minimizing the norm of the Lie bracket was the approach to the existence of a compact real form of a simple complex Lie algebra suggested by Cartan [14] and carried out by Richardson [58]. One signiﬁcant diﬀerence in the method of Donaldson [17], which we follow in the section, is that there is no need to assume that the Killing form is nondegenerate, but that this result emerges as a byproduct of the proof (Theorem 7.6.10). There is also no need to use the structure theory of Lie algebras as in the work of Weyl [77]. Instead one can use the existence of a symmetric inner product as a starting point to develop the structure theory of Lie algebras. Remark 7.6.30. As pointed out by Donaldson [17], a more direct approach to Theorem 7.6.26 would be to carry over the entire program in the present section and §7.5 to the complex setting, starting in §7.5.2 with convex functions on the Hadamard manifold M = Q0(V ) ∼= SL(V )/SU(V ) of positive deﬁnite Hermitian automorphisms with determinant one of a complex vector space V equipped with a Hermitian inner product (Remark 6.5.20). In the complex Lie algebra setting with Q0(g) ∼= SL(g, C)/SU(g, C) the logarithm of the function fg : Q0(g) → R is the log-norm function of Kempf and Ness in geometric invariant theory [38, 20]. Thus the existence of a critical point of fg is the polystability condition in GIT. This approach was developed
by Lauret [44] and he proved that the polystable points are precisely the semisimple Lie algebras. This is the content of Theorem 7.6.20 in the complex setting. Lauret’s proof uses Cartan’s theorem about the compact real form of a semisimple complex Lie algebra. One can also deduce the theorems in the real setting from those in the complex setting by complexifying the relevant real inner product space V to obtain a complex vector space V c = V ⊕ iV with a Hermitian inner product, and embedding the space P0(V ) ∼= SL(V )/SO(V ) as a totally geodesic submanifold into Q0(V c) ∼= SL(V c)/SU(V c). 410 CHAPTER 7. TOPICS IN GEOMETRY Appendix A Notes This appendix explains some notations and standard results from ﬁrst year analysis that are used throughout this book. A.1 Maps and Functions The notation f : X → Y means that f is a function which assigns to every point x in the set X a point f (x) in the set Y. When Y = R we express this by saying that f is a real valued function deﬁned on the set X and, if Y is a vector space, we may say that f is a vector valued function. However in general it is better to say that f is a map from X to Y and call the set X the source of the map and the set Y its target. The graph of f is the set graph(f ) := {(x, yx)}. We always distinguish two maps with the same graph when their targets are diﬀerent. A map f : X → Y is said to be    injective surjective bijective    iﬀ    f (x1) = f (x2) =⇒ x1 = x2 ∀y ∈ Y ∃x ∈ X s.t. y = f (x) it is both injective and surjective.    Then (a) f is injective ⇐⇒ it has a left inverse g : Y → X (i.e. g
◦ f = idX ); (b) f is surjective ⇐⇒ it has a right inverse g : Y → X (i.e. f ◦ g = idY ); (c) f is bijective ⇐⇒ it has a two sided inverse f −1 : Y → X. (Item (b) is the Axiom of Choice.) 411 412 APPENDIX A. NOTES The analogous principle holds for linear maps: if A ∈ Rm×n, then the linear map Rn → Rm : x → Ax is (a) injective ⇐⇒ BA = 1ln for some B ∈ Rn×m; (b) surjective ⇐⇒ AB = 1lm for some B ∈ Rn×m; (c) bijective ⇐⇒ A is invertible (i.e. m = n and det(A) = 0). (Here 1lk is the k × k identity matrix.) However, this principle fails completely for continuous maps: the map f : [0, 2π) → S1 deﬁned by f (θ) = (cos θ, sin θ) is continuous and bijective but its inverse is not continuous. (Here S1 ⊂ R2 is the unit circle x2 + y2 = 1.) A.2 Normal Forms The Fundamental Idea of Diﬀerential Calculus is that near a point x0 ∈ U a smooth map f : U → V behaves like its linear approximation, i.e. f (x) ≈ f (x0) + df (x0)(x − x0). The Normal Form Theorem from Linear Algebra says that if A ∈ Rm×n has rank r, then there are invertible matrices P ∈ Rm×m and Q ∈ Rn×n such that P −1AQ = 1lr 0r×(n−r). 0(m−r)×r 0(m−r)×(n−r) By the Fundamental Idea we can expect an analogous theorem for smooth maps. Theorem A.2.1 (Local Normal Form for Smooth Maps). Let U ⊂ Rn and V ⊂ Rm be open, x0 ∈ U, and f : U → V be smooth. Assume that the
derivative df (x0) ∈ Rm×n has rank r. Then there is an open neighborhood U0 of x0 in U, an open neighborhood V0 of f (x0) in V, a diﬀeomorphism φ : U1 × U2 ⊂ Rr × Rn−r, a diﬀeomorphism ψ : V0 → U1 × V2 ⊂ Rr × Rm−r, such that φ(x0) = (0, 0), ψ(f (x0)) = (0, 0), and ψ−1 ◦ f ◦ φ(x, y) = (x, g(x, y)) and dg(0, 0) = 0 for (x, y) ∈ U1 × U2. The Local Normal Form Theorem is an easy consequence of the Inverse Function Theorem. A.2. NORMAL FORMS 413 Theorem A.2.2 (Inverse Function Theorem). Let U ⊂ Rn, V ⊂ Rm, x0 ∈ U and f : U → V be a smooth map. If df (x0) is invertible, then (m = n and) there are neighborhoods U0 of x0 in U and V0 of f (x0) in V so that the restriction f\|U0 : U0 → V0 is a diﬀeomorphism. Here follow some other consequences of the Inverse Function Theorem. The terms submersion and immersion are deﬁned in §2.6.1 and Deﬁnition 2.3.2 of §2.3. Corollary A.2.3 (Submersion Theorem). When r = m the diﬀeomorphisms φ and ψ in Theorem A.2.1 may be chosen so that the local normal form is ψ−1 ◦ f ◦ φ(x, y) = x. Corollary A.2.4 (Immersion Theorem). When r = n the diﬀeomorphisms φ and ψ in Theorem A.2.1 may be chosen so that the local normal form is ψ−1 ◦ f ◦ φ(x) = (x, 0). Corollary A.2.
5 (Rank Theorem). If the rank of df (x) = r for all x ∈ U, then for every x0 ∈ U the diﬀeomorphisms φ and ψ in Theorem A.2.1 may be chosen so that the local normal form is ψ−1 ◦ f ◦ φ(x) = (x1,..., xr, 0,..., 0). Corollary A.2.6 (Implicit Function Theorem). Let U ⊂ Rm × Rn be an open set, let F : U → Rn be smooth, and let (x0, y0) ∈ U with x0 ∈ Rm and y0 ∈ Rn. Deﬁne the partial derivative d2F (x0, y0) ∈ Rn×n by d2F (x0, y0)v := d dt t=0 F (x0, y0 + tv) for v ∈ Rn. Assume that F (x0, y0) = 0 and that d2F (x0, y0) is invertible. Then there exist neighborhoods U0 of x0 in Rm and V0 of y0 in Rn and a smooth map g : U0 → V0 such that U0 × V0 ⊂ U, g(x0) = y0 and F (x, y) = 0 ⇐⇒ y = g(x) for x ∈ U0 and y ∈ V0. 414 APPENDIX A. NOTES A.3 Euclidean Spaces This is the arena of Euclidean geometry; i.e. every ﬁgure which is studied in Euclidean geometry is a subset of Euclidean space. To deﬁne it one could proceed axiomatically as Euclid did; one would then verify that the axioms characterized Euclidean space by constructing “Cartesian Coordinate Systems” which identify the n-dimensional Euclidean space En with the ndimensional numerical space Rn. This program was carried out rigorously by Hilbert. We shall adopt the mathematically simpler but philosophically less satisfying course of taking the characterization as the deﬁnition. We shall use three closely related spaces: n-dimensional Euclidean aﬃne space En,
n-dimensional Euclidean vector space En, and the space Rn of all n-tuples of real numbers. The distinction among them is a bit pedantic, especially if one views as the purpose of geometry the interpretation of calculations on Rn. The purpose for distinguishing these three spaces is the same as in elementary vector calculus; it aids geometric intuition. Here is the precise deﬁnition. Deﬁnition A.3.1. An n-dimensional Euclidean vector space is a real ndimensional vector space En equipped with a (real valued symmetric positive deﬁnite) inner product En × En → R : (v, w) → v, w. An n-dimensional Euclidean aﬃne space consists of a set En and an n-dimensional Euclidean vector space En and maps En × En → En : (p, q) → p − q, En × En → En : (p, v) → p + v satisfying the axioms p + 0 = p, p + (v + w) = (p + v) + w, q + (p − q) = p for all p, q ∈ En and all v, w ∈ En. The vector p − q ∈ En is called the vector from q to p and the point p + v is called the translate of p by v. It follows easily that each choice of a point o ∈ En determines a bijection v → o + v from En onto En. The inner product on En equips the space En with a metric via the formula \|p − q\| = p − q, p − q, p, q ∈ En. The standard Euclidean space of dimension n is En = En = Rn with the usual matrix algebra operations (x ± y)i = xi ± yi, x, y = i xiyi. A.3. EUCLIDEAN SPACES 415 Lemma A.3.2. Any choice of an origin o ∈ En and an orthonormal basis e1,..., en for En determines an isometric bijection: Rn → En : (x1,..., xn) → o + n i=1 xiei (the inverse of which is called a Cartesian coordinate system on En). Lemma A.3.3. If En → Rn : p → (
x1,..., xn), (y1,..., yn) are two Cartesian coordinate systems, the change of coordinates map has the form yj(p) = n j=1 aj i xi(p) + vi, where the matrix a = (aj i ) ∈ Rn×n is an orthogonal matrix and v ∈ Rn. Example A.3.4. Any n-dimensional aﬃne subspace of some numerical space Rk (with k > n) is an example of a Euclidean space. The corresponding vector space En is the unique vector subspace of Rk for which: En = o + En for o ∈ En. This subspace is independent of the choice of o ∈ En. Note that En contains the “preferred” point 0 while En has no preferred point. Such spaces En and En would arise in linear algebra by taking En to be the space of solutions of k − n independent inhomogeneous linear equations in k unknowns while En is the space of solutions of the corresponding homogeneous equations. The correspondence between En and En illustrates the mantra The general solution of an inhomogeneous system of linear equations is a particular solution plus the general solution of the corresponding homogeneous linear system. This discussion shows that a Euclidean space En is an n-dimensional manifold with its Cartesian coordinate systems whose tangent space at each point is naturally isomorphic to En. Thus it is natural to introduce submanifolds of Euclidean space as submanifolds of En whose tangent spaces are then linear subspaces of the associated vector space En. Instead we have chosen in this book for simplicity of the exposition to describe manifolds as subsets of the vector space Rn equipped with its standard inner product. 416 APPENDIX A. NOTES Bibliography [1] Ralph Herman Abraham & Joel William Robbin, Transversal Mappings and Flows. Benjamin Press, 1967. [2] Lars Valerian Ahlfors & Leo Sario, Riemann Surfaces. Princeton University Press, 1960. [3] Aleksandr Danilovich Alexandrov, ¨Uber eine Verallgemeinerung der Riemannschen Geometrie. Schriftreihe des Forschinstituts f¨ur Mathematik 1 (1957), 33–84. [4] Michael Francis Atiyah & Nigel James
Hitchin & Isodore Manuel Singer, Selfduality in four-dimensional Riemannian geometry. Prococeedings of the Royal Society of London, Series A 362 (1978), 425–461. [5] Thierry ´Emilien Flavien Aubin, ´Equations diﬀ´erentielles non lin´eaires et probl`eme de Yamabe concernant la courbure scalaire. Journal de Math´ematiques Pures et Appliqu´ees 55 (1976), 269–296. [6] Marcel Berger, Les vari´et´es Riemanniennes 1/4-pinc´ees, Annali della Scuola Normale Superiore di Pisa 14 (1960), 161–170. [7] Richard Lawrence Bishop & Barrett O’Neill, Manifolds of negative curvature. Transactions of the American Mathematical Society 145 (1969), 1–49. [8] Simon Brendle, Ricci Flow and the Sphere Theorem. Graduate Studies in Mathematics 111. AMS, Providence, RI, 2010. [9] Simon Brendle & Richard Melvin Schoen, Manifolds with 1/4-pinched curvature are space forms. Journal of the American Mathematical Society 22 (2009), 287–307. https://arxiv.org/pdf/0705.0766.pdf [10] Simon Brendle & Richard Melvin Schoen, Curvature, Sphere Theorems, and the Ricci Flow. Bulletin of the American Mathematical Society 48 (2011), 1–32. https://arxiv.org/pdf/1001.2278.pdf [11] Martin Bridson, & Andr´e Haeﬂiger, Metric spaces of non-positive curvature. Springer, 1999. https://www.math.bgu.ac.il/~barakw/rigidity/bh.pdf 417 418 BIBLIOGRAPHY [12] Theo B¨uhler & Dietmar Arno Salamon, Functional Analysis. Graduate Studies in Mathematics 191, American Mathematical Society, Providence, Rhode Island, 2018. [13] ´Elie Joseph Cartan, Les groupes r´eels simples, ﬁnis et continus. Annales sci- entiﬁques de l’ ´Ecole Normale Sup’erieure 31 (19
14), 263–355. [14] ´Elie Joseph Cartan, Groupes simples clos et ouverts et g´eom´etrie Riemanni- enne. Journal de Math´ematiques Pures et Appliqu´ees 8 (1929) 1–33. [15] ´Elie Joseph Cartan, La th´eorie des groupes ﬁnis et continus et l’Analysis Situs. M´emorial des Sciences Math´ematiques 42 (1930), 1–61. [16] Bill Casselmann, Symmetric spaces of semi-simple groups. Essays on representations of real groups, August, 2012. http://www.math.ubc.ca/~cass/ research/pdf/Cartan.pdf [17] Simon Kirwan Donaldson, Lie algebra theory without algebra. In “Algebra, Arithmetic, and Geometry, In honour of Yu. I. Manin”, edited by Yuri Tschinkel and Yuri Zarhin, Progress in Mathematics, Birkh¨auser 269, 2009, pp. 249–266. https://arxiv.org/pdf/math/0702016.pdf [18] Simon Kirwan Donaldson, Towords enumerative theories for structures on 4manifolds. Seminar, 15 September 2020. https://www.youtube.com/watch? v=44hwQKQggbA [19] David Bernard Alper Epstein, Foliations with all leaves compact. Annales de l’institute Fourier 26 (1976), 265–282. [20] Valentina Georgoulas & Joel William Robbin & Dietmar Arno Salamon, The Moment-Weight Inequality and the Hilbert–Mumford Criterion. Lecture Notes in Mathematics 2297, Springer-Verlag, 2021. https://people.math.ethz. ch/~salamond/PREPRINTS/momentweight-book.pdf [21] Mikhael Leonidovich Gromov, Pseudoholomorphic curves in symplectic man- ifolds. Inventiones Mathematicae 82 (1985), 307–347. [22] Michael Leonidovich Gromov & Herbert Blaine Lawson, The classiﬁcation of simply connected manifolds of positive scalar curvature, Annals of Mathematics 111 (1980), 423–434. [23]
Victor Guillemin & Alan Pollack, Diﬀerential Topology. Prentice-Hall, 1974. [24] Allen Hatcher, Algebraic Topology, Cambridge, 2002. [25] Sigurour Helgason, Diﬀerential Geometry, Lie Groups, and Symmetric Spaces. Graduate Studies in Mathematics 35, AMS, 2001. BIBLIOGRAPHY 419 [26] Noel Justin Hicks, Notes on Diﬀerential Geometry, Van Nostrand Rheinhold Mathematical Studies, 1965. [27] David Hilbert & Stephan Cohn-Vossen, Geometry and the Imagination. Chelsea Publishing Company, 1952. [28] Joachim Hilgert & Karl-Hermann Neeb, Structure and Geometry of Lie groups. Springer Monographs in Mathematics, Springer-Verlag, 2012. [29] Morris William Hirsch, Diﬀerential Topology. Springer, 1994. [30] Nigel James Hitchin, Harmonic spinors. Advances in Mathematics 14 (1974), 1–55. [31] Heinz Hopf, Zum Cliﬀord–Kleinschen Raumproblem. Mathematische Annalen 95 (1926), 313–339. [32] Heinz Hopf, Diﬀerentialgeometrie und topologische Gestalt. Jahresbericht der Deutschen Mathematiker-Vereinigung 41 (1932), 209–229. [33] Ralph Howard, Kuiper’s theorem on conformally ﬂat manifolds. Lecture Notes, University of South Caroline, Columbia, July 1996. http://ralphhoward. github.io/SemNotes/Notes/conformal.pdf [34] Mustafa Kalafat, Locally conformally ﬂat and self-dual structures on simple 4-manifolds. Proceedings of the G¨okova Geometry-Topology Conference 2012, International Press, Somerville, MA, 2013, pp 111–122. http://gokovagt. org/proceedings/2012/ggt12-kalafat.pdf [35] Tosio Kato, Perturbation Theory for Linear Operators. Grundlehen der Math- ematische Wissenschaften 132, Springer-Verlag, 1976. [36] John Leroy Kelley,
General Topology. Graduate Texts in Mathematics, 27. Springer 1975. [37] George Kempf, Instability in invariant theory. Annals of Mathematics 108 (1978), 299–317. [38] George Kempf & Linda Ness, The lengths ov vectors in representation spaces. Springer Lecture Notes 732 (1978), 233–242. [39] Wilhelm Karl Joseph Killing, ¨Uber die Grundlagen der Geometrie. Journal f¨ur die reine und angewandte Mathematik 109 (1892), 121–186. [40] Frances Kirwan, Cohomology of Quotients in Symplectic and Algebraic Geom- etry. Princeton Uuniversity Press, 1984. [41] Wilhelm Klingenberg, ¨Uber Riemannsche Mannigfaltigkeiten mit positiver Kr¨ummung. Commentarii Mathematici Helvetici 35 (1961), 47–54. 420 BIBLIOGRAPHY [42] Nicolaas Hendrik Kuiper, On conformally ﬂat spaces in the large. Annals of Mathematics 50 (1949), 916–924. [43] Wolfgang K¨uhnel, Diﬀerential geometry: Curves-surfaces-manifolds, Second Edition. Student Mathematical Library, 16, AMS, Providence, RI, 2006. [44] Jorge Lauret, On the moment map for the variety of Lie algebras. Journal of Functional Analysis 202 (2003), 392–423. [45] Claude LeBrun, On the topology of self-dual 4-manifolds. Proceedings of the American Mathematical Society 98 (1986), 637–640. [46] Andr´e Lichnerowicz, Spineurs harmoniques, Comptes rendus de l’Acad´emie des Sciences, S´erie A-B 257 (1963), 7–9. [47] Anatolij Ivanovich Malcev, On the theory of the Lie groups in the large. Matematichevskii Sbornik N. S. 16 (1945), 163–189. Corrections to my paper ”On the theory of Lie groups in the large”. Matematichevskii Sbornik N. S. 19 (1946), 523–524. [48] Alina Marian, On the real moment map.
Mathematical Research Letters 8 (2001), 779–788. [49] Dusa McDuﬀ & Dietmar Arno Salamon, Introduction to Symplectic Topology, Third Edition. Oxford University Press, 2017. [50] John Willard Milnor, Topology from the Diﬀerentiable Viewpoint. The Uni- versity Press of Virginia 1969. [51] James Raymond Munkres, Topology, Second Edition. Prentice Hall, 2000. [52] Sumner Byron Myers & Norman Earl Steenrod, The group of isometries of a Riemannian manifold. Annals of Mathematics 40 (1939), 400–416. [53] Linda Ness, A stratiﬁcation of the null cone by the moment map. American Journal of Mathematics 106 (1984), 128–132. [54] John von Neumann, ¨Uber die analytischen Eigenschaften von Gruppen linearer Transformationen und ihrer Darstellungen. Mathematische Zeitschrift 30 (1929), 3–42. [55] Takushiro Ochiai & Tsunero Takahashi, The group of isometries of a left invariant Riemannian metric on a Lie group. Mathematische Annalen 223 (1976), 91–96. [56] Roger Penrose, Nonlinear gravitons and curved twistor theory. The riddle of gravitation – on the occasion of the 60th birthday of Peter G. Bergmann (Proc. Conf., Syracuse Univ., Syracuse, NY., 1975), General Relativity and Gravitation 7 (1976), 31–52. BIBLIOGRAPHY 421 [57] Harry Earnest Rauch, A contribution to diﬀerential geometry in the large. Annals of Mathematics 54 (1951), 38–55. [58] Roger Richardson, Compact real forms of a complex semisimple Lie algebra. Journal of Diﬀerential Geometry 2 (1968), 411–419. [59] Roger Richardson & Peter Slodowy, Minimal vectors for real reductive group actions. Journal of the London Mathematical Society 42 (1990), 409–429. [60] Joel William Robbin, Extrinsic Diﬀerential Geometry. 1982. http://www.math.wisc.edu/~robbin/0geom.pdf [61]
Joel William Robbin & Dietmar Arno Salamon, The spectral ﬂow and the Maslov index. Bulletin of the London Mathematical Society 27 (1995), 1–33. https://people.math.ethz.ch/~salamond/PREPRINTS/spec.pdf [62] Mary Ellen Rudin, A new proof that metric spaces are paracompact. Proceed- ings of the American Mathematical Society 20 (1969), 603. [63] Dietmar Arno Salamon, Analysis I. Lecture Course, ETHZ, HS 2016. [64] Dietmar Arno Salamon, Analysis II. Lecture Course, ETHZ, FS 2017. https://people.math.ethz.ch/~salamon/PREPRINTS/analysis2.pdf [65] Dietmar Arno Salamon, Spin Geometry and Seiberg–Witten invariants. Unpublished Manuscript, 1999. https://people.math.ethz.ch/~salamond/ PREPRINTS/witsei.pdf [66] Dietmar Arno Salamon, Measure and Integration. European Mathematical Society, Textbooks in Mathematics, 2016. https://people.math.ethz.ch/ ~salamond/PREPRINTS/measure.pdf [67] Dietmar Arno Salamon, Uniqueness of symplectic structures. Acta Mathemat- ica Vietnamica 38 (2013), 123–144. [68] Dietmar Arno Salamon & Thomas Walpuski, Notes on the Octonions. Proceedings of the 23rd G¨okova Geometry-Topology Conference 2016, edited by S. Akbulut, D. Auroux, and T. ¨Onder, International Press, Somerville, Massachusetts, 2017, pp 1-85. https://people.math.ethz.ch/~salamon/ PREPRINTS/Octonions.pdf [69] Richard Melvin Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature. Journal of Diﬀererential Geometry 20 (1084), 479– 495. [70] Richard Melvin Schoen & Shing-Tung Yau, On the structure of manifolds with positive scalar curvature. Manuscripta Mathematica 28 (1979), 159–183. [71] Carl Ludwig Siegel, Sy
mplectic Geometry. Academic Press, 1964. 422 BIBLIOGRAPHY [72] Stephen Smale, Diﬀeomorphisms of the 2-sphere. Proceedings of the American Mathematical Society 10 (1959), 621–626. [73] Cliﬀord Henry Taubes, Self-dual Yang-Mills connections on non-self-dual 4- manifolds. Journal of Diﬀerential Geometry 17 (1982), 139–170. [74] Cliﬀord Henry Taubes, SW =⇒ Gr: From the Seiberg–Witten equations to pseudoholomorphic curves. Journal of the American Mathematical Society 9 (1996), 845–918. [75] Neil Sidney Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds. Annali della Scuola Normale Superiore di Pisa 22 (1968), 265–274. [76] Joa Weber, Perturbed closed geodesics are periodic orbits: Transversality. Mathematische Zeitschrift 241 (2002), 45–81. Index and [77] Hermann Weyl, Theorie der Darstellung kontinuierlicher halbeinfacher Gruppen durch lineare Transformationen, I, II, III; Nachtrag zu der Arbeit III. Mathematische Zeitschrift 23 (1925), 271–309; 24 (1926), 328–376, 377–395, 789–791. [78] Hidehiko Yamabe, On a deformation of Riemannian structures on compact manifolds. Osaka Journal of Mathematics 12 (1960), 21–37. Index abelian Lie algebra, 391 aﬃne subspace, 159 atlas, 4, 87 maximal, 9, 11 smooth, 11 topological, 8 automorphism of a Lie algebra, 58, 61 of a Lie group, 58 of F (M ), 69 Axiom of Choice, 411 basic vector ﬁeld, 145, 173 Bianchi identity ﬁrst, 239 in local coordinates, 254 second, 276 in local coordinates, 278 Bonnet–Myers Theorem, 310 Brouwer, 8 bundle frame, 138 principal, 115, 140 tangent, 74, 97 vector, 74 Cartan decomposition, 388, 403 Cartan Fixed Point
points, 258 Hopf ﬁbration, 221 Hopf–Rinow Theorem, 210 horizontal curve, 142 lift, 142 subbundle, 173 tangent space, 142, 173 hyperbolic space, 286 Poincar´e model, 2, 290 upper half plane, 292 hyperboloid, 271 hypersurface, 249 426 ideal in a complex Lie algebra, 404 in a Lie algebra, 391 in F (M ), 70 immersion, 34 Implicit Function Theorem, 79, 413 injectivity radius, 194 inner product symmetric, 382, 391 integrable subbundle of T M, 80 integral curve, 39 intrinsic distance, 183, 217 topology, 86, 88 Invariance of Domain, 8 Inverse Function Theorem, 31 involutive subbundle of T M, 80 isometry, 224 group, 224, 343 local, 230 of a Lie group, 362 isomorphism, 12 of Lie algebras, 58 of Lie groups, 58 isotopy, 45 isotropy subgroup, 114 Jacobi ﬁeld, 266 Jacobi identity, 49 Jacobian matrix, 10 Kato Selection Theorem, 333 Killing form, 242 nondegenerate, 243 Killing form complex, 404 Killing vector ﬁeld, 347 complete, 348, 349 Klein bottle, 112 Kuiper’s Theorem, 322 Laplace–Beltrami operator, 317 INDEX leaf of a foliation, 85, 113 closed, 113 extrinsic topology, 86 intrinsic topology, 86 length of a curve, 176 Levi-Civita connection, 171, 234 axioms, 171 Lie algebra, 50 abelian, 391 automorphism, 58, 61 Cartan involution, 400 center, 66 commutant, 242 complex, 404 derivation, 61 Heisenberg, 58 homomorphism, 50, 58 ideal, 391 inﬁnitesimal action, 68, 114 invariant inner product, 242 isomorphism, 58 of a Lie group, 55 of vector ﬁelds, 50 semisimple, 399 simple, 391 symmetric inner product, 391 Lie bracket of vector ﬁelds, 49, 99 Lie group, 52 automorphism, 58 bi-invariant metric, 242 Cartan decomposition, 403 center, 66 complex, 406 group action, 68 homomorphism, 58 complexi�
drag force is added to a harmonic oscillator, then the equation of motion becomes m¨x = −mω2x − k ˙x, where ω is the angular frequency of the oscillator in the absence of damping. Rewrite as where γ = k/2m > 0. Solutions are x = eλt, where ¨x + 2γ ˙x + ω2x = 0, or λ2 + 2γλ + ω2 = 0, λ = −γ ± γ2 − ω2. If γ > ω, then the roots are real and negative. So we have exponential decay. We call this an overdamped oscillator. If 0 < γ < ω, then the roots are complex with Re(λ) = −γ. So we have decaying oscillations. We call this an underdamped oscillator. For details, refer to Diﬀerential Equations. 26 4 Orbits IA Dynamics and Relativity 4 Orbits The goal of this chapter is to study the orbit of a particle in a central force, m¨r = −∇V (r). While the universe is in three dimensions, the orbit is conﬁned to a plane. This is since the angular momentum L = mr × ˙r is a constant vector, as we’ve previously shown. Furthermore L · r = 0. Therefore, the motion takes place in a plane passing through the origin, and perpendicular to L. 4.1 Polar coordinates in the plane We choose our axes such that the orbital plane is z = 0. To describe the orbit, we introduce polar coordinates (r, θ): x = r cos θ, y = r sin θ. Our object is to separate the motion of the particle into radial and angular components. We do so by deﬁning unit vectors in the directions of increasing r and increasing θ: cos θ sin θ, ˆθ = − sin θ cos θ ˆr = y ˆθ ˆr r θ. x These two unit vectors form an orthonormal basis. However, they are not basis vectors in the normal sense. The directions of these basis vectors depend on time. In particular, we have Proposition. dˆr dθ dˆθ dθ = = − sin θ cos θ
− cos θ − sin θ = ˆθ = −ˆr. Often, we want the derivative with respect to time, instead of θ. By the chain rule, we have dˆr dt = ˙θˆθ, dˆθ dt = − ˙θˆr. We can now express the position, velocity and acceleration in this new polar basis. The position is given by r = rˆr. 27 4 Orbits IA Dynamics and Relativity Taking the derivative gives the velocity as ˙r = ˙rˆr + r ˙θˆθ. The acceleration is then ¨r = ¨rˆr + ˙r ˙θˆθ + ˙r ˙θˆθ + r ¨θˆθ − r ˙θ2ˆr = (¨r − r ˙θ2)ˆr + (r ¨θ + 2 ˙r ˙θ)ˆθ. Deﬁnition (Radial and angular velocity). angular velocity. ˙r is the radial velocity, and ˙θ is the Example (Uniform motion in a circle). If we are moving in a circle, then ˙r = 0 and ˙θ = ω = constant. So ˙r = rωˆθ. The speed is given by and the acceleration is v = \|˙r\| = r\|ω\| = const ¨r = −rω2ˆr. Hence in order to make a particle of mass m move uniformly in a circle, we must supply a centripetal force mv2/r towards the center. 4.2 Motion in a central force ﬁeld Now let’s put in our central force. Since V = V (r), we have F = −∇V = dV dr ˆr. So Newton’s 2nd law in polar coordinates is m(¨r − r ˙θ2)ˆr + m(r ¨θ + 2 ˙r ˙θ)ˆθ = − dV dr ˆr. The
θ component of this equation is We can rewrite it as m(r ¨θ − 2 ˙r ˙θ) = 0. 1 r d dt (mr2 ˙θ) = 0. Let L = mr2 ˙θ. This is the z component (and the only component) of the conserved angular momentum L: L = mr × ˙r = mrˆr × ( ˙rˆr + r ˙θˆθ) = mr2 ˙θ ˆr × ˆθ = mr2 ˙θ ˆz. So the angular component tells us that L is constant, which is the conservation of angular momentum. However, a more convenient quantity is the angular momentum per unit mass: 28 4 Orbits IA Dynamics and Relativity Notation (Angular momentum per unit mass). The angular momentum per unit mass is h = = r2 ˙θ = const. L m Now the radial (r) component of the equation of motion is m(¨r − r ˙θ2) = − dV dr. We eliminate ˙θ using r2 ˙θ = h to obtain where m¨r = − dV dr + mh2 r3 = − dVeﬀ dr, Veﬀ(r) = V (r) + mh2 2r2. We have now reduced the problem to 1D motion in an (eﬀective) potential — as studied previously. The total energy of the particle is E = = 1 2 1 2 m\|˙r\|2 + V (r) m( ˙r2 + r2 ˙θ2) + V (r) (since ˙r = ˙rˆr + r ˙θˆθ, and ˆr and ˆθ are orthogonal) m ˙r2 + mh2 2r2 + V (r) m ˙r2 + Veﬀ(r). = = 1 2 1 2 Example. Consider an attractive force following the inverse-square law (e.g. gravity). Here V = − mk r, for some constant k. So mk r We have two terms
of opposite signs and diﬀerent dependencies on r. For small r, the second term dominates and Veﬀ is large. For large r, the ﬁrst term dominates. Then Veﬀ asymptotically approaches 0 from below. Veﬀ = − mh2 2r2. + Veﬀ Emin r∗ r 29 4 Orbits IA Dynamics and Relativity The minimum of Veﬀ is at r∗ = h2 k, Emin = − mk2 2h2. We have a few possible types of motion: – If E = Emin, then r remains at r∗ and ˙θh/r2 is constant. So we have a uniform motion in a circle. – If Emin < E < 0, then r oscillates and ˙r = h/r2 does also. This is a non-circular, bounded orbit. We’ll now introduce a lot of (pointless) deﬁnitions: Deﬁnition (Periapsis, apoapsis and apsides). The points of minimum and maximum r in such an orbit are called the periapsis and apoapsis. They are collectively known as the apsides. Deﬁnition (Perihelion and aphelion). For an orbit around the Sun, the periapsis and apoapsis are known as the perihelion and aphelion. In particular Deﬁnition (Perigee and apogee). The perihelion and aphelion of the Earth are known as the perigee and apogee. – If E ≥ 0, then r comes in from ∞, reaches a minimum, and returns to inﬁnity. This is an unbounded orbit. We will later show that in the case of motion in an inverse square force, the trajectories are conic sections (circles, ellipses, parabolae and hyperbolae). Stability of circular orbits We’ll now look at circular orbits, since circles are nice. Consider a general potential energy V (r). We have to answer two questions: – Do circular orbits exist? – If they do, are they stable? The conditions for existence and stability are rather straightforward. For a circular orbit, r = r∗ = const for some value of h =
0 (if h = 0, then the object is just standing still!). Since ¨r = 0 for constant r, we require V eﬀ(r∗) = 0. The orbit is stable if r∗ is a minimum of Veﬀ, i.e. V eﬀ(r∗) > 0. In terms of V (r), circular orbit requires V (r∗) = mh2 r3 ∗ 30 4 Orbits IA Dynamics and Relativity and stability further requires V (r∗) + 3mh2 r4 ∗ = V (r∗) + 3 r∗ V (r∗) > 0. In terms of the radial force F (r) = −V (r), the orbit is stable if F (r∗) + 3 r F (r∗) < 0. Example. Consider a central force with V (r) = − mk rp for some k, p > 0. Then V (r) + 3 r V (r) = − p(p + 1) + 3p mk rp+2 = p(2 − p) mk rp+2. So circular orbits are stable for p < 2. This is illustrated by the graphs of Veﬀ(r) for p = 1 and p = 3. Veﬀ p = 3 p = 1 4.3 Equation of the shape of the orbit In general, we could determine r(t) by integrating the energy equation E = 1 2 t = ± m ˙r2 + Veﬀ(r) m 2 dr E − Veﬀ(r) However, this is usually not practical, because we can’t do the integral. Instead, it is usually much easier to ﬁnd the shape r(θ) of the orbit. Still, solving for r(θ) is also not easy. We will need a magic trick — we introduce the new variable Notation. u = 1 r. 31 4 Orbits Then and IA Dynamics and Relativity ˙r = dr dθ ˙θ = dr dθ h r2 = −h du dθ, ¨r = d dt −h du dθ = −h d2u dθ2 ˙θ = −h d2u dθ2 h r2 =
−h2u2 d2u dθ2. This doesn’t look very linear with u2, but it will help linearizing the equation when we put in other factors. The radial equation of motion m¨r − mh2 r3 = F (r) then becomes Proposition (Binet’s equation). −mh2u2 d2u dθ2 + u = F. 1 u This still looks rather complicated, but observe that for an inverse square force, F (1/u) is proportional to u2, and then the equation is linear! In general, given an arbitrary F (r), we aim to solve this second order ODE for u(θ). If needed, we can then work out the time-dependence via ˙θ = hu2. 4.4 The Kepler problem The Kepler problem is the study of the orbits of two objects interacting via a central force that obeys the inverse square law. The goal is to classify the possible orbits and study their properties. One of the most important examples of the Kepler problem is the orbit of celestial objects, as studied by Kepler himself. Shapes of orbits For a planet orbiting the sun, the potential and force are given by V (r) = mk r, F (r) = − mk r2 with k = GM (for the Coulomb attraction of opposite charges, we have the same equation with k = − Qq 4πε0m Binet’s equation then becomes linear, and ). d2u dθ2 + u = k h2. We write the general solution as u = k h2 + A cos(θ − θ0), 32 4 Orbits IA Dynamics and Relativity where A ≥ 0 and θ0 are arbitrary constants. If A = 0, then u is constant, and the orbit is circular. Otherwise, u reaches a maximum at θ = θ0. This is the periapsis. We now re-deﬁne polar coordinates such that the periapsis is at θ = 0. Then Proposition. The orbit of a planet around the sun is given by r = 1 + e cos θ, (∗) with = h2/k and e = Ah2/k. This is the polar equation of a conic, with a focus (the sun) at the origin. Deﬁnition (Eccentricity).
The dimensionless parameter e ≥ 0 in the equation of orbit is the eccentricity and determines the shape of the orbit. We can rewrite (∗) in Cartesian coordinates with x = r cos θ and y = r sin θ. Then we obtain (1 − e2)x2 + 2ex + y2 = 2. (†) There are three diﬀerent possibilities: – Ellipse: (0 ≤ e < 1). r is bounded by 1 + e ≤ r ≤ 1 − e. (†) can be put into the equation of an ellipse centered on (−ea, 0), (x + ea)2 a2 + y2 b2 = 1, where a = 1 − e2 and b = √ 1 − e2 ≤ a. b O a ae a and b are the semi-major and semi-minor axis. is the semi-latus rectum. One focus of the ellipse is at the origin. If e = 0, then a = b = and the ellipse is a circle. – Hyperbola: (e > 1). For e > 1, r → ∞ as θ → ±α, where α = cos−1(1/e) ∈ (π/2, π). Then (†) can be put into the equation of a hyperbola centered on (ea, 0), (x − ea)2 a2 − y2 b2 = 1, with a = e2 − 1, b = √ e2 − 1. 33 4 Orbits IA Dynamics and Relativity O b a This corresponds to an unbound orbit that is deﬂected (scattered) by an attractive force. b is both the semi-minor axis and the impact parameter. It is the distance by which the planet would miss the object if there were no attractive force. The asymptote is y = b a (x − ea), or x e2 − 1 − y = eb. Alternatively, we can write the equation of the asymptote as (x, y) · √ e2 − 1 e, − 1 e = b or r · n = b, the equation of a line at a distance b from the origin. – Parabola: (e = 1). Then (∗) becomes r = 1 + cos θ. We see that r → ∞ as θ
→ ±π. (†) becomes the equation of a parabola, y2 = ( − 2x). The trajectory is similar to that of a hyperbola. Energy and eccentricity We can ﬁgure out which path a planet follows by considering its energy. E = = 1 2 1 2 m( ˙r2 + r2 ˙θ2) − du dθ mh2 2 mk r + u2 − mku Substitute u = 1 (1 + e cos θ) and =, and it simpliﬁes to h2 k mk 2 E = (e2 − 1), which is independent of θ, as it must be. Orbits are bounded for e < 1. This corresponds to the case E < 0. Unbounded orbits have e > 1 and thus E > 0. A parabolic orbit has e = 1, E = 0, and is “marginally bound”. 2GM r = vesc, which Note that the condition E > 0 is equivalent to \|˙r\| > means you have enough kinetic energy to escape orbit. 34 4 Orbits IA Dynamics and Relativity Kepler’s laws of planetary motion When Kepler ﬁrst studied the laws of planetary motion, he took a telescope, observed actual planets, and came up with his famous three laws of motion. We are now going to derive the laws with pen and paper instead. Law (Kepler’s ﬁrst law). The orbit of each planet is an ellipse with the Sun at one focus. Law (Kepler’s second law). The line between the planet and the sun sweeps out equal areas in equal times. Law (Kepler’s third law). The square of the orbital period is proportional to the cube of the semi-major axis, or P 2 ∝ a3. We have already shown that Law 1 follows from Newtonian dynamics and the inverse-square law of gravity. In the solar system, planets generally have very low eccentricity (i.e. very close to circular motion), but asteroids and comets can have very eccentric orbits. In other solar systems, even planets have have highly eccentric orbits. As we’ve previously shown, it is also possible for the object to have a parabolic or hyperbolic orbit. However, we tend not to call these “planets”. Law 2 follows simply from the
conservation of angular momentum: The area swept out by moving dθ is dA = 1 2 r2 dθ (area of sector of circle). So dA dt = 1 2 r2 ˙θ = h 2 = const. and is true for any central force. Law 3 follows from this: the total area of the ellipse is A = πab = h 2 P (by the second law). But b2 = a2(1 − e2) and h2 = k = ka(1 − e2). So P 2 = (2π)2a4(1 − e2) ka(1 − e2) = (2π)2a3 k. Note that the third law is very easy to prove directly for circular orbits. Since the radius is constant, ¨r = 0. So the equations of motion give So −r ˙θ2 = − k r2 r3 ˙θ2 = k Since ˙θ ∝ P −1, the result follows. 4.5 Rutherford scattering Finally, we will consider the case where the force is repulsive instead of attractive. An important example is the Rutherford gold foil experiment, where Rutherford bombarded atoms with alpha particles, and the alpha particles are repelled by the nucleus of the atom. 35 4 Orbits IA Dynamics and Relativity Under a repulsive force, the potential and force are given by V (r) = + mk r, F (r) = + mk r2. For Coulomb repulsion of like charges, The solution is now k = Qq 4πε0m > 0. u = − k h2 + A cos(θ − θ0). wlog, we can take A ≥ 0, θ0 = 0. Then with r = e cos θ − 1 = h2 k, e = Ah2 k. We know that r and are positive. So we must have e ≥ 1. Then r → ∞ as θ → ±α, where α = cos−1(1/e). The orbit is a hyperbola, again given by (x − ea)2 a2 − y2 b2 = 1,. However, this time, the trajectory is the other with a = branch of the hyperbola. e2−1 and b = √ e2−1 b O β 2α It seems as if the particle is
deﬂected by O. We can characterize the path of the particle by the impact parameter b and the incident speed v (i.e. the speed when far away from the origin). We know that the angular momentum per unit mass is h = bv (velocity × perpendicular distance to O). How does the scattering angle β = π − 2α depend on the impact parameter b and the incident speed v? Recall that the angle α is given by α = cos−1(1/e). So we obtain 1 e = cos α = cos π 2 − β 2 = sin, β 2 36 4 Orbits IA Dynamics and Relativity So So b = √ e2 − 1 = (bv)2 k tan β 2. β = 2 tan−1 k bv2. We see that if we have a small impact parameter, i.e. b k/v2, then we can have a scattering angle approaching π. 37 5 Rotating frames IA Dynamics and Relativity 5 Rotating frames Recall that Newton’s laws hold only in inertial frames. However, sometimes, our frames are not inertial. In this chapter, we are going to study a particular kind of non-inertial frame — a rotating frame. An important rotating frame is the Earth itself, but there are also other examples such as merry-go-rounds. 5.1 Motion in rotating frames Now suppose that S is an inertial frame, and S is rotating about the z axis with angular velocity ω = ˙θ with respect to S. Deﬁnition (Angular velocity vector). The angular velocity vector of a rotating frame is ω = ωˆz, where ˆz is the axis of rotation and ω is the angular speed. First we wish to relate the basis vectors {ei} and {e Consider a particle at rest in S. From the perspective of S, its velocity is i} of S and S respectively. dr dt S = ω × r, where ω = ωˆz is the angular velocity vector (aligned with the rotation axis). This formula also applies to the basis vectors of S. de i dt S = ω × e i. Now given a general time-dependent vector a, we can express it in the {e as follows: i} basis a = i(t)e a i. From the perspective of S, e by i
is constant and the time derivative of a is given da dt S = da i dt e i. In S, however, e time derivative of a: i is not constant. So we apply the product rule to obtain the da dt S = dai dt e i + iω × e a i = da dt S + ω × a. This key identity applies to all vectors and can be written as an operator equation: Proposition. If S is an inertial frame, and S is rotating relative to S with angular velocity ω, then d dt S = d dt S + ω ×. Applied to the position vector r(t) of a particle, it gives dr dt = S dr dt S + ω × r. 38 5 Rotating frames IA Dynamics and Relativity We can interpret this as saying that the diﬀerence in velocity measured in the two frames is the relative velocity of the frames. We apply this formula a second time, and allow ω to depend on time. Then we have d2r dt2 S = = d dt d2r dt2 S S + ω× + 2ω × dr dt dr dt S + ω × r. S + ˙ω × r + ω × (ω × r) Since S is inertial, Newton’s Second Law is m d2r dt2 S = F. So Proposition. m d2r dt2 S = F − 2mω × dr dt S − m ˙ω × r − mω × (ω × r). Deﬁnition (Fictious forces). The additional terms on the RHS of the equation of motion in rotating frames are ﬁctitious forces, and are needed to explain the motion observed in S. They are – Coriolis force: −2mω × dr S. dt – Euler force: −m ˙ω × r – Centrifugal force: −mω × (ω × r). In most cases, ω is constant and can neglect the Euler force. 5.2 The centrifugal force What exactly does the centrifugal force do? Let ω = ω ˆω, where \| ˆω\| = 1. Then −mω × (ω × r) = −m(ω · r)ω − (ω · ω)
r = mω2r⊥, where r⊥ = r−(r· ˆω) ˆω is the projection of the position on the plane perpendicular to ω. So the centrifugal force is directed away from the axis of rotation, and its magnitude is mω2 times the distance form the axis. ω r⊥ r 39 5 Rotating frames IA Dynamics and Relativity Note that r⊥ · r⊥ = r · r − (r · ˆω)2 ∇(\|r⊥\|2) = 2r − 2(r · ˆω) ˆω = 2r⊥. So −mω × (ω × r) = −∇ − mω2\|r⊥\|2 = −∇ − m\|ω × r\|2. 1 2 1 2 Thus the centrifugal force is a conservative (ﬁctitious) force. On a rotating planet, the gravitational and centrifugal forces per unit mass combine to make the eﬀective gravity, geﬀ = g + ω2r⊥. This gravity will not be vertically downwards. Consider a point P at latitude λ on the surface of a spherical planet of radius R. We construct orthogonal axes: ω ˆy ˆz P λ O with ˆx into the page. So ˆz is up, ˆy is North, and ˆx is East. At P, we have r = Rˆz g = −gˆz ω = ω(cos λˆy + sin λˆz) So geﬀ = g + ω2r⊥ = −gˆz + ω2R cos λ(cos λˆz − sin λˆy) = −ω2R cos λ sin λˆy − (g − ω2R cos2 λ)ˆz. So the angle α between g and geﬀ is given by tan α = ω2R cos λ sin λ g − ω2R cos2 λ. This is 0 at the equator and the poles, and greatest when you are halfway between. However, this is still tiny on Earth, and does not a
ﬀect our daily life. 40 5 Rotating frames IA Dynamics and Relativity 5.3 The Coriolis force The Coriolis force is a more subtle force. Writing v = dr force as dt F = −2mω × v. S, we can write the Note that this has the same form as the Lorentz force caused by a magnetic ﬁeld, and is velocity-dependent. However, unlike the eﬀects of a magnetic ﬁeld, particles do not go around in circles in a rotating frame, since we also have the centrifugal force in play. Since this force is always perpendicular to the velocity, it does no work. Consider motion parallel to the Earth’s surface. We only care about the eﬀect of the Coriolis force on the horizontal trajectory, and ignore the vertical component that is tiny compared to gravity. So we only take the vertical component of ω, ω sin λˆz. The horizontal velocity v = vx ˆx + vy ˆy generates a horizontal Coriolis force: −2mω sin λˆz × v = −2mω sin λ(vy ˆx − vx ˆy). In the Northern hemisphere (0 < λ < π/2), this causes a deﬂection towards the right. In the Southern Hemisphere, the deﬂection is to the left. The eﬀect vanishes at the equator. Note that only the horizontal eﬀect of horizontal motion vanishes at the equator. The vertical eﬀects or those caused by vertical motion still exist. Example. Suppose a ball is dropped from a tower of height h at the equator. Where does it land? In the rotating frame, ¨r = g − 2ω × ˙r − ω × (ω × r). We work to ﬁrst order in ω. Then ¨r = g − 2ω × ˙r + O(ω2). Integrate wrt t to obtain ˙r = gt − 2ω × (r − r0) + O(ω2), where r0 is the initial position. We substitute into the original equation to obtain ¨r = g − 2ω × gt + O(ω2). (
where some new ω2 terms are thrown into O(ω2)). We integrate twice to obtain r = r0 + 1 2 gt2 − 1 3 ω × gt3 + O(ω2). In components, we have g = (0, 0, −g), ω = (0, ω, 0) and r0 = (0, 0, R + h). So r = 1 3 ωgt3, 0, R + h − 1 2 gt2 + O(ω2). So the particle hits the ground at t = 2h/g, and its eastward displacement is 1 3 wg 2h g 3/2. 41 5 Rotating frames IA Dynamics and Relativity This can be understood in terms of angular momentum conservation in the non-rotating frame. At the beginning, the particle has the same angular velocity with the Earth. As it falls towards the Earth, to maintain the same angular momentum, the angular velocity has to increase to compensate for the decreased radius. So it spins faster than the Earth and drifts towards the East, relative to the Earth. Example. Consider a pendulum that is free to swing in any plane, e.g. a weight on a string. At the North pole, it will swing in a plane that is ﬁxed in an inertial frame, while the Earth rotates beneath it. From the perspective of the rotating frame, the plane of the pendulum rotates backwards. This can be explained as a result of the Coriolis force. In general, at latitude λ, the plane rotates rightwards with period 1 day sin λ. 42 6 Systems of particles IA Dynamics and Relativity 6 Systems of particles Now suppose we have N interacting particles. We adopt the following notation: particle i has mass mi, position ri, and momentum pi = mi ˙ri. Note that the subscript denotes which particle it is referring to, not vector components. Newton’s Second Law for particle i is where Fi is the total force acting on particle i. We can write Fi as mi¨ri = ˙pi = Fi, Fi = Fext i + N j=1 Fij, where Fij is the force on particle i by particle j, and Fext on i, which comes from particles outside the system. i is the external force Since a particle cannot exert a force on itself, we have Fii = 0. Also, Newton’s
third law requires that Fij = −Fji. For example, if the particles interact only via gravity, then we have Fij = − Gmimj(ri − rj) \|ri − rj\|3 = −Fji. 6.1 Motion of the center of mass Sometimes, we are interested in the collection of particles as a whole. For example, if we treat a cat as a collection of particles, we are more interested in how the cat as a whole falls, instead of tracking the individual particles of the cat. Hence, we deﬁne some aggregate quantities of the system such as the total mass and investigate how these quantities relate. Deﬁnition (Total mass). The total mass of the system is M = mi. Deﬁnition (Center of mass). The center of mass is located at R = 1 M N i=1 miri. This is the mass-weighted average position. Deﬁnition (Total linear momentum). The total linear momentum is P = N i=1 pi = N i=1 mi ˙ri = M ˙R. Note that this is equivalent to the momentum of a single particle of mass M at the center of mass. 43 6 Systems of particles IA Dynamics and Relativity Deﬁnition (Total external force). The total external force is F = N i=1 Fext i. We can now obtain the equation of motion of the center of mass: Proposition. Proof. M ¨R = F. M ¨R = ˙P N = ˙pi i=1 N i=1 = = F + 1 2 = F Fext i + N N Fij j=1 i=1 (Fij + Fji) i j This means that if we don’t care about the internal structure, we can treat the system as a point particle of mass M at the center of mass R. This is why Newton’s Laws apply to macroscopic objects even though they are not individual particles. Law (Conservation of momentum). If there is no external force, i.e. F = 0, then ˙P = 0. So the total momentum is conserved. If there is no external force, then the center of mass moves uniformly in a straight line. In this case, we can pick a particularly nice frame of reference, known as the center of mass frame. De�
�nition (Center of mass frame). The center of mass frame is an inertial frame in which R = 0 for all time. Doing calculations in the center of mass frame is usually much more convenient than using other frames, After doing linear motion, we can now look at angular motion. Deﬁnition (Total angular momentum). The total angular momentum of the system about the origin is L = ri × pi. How does the total angular momentum change with time? Here we have to assume a stronger version of Newton’s Third law, saying that i Fij = −Fji and is parallel to (ri − rj). This is true, at least, for gravitational and electrostatic forces. 44 6 Systems of particles IA Dynamics and Relativity Then we have ˙L = ri × ˙pi + ˙ri × pi i i i i = = =  Fext i + Fij   + m(˙ri × ˙ri) ri × ri × Fext i + j ri × Fij Gext i + i i j 1 2 j (ri × Fij + rj × Fji) (ri − rj) × Fij i j = G + 1 2 = G, where Deﬁnition (Total external torque). The total external torque is G = i ri × Fext i. So the total angular momentum is conserved if G = 0, ie the total external torque vanishes. 6.2 Motion relative to the center of mass So far, we have shown that externally, a multi-particle system behaves as if it were a point particle at the center of mass. But internally, what happens to the individual particles themselves? We write ri = R + rc i, where rc i is the position of particle i relative to the center of mass. We ﬁrst obtain two useful equalities: mirc i = miri − miR = M R − M R = 0. i Diﬀerentiating gives mi ˙rc i = 0. i Using these equalities, we can express the angular momentum and kinetic energy in terms of R and rc i only: L = = i i mi(R + rc i ) × ( ˙R + ˙rc i ) miR × ˙R + R × mi ˙rc i + =
M R × ˙R + i i × ˙rc i mirc i 45 mirc i × ˙R + i mirc i × ˙rc i i 6 Systems of particles IA Dynamics and Relativity mi\|˙ri\|2 mI ( ˙R + ˙ri c) · ( ˙R + ˙rc i ) mi ˙R · ˙R + ˙R · mi ˙rc i + i 1 2 i mi ˙rc i · ˙rc i M \| ˙R\|2 + 1 2 i mi\|˙rc i \|2 We see that each item is composed of two parts — that of the center of mass and that of motion relative to center of mass. If the forces are conservative in the sense that and Fext i = −∇iVi(ri), Fij = −∇iVij(ri − rj), where ∇i is the gradient with respect to ri, then energy is conserved in the from E = T + i Vi(ri) + 1 2 i j Vij(ri − rj) = const. 6.3 The two-body problem The two-body problem is to determine the motion of two bodies interacting only via gravitational forces. The center of mass is at R = 1 M (m1r1 + m2r2), where M = m1 + m2. The magic trick to solving the two-body problem is to deﬁne the separation vector (or relative position vector) Then we write everything in terms of R and r. r = r1 − r2. r1 = R + m2 M r, r2 = R − m1 M r. r1 R r r2 Since the external force F = 0, we have ¨R = 0, i.e. the center of mass moves uniformly. 46 6 Systems of particles IA Dynamics and Relativity Meanwhile, F12 − = ¨r = ¨r1 − ¨r2 1 m1 1 m1 = + 1 m2 1 m2 F21 F12 We can write this as where µ¨r = F12(r), µ = m1m2 m1 + m2 is the reduced mass. This is the same as the equation of motion for one particle of mass µ with position vector r relative to a ﬁxed
origin — as studied previously. For example, with gravity, So µ¨r = − Gm1m2ˆr \|r\|2. ¨r = − GMˆr \|r\|2. For example, give a planet orbiting the Sun, both the planet and the sun moves in ellipses about their center of mass. The orbital period depends on the total mass. It can be shown that L = M R × ˙R + µr × ˙r T = 1 2 M \| ˙R\|2 + 1 2 µ\|˙r\|2 by expressing r1 and r2 in terms of r and R. 6.4 Variable-mass problem All systems we’ve studied so far have ﬁxed mass. However, in real life, many objects have changing mass, such as rockets, ﬁreworks, falling raindrops and rolling snowballs. Again, we will use Newton’s second law, which states that dp dt = F, with p = m˙r. We will consider a rocket moving in one dimension with mass m(t) and velocity v(t). The rocket propels itself forwards by burning fuel and ejecting the exhaust at velocity −u relative to the rocket. At time t, the rocket looks like this: v(t) m(t) 47 6 Systems of particles IA Dynamics and Relativity At time t + δt, it ejects exhaust of mass m(t) − m(t + δt) with velocity v(t) − u + O(δt). v(t) − u m v(t) m(t) The change in total momentum of the system (rocket + exhaust) is δp = m(t + δt)v(t + δt) + [m(t) − m(t + δt)][v(t) − u(t) + O(δt)] − m(t)v(t) = (m + ˙mδt + O(δt2))(v + ˙vδt + O(δt2)) − ˙mδt(v − u) + O(δt2) − mv = ( ˙mv + m ˙v − ˙mv + ˙
mu)δt + O(δt2) = (m ˙v + ˙mu)δt + O(δt2). Newton’s second law gives lim δ→0 δp δt = F where F is the external force on the rocket. So we obtain Proposition (Rocket equation). m dv dt + u dm dt = F. Example. Suppose that we travel in space with F = 0. Assume also that u is constant. Then we have m dv dt = −u dm dt. So v = v0 + u log m0 m(t), Note that we are expressing things in terms of the mass remaining m, not time t. Note also that the velocity does not depend on the rate at which mass is ejected, only the velocity at which it is ejected. Of course, if we expressed v as a function of time, then the velocity at a speciﬁc time does depend on the rate at which mass is ejected. Example. Consider a falling raindrop of mass m(t), gathering mass from a stationary cloud. In this case, u = v. So m dv dt + v dm dt = d dt (mv) = mg, with v measured downwards. To obtain a solution of this, we will need a model to determine the rate at which the raindrop gathers mass. 48 7 Rigid bodies IA Dynamics and Relativity 7 Rigid bodies This chapter is somewhat similar to the previous chapter. We again have a lot of particles and we study their motion. However, instead of having forces between the individual particles, this time the particles are constrained such that their relative positions are ﬁxed. This corresponds to a solid object that cannot deform. We call these rigid bodies. Deﬁnition (Rigid body). A rigid body is an extended object, consisting of N particles that are constrained such that the distance between any pair of particles, \|ri − rj\|, is ﬁxed. The possible motions of a rigid body are the continuous isometries of Euclidean space, i.e. translations and rotations. However, as we have previously shown, pure translations of rigid bodies are uninteresting — they simply correspond to the center of mass moving under an external force. Hence we will ﬁrst study rotations. Later, we will
combine rotational and translational eﬀects and see what happens. 7.1 Angular velocity We’ll ﬁrst consider the cases where there is just one particle, moving in a circle of radius s about the z axis. Its position and velocity vectors are r = (s cos θ, s sin θ, z) ˙r = (−s ˙θ sin θ, s ˙θ cos θ, 0). We can write where is the angular velocity vector. In general, we write ˙r = ω × r, ω = ˙θˆz ω = ˙θˆn = ω ˆn, where ˆn is a unit vector parallel to the rotation axis. The kinetic energy of this particle is thus T = m\|˙r\|2 1 2 1 2 1 2 where I = ms2 is the moment of inertia. This is the counterpart of “mass” in rotational motion. ms2 ˙θ2 Iω2 = = Deﬁnition (Moment of inertia). The moment of inertia of a particle is where s is the distance of the particle from the axis of rotation. I = ms2 = m\|ˆn × r\|2, 49 7 Rigid bodies IA Dynamics and Relativity 7.2 Moment of inertia In general, consider a rigid body in which all N particles rotate about the same axis with the same angular velocity: ˙ri = ω × ri. This ensures that d dt \|ri − rj\|2 = 2(˙ri − ˙rj) · (ri − rj) = 2ω × (ri − rj) · (ri − rj) = 0, as required for a rigid body. Similar to what we had above, the rotational kinetic energy is T = 1 2 N i=1 mi\| ˙ri\|2 = 1 2 Iω2, where Deﬁnition (Moment of inertia). The moment of inertia of a rigid body about the rotation axis ˆn is I = N i=1 mis2 i = N i=1 mi\|ˆn × ri\|2. Again, we deﬁne the angular momentum of the system: Deﬁnition. The angular momentum is L
= miri × ˙ri = i i miri × (ω × ri). Note that our deﬁnitions for angular motion are analogous to those for linear motion. The moment of inertia I is deﬁned such that T = 1 2 Iω2. Ideally, we would want the momentum to be L = Iω. However, this not true. In fact, L need not be parallel to ω. What is true, is that the component of L parallel to ω is equal to Iω. Write ω = ω ˆn. Then we have L · ˆn = ω = ω i i = Iω. mi ˆn · (ri × (ˆn × ri)) m(ˆn × ri) · (ˆn × ri) What does L itself look like? Using vector identities, we have L = (ri · ri)ω − (ri · ω)ri mi i Note that this is a linear function of ω. So we can write L = Iω, 50 7 Rigid bodies IA Dynamics and Relativity where we abuse notation to use I for the inertia tensor. This is represented by a symmetric matrix with components Ijk = i mi(\|ri\|2δjk − (ri)j(ri)k), where i refers to the index of the particle, and j, k are dummy suﬃxes. If the body rotates about a principal axis, i.e. one of the three orthogonal eigenvectors of I, then L will be parallel to ω. Usually, the principal axes lie on the axes of rotational symmetry of the body. 7.3 Calculating the moment of inertia For a solid body, we usually want to think of it as a continuous substance with a mass density, instead of individual point particles. So we replace the sum of particles by a volume integral weighted by the mass density ρ(r). Deﬁnition (Mass, center of mass and moment of inertia). The mass is M = ρ dV. R = 1 M ρr dV The center of mass is The moment of inertia is I = ρs2 dV = ρ\|ˆn × r\|2 dV. In theory, we can study inhomogeneous bodies with varying ρ, but usually we mainly consider
homogeneous ones with constant ρ throughout. Example (Thin circular ring). Suppose the ring has mass M and radius a, and a rotation axis through the center, perpendicular to the plane of the ring. a Then the moment of inertia is I = M a2. Example (Thin rod). Suppose a rod has mass M and length. It rotates through one end, perpendicular to the rod. 51 7 Rigid bodies IA Dynamics and Relativity The mass per unit length is M/. So the moment of inertia is I = 0 M x2 dx = 1 3 M 2. Example (Thin disc). Consider a disc of mass M and radius a, with a rotation axis through the center, perpendicular to the plane of the disc. a Then I = 2π a 0 0 r2 s2 r dr dθ area element M πa2 mass per unit length 2π r3 dr dθ 0 = = = a 0 M πa2 M πa2 1 2 1 4 M a2. a4(2π) Now suppose that the rotation axis is in the plane of the disc instead (also rotating through the center). Then M πa2 mass per unit length 2π r3 dr sin2 θ dθ (r sin θ)2 s2 r dr dθ area element 0 I = 2π a 0 0 = = = a 0 M πa2 M πa2 1 4 1 4 M a2. a4π Example. Consider a solid sphere with mass M, radius a, with a rotation axis though the center. a 52 7 Rigid bodies IA Dynamics and Relativity Using spherical polar coordinates (r, θ, φ) based on the rotation axis, (r sin θ)2 s2 r2 sin θ dr dθ dφ volume element (1 − cos2) sin θ dθ 2π 0 dφ I = 2π π a 0 0 0 M 4 3 πa3 ρ π a r4 dr 0 1 5 a5 · 4 3 0 · 2π = = = M 4 3 πa3 M 4 3 πa3 2 M a2. 5 · Usually, ﬁnding the moment of inertia involves doing complicated integrals. We will now come up with two theorems that help us ﬁnd moments of inertia. Theorem (Perpendicular axis theorem).
For a two-dimensional object (a lamina), and three perpendicular axes x, y, z through the same spot, with z normal to the plane, where Iz is the moment of inertia about the z axis. Iz = Ix + Iy, z y x Note that this does not apply to 3D objects! For example, in a sphere, Ix = Iy = Iz. Proof. Let ρ be the mass per unit volume. Then Ix = Iy = Iz = ρy2 dA ρx2 dA ρ(x2 + y2) dA = Ix + Iy. Example. For a disc, Ix = Iy by symmetry. So Iz = 2Ix. Theorem (Parallel axis theorem). If a rigid body of mass M has moment of inertia I C about an axis passing through the center of mass, then its moment of inertia about a parallel axis a distance d away is I = I C + M d2. 53 7 Rigid bodies IA Dynamics and Relativity d CM Proof. With a convenient choice of Cartesian coordinates such that the center of mass is at the origin and the two rotation axes are x = y = 0 and x = d, y = 0, I C = ρ(x2 + y2) dV, ρr dV = 0. and So I = = ρ((x − d)2 + y2) dV ρ(x2 + y2) dV − 2d ρx dV + d2ρ dV = I c + 0 + M d2 = I c + M d2. Example. Take a disc of mass M and radius a, and rotation axis through a point on the circumference, perpendicular to the plane of the disc. Then a I = I c + M a2 = 1 2 M a2 + M a2 = 3 2 M a2. 7.4 Motion of a rigid body The general motion of a rigid body can be described as a translation of its centre of mass, following a trajectory R(t), together with a rotation about an axis through the center of mass. As before, we write ri = R + rc i. 54 7 Rigid bodies IA Dynamics and Relativity Then ˙ri = ˙R + ˙rc i. Using this, we can break down the velocity and kinetic energy into translational and rotational parts. If the body rot

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