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p ∈ F (M ), one for every p ∈ M, such p (R \ {0}) is that fp(p) = 0 = ε(fp) for all p ∈ M. Then the set Up := f −1 an M -open neighborhood of p for every p ∈ M. Choose a sequence of compact sets Kn ⊂ M such that Kn ⊂ intM (Kn+1) for all n and M = n Kn. Then, for each n, there is a gn ∈ F (M ) (a ﬁnite sum of the form i f 2 pi) such that ε(gn) = 0 and gn(q) > 0 for all q ∈ Kn. If M is compact, this is already a contradiction because a positive function cannot belong to the kernel of ε. Otherwise, choose f ∈ F (M ) such that f (q) ≥ n for all q ∈ M \ Kn and all n ∈ N. Then ε(f ) ∈ f (g−1 n (0)) ⊂ f (M \ Kn) ⊂ [n, ∞) by the claim and so ε(f ) ≥ n for all n. This is a contradiction and proves Theorem 2.5.43. Now let N be another smooth submanifold (say of R) and let C∞(M, N ) denote the space of smooth maps from M to N. A homomorphism from F (N ) to F (M ) is a (real) linear map Φ : F (N ) → F (M ) that satisﬁes Φ(f g) = Φ(f )Φ(g), Φ(1) = 1. Let Hom(F (N ), F (M )) denote the space of homomorphisms from F (N ) to F (M ). An automorphism of the algebra F (M ) is a bijective homomorphism Φ : F (M ) → F (M ). The automorphisms of F (M ) form a group denoted by Aut(F (M )). 70 CHAPTER 2. FOUNDATIONS Corollary 2.5.44. The pullback operation C∞(M, N ) → Hom(F (N ), F (M )) : φ → φ∗ is bijective.
In particular, the map Diﬀ(M ) → Aut(F (M )) : φ → φ∗ is an anti-isomorphism of groups. Proof. This is an exercise with hint. Let Φ : F (N ) → F (M ) be a unital algebra homomorphism. By Theorem 2.5.43 there exists a map φ : M → N such that εp ◦ Φ = εφ(p) for all p ∈ M. Prove that f ◦ φ : M → R is smooth for every smooth map f : N → R and deduce that φ is smooth. Remark 2.5.45. The pullback operation is functorial, i.e. (ψ ◦ φ)∗ = φ∗ ◦ ψ∗, id∗ M = idF (M ). for φ ∈ C∞(M, N ) and ψ ∈ C∞(N, P ). Here id denotes the identity map of the space indicated in the subscript. Hence Corollary 2.5.44 may be summarized by saying that the category of smooth manifolds and smooth maps is anti-isomorphic to a subcategory of the category of commutative unital algebras and unital algebra homomorphisms. Exercise 2.5.46. If M is compact, then there is a slightly diﬀerent way to prove Theorem 2.5.43. An ideal in F (M ) is a linear subspace J ⊂ F (M ) satisfying the condition f ∈ F (M ), g ∈ J =⇒ f g ∈ J. A maximal ideal in F (M ) is an ideal J F (M ) such that every ideal J F (M ) containing J is equal to J. Prove that, if M is compact and J ⊂ F (M ) is an ideal with the property that for every p ∈ M there is an f ∈ J with f (p) = 0, then J = F (M ). Deduce that each maximal ideal in F (M ) has the form Jp := {f ∈ F (M ) \| f (p) = 0} for some p ∈ M. Exercise 2.5.47. If M is compact, give another proof of Corollary 2.
5.44 as follows. The set Φ−1(Jp) is a maximal ideal in F (N ) for each p ∈ M. Use Exercise 2.5.46 to deduce that there is a unique map φ : M → N such that Φ−1(Jp) = Jφ(p) for all p ∈ M. Show that φ is smooth and φ∗ = Φ. Exercise 2.5.48. It is a theorem of ring theory that, when I ⊂ R is an ideal in a ring R, the quotient ring R/I is a ﬁeld if and only if the ideal I is maximal. Show that the kernel of the ring homomorphism εp : F (M ) → R of Theorem 2.5.43 is the ideal Jp of Exercise 2.5.46. Conclude that M is compact if and only if every maximal ideal J in F (M ) is of the form J = Jp for some p ∈ M. Hint: The functions of compact support form an ideal. It can be shown that if M is not compact and J is a maximal ideal containing all functions of compact support, then the quotient ﬁeld F (M )/J is a non-Archimedean ordered ﬁeld which properly contains R. 2.5. LIE GROUPS 71 2.5.7 Vector Fields and Derivations A derivation of F (M ) is a linear map δ : F (M ) → F (M ) that satisﬁes δ(f g) = δ(f )g + f δ(g). and the derivations form a Lie algebra denoted by Der(F (M )). We may think of Der(F (M )) as the Lie algebra of Aut(F (M )) with the Lie bracket given by the commutator. By Theorem 2.5.43 the pullback operation Diﬀ(M ) → Aut(F (M )) : φ → φ∗ (2.5.28) can be thought of as a Lie group anti-isomorphism. Diﬀerentiating it at the identity φ = id gives a linear map Vect(M ) → Der(F (M )) : X → LX. (2.5.29) Here
the operator LX : F (M ) → F (M ) is given by the derivative of a function f in the direction of the vector ﬁeld X, i.e. LX f := df · X = d dt t=0 f ◦ φt, where φt denotes the ﬂow of X. Since the map (2.5.29) is the derivative of the “Lie group” anti-homomorphism (2.5.28) we expect it to be a Lie algebra anti-homomorphism. Indeed, one can show that L[X,Y ] = LY LX − LX LY = −[LX, LY ] (2.5.30) for X, Y ∈ Vect(M ). This conﬁrms that our sign in the deﬁnition of the Lie bracket in §2.4.3 is consistent with the standard conventions in the theory of Lie groups. In the literature the diﬀerence between a vector ﬁeld and the associated derivation LX is sometimes neglected in the notation and many authors write Xf := df · X = LX f, thus thinking of a vector ﬁeld on a manifold M as an operator on the space of functions. With this notation one obtains the equation [X, Y ]f = Y (Xf ) − X(Y f ) and here lies the origin for the use of the opposite sign for the Lie bracket in many books on diﬀerential geometry. Exercise 2.5.49. Prove that the map (2.5.29) is bijective. Hint: Fix a derivation δ ∈ Der(F (M )) and prove the following. Fact 1: If U ⊂ M is an open set and f ∈ F (M ) vanishes on U, then δ(f ) vanishes on U. Fact 2: If p ∈ M and the derivative df (p) : TpM → R is zero, then (δ(f ))(p) = 0. (By Fact 1, the proof of Fact 2 can be reduced to an argument in local coordinates.) Exercise 2.5.50. Verify the formula (2.5.30). 72 CHAPTER 2. FOUNDATIONS 2.6 Vector Bundles and Submersions This section characterizes submersions (§
2.6.1) and introduces the concept of a smooth vector bundle in the extrinsic setting (§2.6.2). 2.6.1 Submersions Let M ⊂ Rk be a smooth m-manifold and N ⊂ R be a smooth n-manifold. A smooth map f : N → M is called a submersion iﬀ its derivative df (q) : TqN → Tf (q)M is surjective for every q ∈ N. Figure 2.12: A local right inverse of a submersion. Lemma 2.6.1. Let M ⊂ Rk be a smooth m-manifold, N ⊂ R be a smooth n-manifold, and f : N → M be a smooth map. The following are equivalent. (i) f is a submersion. (ii) For every q0 ∈ N there is an M -open neighborhood U of p0 := f (q0) and a smooth map g : U → N such that g(f (q0)) = q0 and f ◦ g = id : U → U. Thus f has a local right inverse near every point in N (see Figure 2.12). Proof. We prove that (i) implies (ii). Since the derivative df (q0) : Tq0N → Tp0M is surjective we have n ≥ m and dim ker df (q0) = n − m. Hence there is a linear map A : R → Rn−m whose restriction to the kernel of df (q0) is bijective. Now deﬁne the map ψ : N → M × Rn−m by ψ(q) := (f (q), A(q − q0)) q0NgfUMp0 2.6. VECTOR BUNDLES AND SUBMERSIONS 73 for q ∈ N. Then ψ(q0) = (p0, 0) and its derivative dψ(q0) : Tq0N → Tp0M × Rn−m sends w ∈ Tq0N to (df (q0)w, Aw) and is therefore bijective. Hence it follows from the inverse function theorem for manifolds (Theorem 2.2.17) that there exists
an N -open neighborhood V ⊂ N of q0 such that the set W := ψ(N ) ⊂ M × Rn−m is an open neighborhood of (p0, 0) and ψ\|V : V → W is a diﬀeomorphism. Let U := {p ∈ M \| (p, 0) ∈ W } and deﬁne the map g : U → N by g(p) := ψ−1(p, 0). Then p0 ∈ U, g is smooth and (p, 0) = ψ(g(p)) = (f (g(p)), A(g(p) − q0)). Hence f (g(p)) = p for all p ∈ U and g(p0) = ψ−1(p0, 0) = q0. This shows that (i) implies (ii). The converse is an easy consequence of the chain rule and is left to the reader. This proves Lemma 2.6.1 Corollary 2.6.2. The image of a submersion f : N → M is open. Proof. If p0 = f (q0) ∈ f (N ), then the neighborhood U ⊂ M of p0 in Lemma 2.6.1 (ii) is contained in the image of f. Corollary 2.6.3. If N is a nonempty compact manifold, M is a connected manifold, and f : N → M is a submersion, then f is surjective and M is compact. Proof. The image f (N ) is an open subset of M by Corollary 2.6.2, it is a relatively closed subset of M because N is compact, and it is nonempty because N is nonempty. Since M is connected this implies that f (N ) = M. In particular, M is compact. Exercise 2.6.4. Let f : N → M be a smooth map. Prove that the sets {q ∈ N \| df (q) is injective} and {q ∈ N \| df (q) is surjective} are open (in the relative topology of N ). 74 CHAPTER 2. FOUNDATIONS 2.6.2 Vector Bundles Let M ⊂ Rk be an m-dimensional smooth manifold. Deﬁn
ition 2.6.5. A (smooth) vector bundle (over M of rank n) is a smooth submanifold E ⊂ M × R of dimension m + n such that, for every point p ∈ M, the set Ep := v ∈ R \| (p, v) ∈ E is an n-dimensional linear subspace of R (called the ﬁber of E over p). A vector bundle E over M is equipped with a smooth map π : E → M deﬁned by π(p, v) := p. This map is called the canonical projection of E. If E ⊂ M × R is a vector bundle, then a (smooth) section of E is a smooth map s : M → R such that s(p) ∈ Ep for every p ∈ M. A section s : M → R of a vector bundle E over M determines a smooth map σ : M → E which sends the point p ∈ M to the pair (p, s(p)) ∈ E. This map satisﬁes π ◦ σ = id. It is sometimes convenient to abuse notation and eliminate the distinction between s and σ. Thus we will sometimes use the same letter s for the map from M to R and the map from M to E. Deﬁnition 2.6.6. Let M ⊂ Rk be a smooth m-manifold. The set T M := {(p, v) \| p ∈ M, v ∈ TpM } is called the tangent bundle of M. The tangent bundle is a subset of M × Rk and, for every p ∈ M, its ﬁber TpM is an m-dimensional linear subspace of Rk by Theorem 2.2.3. However, it is not immediately obvious from the deﬁnition that T M is a submanifold of M × Rk. This will be proved below. The sections of T M are the vector ﬁelds on M. Exercise 2.6.7. Let f : M → N be a smooth map between manifolds. Prove that the tangent map T M → T N : (p, v) → (f (p), df (p)v) is smooth. Exercise 2.6.8. Let M
⊂ Rk be a smooth m-manifold and let φ : U → Ω be a smooth coordinate chart on an M -open set U ⊂ M with values in an open set Ω = φ(U ) ⊂ Rm. Prove that the map φ : T U → Ω × Rm deﬁned by φ(p, v) := (φ(p), dφ(p)) for p ∈ U and v ∈ TpM is a diﬀeomorphism. It is called a standard coordinate chart on T M. Deduce that T M is a smooth 2m-dimensional submanifold of M × Rk and hence is a smooth vector bundle over M. (See also Corollary 2.6.12 below.) 2.6. VECTOR BUNDLES AND SUBMERSIONS 75 Exercise 2.6.9. Let V ⊂ R be an n-dimensional linear subspace. The orthogonal projection of R onto V is the matrix Π ∈ R× that satisﬁes Π = Π2 = ΠT, im Π = V. (2.6.1) Prove that there is a unique matrix Π ∈ R× satisfying (2.6.1). Prove that, for every symmetric matrix S = ST ∈ R×, the kernel of S is the orthogonal complement of the image of S. If D ∈ R×n is any injective matrix whose image is V, prove that det(DTD) = 0 and Π = D(DTD)−1DT. (2.6.2) Theorem 2.6.10 (Vector bundles). Let M ⊂ Rk be a smooth m-manifold and let E ⊂ M × R be a subset. Assume that, for every p ∈ M, the set Ep := v ∈ R \| (p, v) ∈ E (2.6.3) is an n-dimensional linear subspace of R. Let Π : M → R× be the map that assigns to each p ∈ M the orthogonal projection of R onto Ep, i.e. Π(p) = Π(p)2 = Π(p)T, im Π(
p) = Ep. (2.6.4) Then the following are equivalent. (i) E is a vector bundle. (ii) For every p0 ∈ M and every v0 ∈ Ep0 there is a smooth map s : M → R such that s(p0) = v0 and s(p) ∈ Ep for all p ∈ M. (iii) The map Π : M → R× is smooth. (iv) For every p0 ∈ M there is an open neighborhood U ⊂ M of p0 and a diﬀeomorphism π−1(U ) → U × Rn : (p, v) → Φ(p, v) = (p, Φp(v)) such that the map Φp : Ep → Rn is an isometric isomorphism for all p ∈ U. (v) For every p0 ∈ M there is an open neighborhood U ⊂ M of p0 and a diﬀeomorphism π−1(U ) → U × Rn : (p, v) → Φ(p, v) = (p, Φp(v)) such that the map Φp : Ep → Rn is a vector space isomorphism for all p ∈ U. Condition (i) implies that the projection π : E → M is a submersion. In (ii) the section s can be chosen to have compact support, i.e. there exists a compact subset K ⊂ M such that s(p) = 0 for all p ∈ M \ K. Before giving the proof of Theorem 2.6.10 we explain some of its conse- quences. 76 CHAPTER 2. FOUNDATIONS Deﬁnition 2.6.11. The maps Φ : π−1(U ) → U × Rn in Theorem 2.6.10 are called local trivializations of E. They ﬁt into commutative diagrams π−1(U ) π U × Rn pr1. Φ U Corollary 2.6.12. Let M ⊂ Rk be a smooth m-manifold. Then T M is a vector bundle over M and hence is a smooth 2m-manifold in Rk × Rk. Proof. Let φ
: U → Ω be a coordinate chart on an M -open set U ⊂ M with values in an open subset Ω ⊂ Rm. Denote its inverse by ψ := φ−1 : Ω → M. By Theorem 2.2.3 the linear map dψ(x) : Rm → Rk is injective and its image is Tψ(x)M for every x ∈ Ω. Hence the map D : U → Rk×m deﬁned by D(p) := dψ(φ(p)) ∈ Rk×m is smooth and, for every p ∈ U, the linear map D(p) : Rm → Rk is injective and its image is TpM. Thus the function ΠT M : M → Rk×k deﬁned by (2.6.4) with Ep = TpM is given by ΠT M (p) = D(p) D(p)TD(p) −1 D(p)T for p ∈ U. Hence ΠT M is smooth and so T M is a vector bundle by Theorem 2.6.10. Let M ⊂ Rk be an m-manifold, N ⊂ R be an n-manifold, f : N → M be a smooth map, and E ⊂ M × Rd be a vector bundle. The pullback bundle is the vector bundle f ∗E → N deﬁned by f ∗E := (q, v) ∈ N × Rd \| v ∈ Ef (q) and the normal bundle of E is the vector bundle E⊥ → M deﬁned by E⊥ := (p, w) ∈ M × Rd \| v, w = 0 ∀ v ∈ Ep. Corollary 2.6.13. The pullback and normal bundles are vector bundles. Proof. Let Π = ΠE : M → Rd×d be the map deﬁned by (2.6.4). This map is smooth by Theorem 2.6.10. Moreover, the corresponding maps for f ∗E and E⊥ are given by Πf ∗E = ΠE ◦ f : N → Rd×d
, ΠE⊥ = 1l − ΠE : M → Rd×d. These maps are smooth and hence it follows from Theorem 2.6.10 that f ∗E and E⊥ are vector bundles. 2.6. VECTOR BUNDLES AND SUBMERSIONS 77 Proof of Theorem 2.6.10. We ﬁrst assume that E is a vector bundle and prove that π : E → M is a submersion. Let σ : M → E denote the zero section given by σ(p) := (p, 0). Then π ◦σ = id and hence it follows from the chain rule that the derivative dπ(p, 0) : T(p,0)E → TpM is surjective. Now it follows from Exercise 2.6.4 that for every p ∈ M there is an ε > 0 such that the derivative dπ(p, v) : T(p,v)E → TpM is surjective for every v ∈ Ep with \|v\| < ε. Consider the map fλ : E → E deﬁned by This map is a diﬀeomorphism for every λ > 0. It satisﬁes fλ(p, v) := (p, λv). and hence π = π ◦ fλ dπ(p, v) = dπ(p, λv) ◦ dfλ(p, v) : T(p,v)E → TpM. Since dfλ(p, v) is bijective and dπ(p, λv) is surjective for λ < ε/ \|v\| it follows that dπ(p, v) is surjective for every p ∈ M and every v ∈ Ep. Thus the projection π : E → M is a submersion for every vector bundle E over M. We prove that (i) implies (ii). Let p0 ∈ M and v0 ∈ Ep0. We have already proved that π is a submersion. Hence it follows from Lemma 2.6.1 that there exists an M -open neighborhood U ⊂ M of p0 and a smooth map σ0 : U → E such that π ◦ σ
0 = id : U → U, σ0(p0) = (p0, v0). Deﬁne the map s0 : U → R by (p, s0(p)) := σ0(p) for p ∈ U. Then s0(p0) = v0 and s0(p) ∈ Ep for all p ∈ U. Now choose ε > 0 such that {p ∈ M \| \|p − p0\| < ε} ⊂ U and choose a smooth cutoﬀ function β : Rk → [0, 1] such that β(p0) = 1 and β(p) = 0 for \|p − p0\| ≥ ε. Deﬁne s : M → R by s(p) := β(p)s0(p), 0, if p ∈ U, if p /∈ U. This map satisﬁes the requirements of (ii). 78 CHAPTER 2. FOUNDATIONS We prove that (ii) implies (iii). Thus we assume that E satisﬁes (ii). Choose p0 ∈ M and a basis v1,..., vn of Ep0. By (ii) there exists smooth sections s1,..., sn : M → R of E such that si(p0) = vi for i = 1,..., n. Now there exists an M -open neighborhood U ⊂ M of p0 such that the vectors s1(p),..., sn(p) are linearly independent, and hence form a basis of Ep for every p ∈ U. Hence, for every p ∈ U, we have Ep = imD(p), D(p) := [s1(p) · · · sn(p)] ∈ R×n. By Exercise 2.6.9, this implies Π(p) = D(p)(D(p)TD(p))−1D(p)T for every p ∈ U. Thus every p0 ∈ M has a neighborhood U such that the restriction of Π to U is smooth. This shows that (ii) implies (iii). We prove that (iii) implies (iv). Fix a point p0 ∈ M and choose a ba- sis v1,..., v
n of Ep0. For p ∈ M deﬁne D(p) := [Π(p)v1 · · · Π(p)vn] ∈ R×n Then D : M → R×n is a smooth map and D(p0) has rank n. Hence the set U := {p ∈ M \| rankD(p) = n} ⊂ M is an open neighborhood of p0 and Ep = imD(p) for all p ∈ U. Thus π−1(U ) = {(p, v is an open set containing π−1(p0). Deﬁne the map Φ : π−1(U ) → U × Rn by Φ(p, v) := p, Φp(v), Φp(v) := D(p)TD(p) −1/2 D(p)Tv for p ∈ U and v ∈ Ep. This map is bijective and its inverse is given by Φ−1(p, ξ) = p, Φ−1 p (ξ), Φ−1 p (ξ) = D(p) D(p)TD(p) −1/2 ξ for p ∈ U and ξ ∈ Rn. Thus Φ is a diﬀeomorphism and \|Φp(v)\| = \|v\| for all p ∈ U and all v ∈ Ep. This shows that (iii) implies (iv). That (iv) implies (v) is obvious. We prove that (v) implies (i). Shrinking U if necessary, we may assume that there exists a coordinate chart φ : U → Ω with values in an open set Ω ⊂ Rm. Then the composition (φ × id) ◦ Φ : π−1(U ) → Ω × Rn is a diffeomorphism. Thus E ⊂ Rk × R is a manifold of dimension m + n and this proves Theorem 2.6.10. Exercise 2.6.14. Deﬁne the notion of an isomorphism between two vector bundles E and F over M. Construct a vector bundle E ⊂ S1 × R2 of rank 1 that does not admit a global trivialization
, i.e. that is not isomorphic to the trivial bundle S1 × R. Such a vector bundle is called a M¨obius strip. 2.6. VECTOR BUNDLES AND SUBMERSIONS 79 The Implicit Function Theorem Next we carry over the Implicit Function Theorem in Corollary A.2.6 to smooth maps on vector bundles. Theorem 2.6.15 (Implicit Function Theorem). Let M ⊂ Rk be a smooth m-manifold, let N ⊂ Rk be a smooth n-manifold, let E ⊂ M × R be a smooth vector bundle of rank n, let W ⊂ E be open, and let f : W → N be a smooth map. For p ∈ M deﬁne fp : Wp → N by Wp := {v ∈ Ep \| (p, v) ∈ W }, fp(v) := f (p, v). Let p0 ∈ M such that 0 ∈ Wp0 and dfp0(0) : Ep0 → Tq0N is bijective, where q0 := f (p0, 0) ∈ N. Then there exists a constant ε > 0, open neighborhoods U0 ⊂ M of p0 and V0 ⊂ N of q0, and a smooth map h : U0 × V0 → R such that {(p, v) ∈ E \| p ∈ U0, \|v\| < ε} ⊂ W and h(p, q) ∈ Ep, \|h(p, q)\| < ε (2.6.5) for all (p, q) ∈ U0 × V0 and fp(v) = 0 ⇐⇒ v = h(p, q) (2.6.6) for all (p, q) ∈ U0 × V0, and all v ∈ Ep with \|v\| < ε. Proof. Choose a coordinate chart ψ : V → Rn on an open set V ⊂ N containing q0. Choose an open neighborhood U ⊂ M of p0 such that (p, 0) ∈ W and f (p, 0) ∈ V for all p ∈ U, there is a coordinate chart φ : U →
Ω ⊂ Rm, and there is a local trivialization Φ : π−1(U ) → U × Rn as in Theorem 2.6.10 with \|Φp(v)\| = \|v\| for p ∈ U and v ∈ Ep. Deﬁne Br := {ξ ∈ Rn \| \|ξ\| < r} and choose r > 0 so small that Φ−1(U × Br) ⊂ W and f ◦ Φ−1(U × Br) ⊂ V. Deﬁne the map F : Ω × Rn × Br → Rn by F (x, y, ξ) := ψ ◦ f ◦ Φ−1 φ−1(x), ξ − y for (x, y) ∈ Ω × Rn and ξ ∈ Br. Let x0 := φ(p0) and y0 := ψ(q0). Then we have F (x0, y0, 0) = 0 and the derivative d3F (x0, y0, 0) : Rn → Rn of F with respect to ξ at (x0, y0, 0) is bijective. Hence Corollary A.2.6 asserts that there exist open neighborhoods U0 ⊂ U of p0 and V0 ⊂ V of q0, a constant 0 < ε < r, and a smooth map g : φ(U0) × ψ(V0) → Bε such that F (x, y, ξ) = 0 ⇐⇒ g(x, y) = ξ for all (x, y) ∈ φ(U0) × ψ(V0) and all ξ ∈ Bε. Thus the map h : U0 × V0 → R, h(p, q) := Φ−1 p g(φ(p), ψ(q)), satisﬁes the requirements of Theorem 2.6.15. 80 CHAPTER 2. FOUNDATIONS 2.7 The Theorem of Frobenius Let M ⊂ Rk be an m-dimensional manifold and n be a nonnegative integer. A subbundle of rank n of the tangent bundle T M is a subset E
⊂ T M that is itself a vector bundle of rank n over M, i.e. it is a submanifold of T M and the ﬁber Ep = {v ∈ TpM \| (p, v) ∈ E} is an n-dimensional linear subspace of TpM for every p ∈ M. Note that the rank n of a subbundle is necessarily less than or equal to m. In the literature a subbundle of the tangent bundle is sometimes called a distribution on M. We shall, however, not use this terminology in order to avoid confusion with the concept of a distribution in the functional analytic setting. Deﬁnition 2.7.1. Let M ⊂ Rk be an m-dimensional manifold and E ⊂ T M be a subbundle of rank n. The subbundle E is called involutive if, for any two vector ﬁelds X, Y ∈ Vect(M ), we have X(p), Y (p) ∈ Ep ∀ p ∈ M =⇒ [X, Y ](p) ∈ Ep ∀ p ∈ M. (2.7.1) The subundle E is called integrable if, for every p0 ∈ M, there exists a submanifold N ⊂ M such that p0 ∈ N and TpN = Ep for every p ∈ N. A foliation box for E (see Figure 2.13) is a coordinate chart φ : U → Ω on an M -open subset U ⊂ M with values in an open set Ω ⊂ Rn × Rm−n such that the set Ω ∩ (Rn × {y}) is connected for every y ∈ Rm−n and, for every p ∈ U and every v ∈ TpM, we have v ∈ Ep ⇐⇒ dφ(p)v ∈ Rn × {0}. Figure 2.13: A foliation box. Theorem 2.7.2 (Frobenius). Let M ⊂ Rk be an m-dimensional manifold and E ⊂ T M be a subbundle of rank n. Then the following are equivalent. (i) E is involutive. (ii) E is integrable.
(iii) For every p0 ∈ M there exists a foliation box φ : U → Ω with p0 ∈ U. MUΩφ 2.7. THE THEOREM OF FROBENIUS 81 It is easy to show that (iii) =⇒ (ii) =⇒ (i) (see below). The hard part of the theorem is to prove that (i) =⇒ (iii). This requires the following lemma. Lemma 2.7.3. Let E ⊂ T M be an involutive subbundle and X ∈ Vect(M ) be a complete vector ﬁeld such that X(p) ∈ Ep for every p ∈ M. Denote by R → Diﬀ(M ) : t → φt the ﬂow of X. Then, for all t ∈ R and all p ∈ M, we have dφt(p)Ep = Eφt(p). (2.7.2) We show ﬁrst how Theorem 2.7.2 follows from Lemma 2.7.3. Lemma 2.7.3 implies Theorem 2.7.2. We prove ﬁrst that (iii) implies (ii). Let p0 ∈ M, choose a foliation box φ : U → Ω for E with p0 ∈ U, and deﬁne N := (p ∈ U \| φ(p) ∈ Rn × {y0}} where (x0, y0) := φ(p0) ∈ Ω. Then N satisﬁes the requirements of (ii). We prove that (ii) implies (i). Choose two vector ﬁelds X, Y ∈ Vect(M ) that satisfy X(p), Y (p) ∈ Ep for all p ∈ M and ﬁx a point p0 ∈ M. Then, by (ii), there exists a submanifold N ⊂ M containing p0 such that TpN = Ep for every p ∈ N. Hence the restrictions X\|N and Y \|N are vector ﬁelds on N and so is the restriction of the Lie bracket [X, Y ] to N. Thus we
have [X, Y ](p0) ∈ Tp0N = Ep0 as claimed. We prove that (i) implies (iii). Thus we assume that E is an involutive subbundle of T M and ﬁx a point p0 ∈ M. By Theorem 2.6.10 there exist vector ﬁelds X1,..., Xn ∈ Vect(M ) such that Xi(p) ∈ Ep for all i and p and the vectors X1(p0),..., Xn(p0) form a basis of Ep0. Using Theorem 2.6.10 again we ﬁnd vector ﬁelds Y1,..., Ym−n ∈ Vect(M ) such that the vectors X1(p0),..., Xn(p0), Y1(p0),..., Ym−n(p0) form a basis of Tp0M. Using cutoﬀ functions as in the proof of Theorem 2.6.10 we may assume without loss of generality that the vector ﬁelds Xi and Yj have compact support and hence are complete (see Exercise 2.4.13). Denote by φt n the ﬂows of the vector ﬁelds X1,..., Xn, respectively, and by ψt m−n the ﬂows of the vector ﬁelds Y1,..., Ym−n. Deﬁne the map 1,..., ψt 1,..., φt ψ : Rn × Rm−n → M by ψ(x, y) := φx1 1 ◦ · · · ◦ φxn n ◦ ψy1 1 ◦ · · · ◦ ψym−n m−n (p0). 82 CHAPTER 2. FOUNDATIONS By Lemma 2.7.3, this map satisﬁes ∂ψ ∂xi (x, y) ∈ Eψ(x,y) (2.7.3) for all x ∈ Rn and y ∈ Rm−n. Moreover, ∂ψ ∂xi (
0, 0) = Xi(p0), ∂ψ ∂yj (0, 0) = Yj(p0), and so the derivative dψ(0, 0) : Rn × Rm−n → Tp0M is bijective. Hence, by the Inverse Function Theorem 2.2.17, there exists an open neighborhood Ω ⊂ Rn × Rm−n of the origin such that the set U := ψ(Ω) ⊂ M is an M -open neighborhood of p0 and ψ\|Ω : Ω → U is a diﬀeomorphism. Thus the vectors ∂ψ/∂xi(x, y) are linearly independent for every (x, y) ∈ Ω and, by (2.7.3), form a basis of Eψ(x,y). Hence φ := (ψ\|Ω)−1 : U → Ω is a foliation box and this proves Theorem 2.7.2, assuming Lemma 2.7.3. To complete the proof of the Frobenius theorem it remains to prove Lemma 2.7.3. This requires the following result. Lemma 2.7.4. Let E ⊂ T M be an involutive subbundle. If β : R2 → M is a smooth map such that ∂β ∂s (s, 0) ∈ Eβ(s,0), ∂β ∂t (s, t) ∈ Eβ(s,t), (2.7.4) for all s, t ∈ R, then for all s, t ∈ R. ∂β ∂s (s, t) ∈ Eβ(s,t), (2.7.5) We ﬁrst show how Lemma 2.7.3 follows from Lemma 2.7.4. 2.7. THE THEOREM OF FROBENIUS 83 Lemma 2.7.4 implies Lemma 2.7.3. Let X ∈ Vect(M ) be a complete vector ﬁeld satisfying X(p) ∈ Ep for every p ∈ M and let φt be the ﬂow of X. Choose a point p0 ∈ M
and a vector v0 ∈ Ep0. By Theorem 2.6.10 there is a vector ﬁeld Y ∈ Vect(M ) with values in E such that Y (p0) = v0. Moreover this vector ﬁeld may be chosen to have compact support and hence it is complete (see Exercise 2.4.13). Thus there is a solution γ : R → M of the initial value problem ˙γ(s) = Y (γ(s)), γ(0) = p0. β(s, t) := φt(γ(s)) Deﬁne β : R2 → M by for s, t ∈ R. Then ∂β ∂s ∂β ∂t (s, 0) = ˙γ(s) = Y (γ(s)) ∈ Eβ(s,0), (s, t) = X(β(s, t)) ∈ Eβ(s,t) for all s, t ∈ R. Hence it follows from Lemma 2.7.4 that dφt(p0)v0 = dφt(γ(0)) ˙γ(0) = ∂β ∂s (0, t) ∈ Eφt(p0) for every t ∈ R. This proves Lemma 2.7.3, assuming Lemma 2.7.4. Proof of Lemma 2.7.4. Given any point p0 ∈ M we choose a coordinate chart φ : U → Ω, deﬁned on an M -open set U ⊂ M with values in an open set Ω ⊂ Rn × Rm−n, such that p0 ∈ U and dφ(p0)Ep0 = Rn × {0}. Shrinking U, if necessary, we ﬁnd that for every p ∈ U the linear subspace dφ(p)Ep ⊂ Rn × Rm−n is the graph of a matrix A ∈ R(m−n)×n. Thus there exists a smooth map A : Ω → R(m−n)×n such that, for every p ∈ U, dφ(p)Ep = {(ξ, A(x
, y)ξ) \| ξ ∈ Rn}, (x, y) := φ(p) ∈ Ω. (2.7.6) For (x, y) ∈ Ω deﬁne the linear maps ∂A ∂x (x, y) : Rn → R(m−n)×n, ∂A ∂y (x, y) : Rm−n → R(m−n)×n by ∂A ∂x (x, y) · ξ := n i=1 ξi ∂A ∂xi (x, y), ∂A ∂y (x, y) · η := m−n j=1 ηj ∂A ∂yj (x, y), for ξ = (ξ1,..., ξn) ∈ Rn and η = (η1,..., ηm−n) ∈ Rm−n. We prove the following. 84 CHAPTER 2. FOUNDATIONS Claim 1. Let (x, y) ∈ Ω, ξ, ξ ∈ Rn and deﬁne η, η ∈ Rm−n by η := A(x, y)ξ and η := A(x, y)ξ. Then ∂A ∂x (x, y) · ξ + ∂A ∂y (x, y) · η ξ = ∂A ∂x (x, y) · ξ + (x, y) · η ξ. ∂A ∂y The graphs of the matrices A(z) determine a subbundle E ⊂ Ω × Rm with the ﬁbers Ez := (ξ, η) ∈ Rn × Rm−n \| η = A(x, y)ξ for z = (x, y) ∈ Ω. This subbundle is the image of the restriction E\|U := {(p, v) \| p ∈ U, v ∈ Ep} under the diﬀeomorphism T M \|U → Ω × Rm : (p, v) → (φ(p), dφ(
p)v) and hence it is involutive. Now ﬁx two elements ξ, ξ ∈ Rn and deﬁne the vector ﬁelds ζ, ζ : Ω → Rm by ζ(z) := (ξ, A(z)ξ), ζ (z) := (ξ, A(z)ξ), z ∈ Ω. Then ζ and ζ are sections of E and their Lie bracket [ζ, ζ ] is given by [ζ, ζ ](z) = 0, dA(z)ζ (z) ξ − (dA(z)ζ(z)) ξ. Since E is involutive the Lie bracket [ζ, ζ ] must take values in the graph of A. Hence the right hand side vanishes and this proves Claim 1. Claim 2. Let I, J ⊂ R be open intervals and let z = (x, y) : I × J → Ω be a smooth map. Fix two points s0 ∈ I and t0 ∈ J and assume that ∂y ∂s (s0, t0) = Ax(s0, t0), y(s0, t0) ∂x ∂s (s, t) = Ax(s, t), y(s, t) ∂x ∂y ∂t ∂t (s, t) (s0, t0), for all s ∈ I and t ∈ J. Then ∂y ∂s (s0, t) = Ax(s0, t), y(s0, t) ∂x ∂s (s0, t) for all t ∈ J. (2.7.7) (2.7.8) (2.7.9) 2.7. THE THEOREM OF FROBENIUS 85 ∂ ∂t − A · ∂x ∂s ∂y ∂s Equation (2.7.9) holds by assumption for t = t0. Moreover, dropping the argument z(s0, t) = z = (x, y) for notational convenience we obtain ∂x ∂s ∂x ∂s ∂x ∂t ∂y ∂A �
�y ∂A ∂y ∂A ∂y ∂A ∂y ∂x ∂t ∂x ∂s ∂x ∂t · ∂y ∂s ∂2y ∂s∂t ∂2y ∂s∂t ∂2y ∂s∂t ∂2y ∂s∂t = − A − + ∂2x ∂s∂t ∂2x ∂s∂t ∂2x ∂s∂t ∂2x ∂s∂t ∂y ∂s · · ∂x ∂t ∂x ∂t ∂x ∂s ∂x ∂s ∂x ∂t ∂A ∂x ∂A ∂x ∂A ∂x ∂A ∂x ∂x ∂s ∂x ∂t. · + ∂A ∂y · − A = ∂A ∂y ∂y ∂s · − A ∂x ∂s Here the second step follows from (2.7.8), the third step follows from Claim 1, and the last step follows by diﬀerentiating equation (2.7.8) with respect to s. Deﬁne the curve η : J → Rm−n by η(t) := ∂y ∂s (s0, t) − Ax(s0, t), y(s0, t) ∂x ∂s (s0, t). By (2.7.7) and what we have just proved, the curve η satisﬁes the linear diﬀerential equation ∂A ∂y x(s0, t), y(s0, t) · η(t) ∂x ∂t η(t0) = 0. (s0, t), ˙η(t) = Hence η(t) = 0 for all t ∈ J. This proves (2.7.9) and Claim 2. Now let β : R2 → M be a smooth map satisfying (2.7.4) and ﬁx a real number s0. Consider the set W := {t ∈ R \| ∂sβ(s
0, t) ∈ Eβ(s0,t)}. By going to local coordinates, we obtain from Claim 2 that W is open. Moreover, W is obviously closed, and W = ∅ because 0 ∈ W by (2.7.4). Hence W = R. Since s0 ∈ R was chosen arbitrarily, this proves (2.7.5) and Lemma 2.7.4. Any subbundle E ⊂ T M determines an equivalence relation on M via p0 ∼ p1 ⇐⇒ there is a smooth curve γ : [0, 1] → M such that γ(0) = p0, γ(1) = p1, ˙γ(t) ∈ Eγ(t) ∀ t. (2.7.10) If E is integrable, this equivalence relation is called a foliation and the equivalence class of p0 ∈ M is called the leaf of the foliation through p0. The next example shows that the leaves do not need to be submanifolds. 86 CHAPTER 2. FOUNDATIONS Example 2.7.5. Consider the torus M := S1 × S1 ⊂ C2 with the tangent bundle T M = (z1, z2, iλ1z1, iλ2z2) ∈ C4 \| \|z1\| = \|z2\| = 1, λ1, λ2 ∈ R. Let ω1, ω2 be real numbers and consider the subbundle E := (z1, z2, itω1z1, itω2z2) ∈ C4 \| \|z1\| = \|z2\| = 1, t ∈ R. The leaf of this subbundle through z = (z1, z2) ∈ T2 is given by L = eitω1z1, eitω2z2 t ∈ R. It is a submanifold if and only if the quotient ω1/ω2 is a rational number (or ω2 = 0). Otherwise each leaf is a dense subset of T2. Exercise 2.7.6. Prove that (2.7.10) deﬁnes an equivalence relation for every subbundle E ⊂ T M. Exercise 2.7.7.
Each subbundle E ⊂ T M of rank 1 is integrable. Exercise 2.7.8. Consider the manifold M = R3. Prove that the subbundle E ⊂ T M = R3 × R3 with ﬁber Ep = (ξ, η, ζ) ∈ R3 \| ζ − yξ = 0 over p = (x, y, z) ∈ R3 is not integrable and that any two points in R3 can be joined by a path tangent to E. Exercise 2.7.9. Consider the manifold M = S3 ⊂ R4 = C2 and deﬁne E := (z, ζ) ∈ C2 × C2 \| \|z\| = 1, ζ ⊥ z, iζ ⊥ z. Thus the ﬁber Ez ⊂ TzS3 = z⊥ is the maximal complex linear subspace of TzS3. Prove that E has real rank 2 and is not integrable. Exercise 2.7.10. Let E ⊂ T M be an involutive subbundle of rank n and let L ⊂ M be a leaf of the foliation determined by E. A subset V ⊂ L is called L-open iﬀ it can be written as a union of submanifolds N of M with tangent spaces TpN = Ep for p ∈ N. Prove that the L-open sets form a topology on L (called the intrinsic topology). Prove that the obvious inclusion ι : L → M is continuous with respect to the intrinsic topology on L. Prove that the inclusion ι : L → M is proper if and only if the intrinsic topology on L agrees with the relative topology inherited from M (called the extrinsic topology). Remark 2.7.11. It is surprisingly diﬃcult to prove that each closed leaf L of a foliation is a submanifold of M. A proof due to David Epstein [19] is sketched in §2.9.4 below. 2.8. THE INTRINSIC DEFINITION OF A MANIFOLD* 87 2.8 The Intrinsic Deﬁnition of a Manifold* It is somewhat restrictive to only consider manifolds
that are embedded in some Euclidean space. Although we shall see that (at least) every compact manifold admits an embedding into a Euclidean space, such an embedding is in many cases not a natural part of the structure of a manifold. In particular, we encounter manifolds that are described as quotient spaces and there are manifolds that are embedded in certain inﬁnite-dimensional Hilbert spaces. For this reason it is convenient, at this point, to introduce a more general intrinisc deﬁnition of a manifold. (See Chapter 1 for an overview.) This requires some background from point set topology that is not covered in the ﬁrst year analysis courses. We shall then see that all the deﬁnitions and results of this chapter carry over in a natural manner to the intrinsic setting. We begin by recalling the intrinsing deﬁnition of a smooth manifold in §1.4. 2.8.1 Deﬁnition and Examples Figure 2.14: Coordinate charts and transition maps. Deﬁnition 2.8.1 (Smooth m-manifold). Let m ∈ N0 and M be a set. A chart on M is a pair (φ, U ) where U ⊂ M and φ is a bijection from U to an open set φ(U ) ⊂ Rm. Two charts (φ1, U1), (φ2, U2) are called compatible iﬀ φ1(U1 ∩ U2) and φ2(U1 ∩ U2) are open and the transition map φ21 = φ2 ◦ φ−1 1 : φ1(U1 ∩ U2) → φ2(U1 ∩ U2) (2.8.1) is a diﬀeomorphism (see Figure 2.14). A smooth atlas on M is a collection A of charts on M any two of which are compatible and such that the sets U, as (φ, U ) ranges over A, cover M (i.e. for every p ∈ M there exists a chart (φ, U ) ∈ A with p ∈ U ). A maximal smooth atlas is an atlas which contains every chart which is compatible with each of its members. A smooth m-
manifold is a pair consisting of a set M and a maximal atlas A on M. MUαβUβαφβφαφ 88 CHAPTER 2. FOUNDATIONS In Lemma 1.4.3 it was shown that, if A is an atlas, then so is the collection A of all charts compatible with each member of A. Moreover, the atlas A is maximal, so every atlas extends uniquely to a maximal atlas. For this reason, a manifold is usually speciﬁed by giving its underlying set M and some atlas on M. Generally, the notation for the atlas is suppressed and the manifold is denoted simply by M. The members of the atlas are called coordinate charts or simply charts on M. By Lemma 1.3.3 a smooth m-manifold admits a unique topology such that, for each chart (φ, U ) of the smooth atlas, the set U ⊂ M is open and the bijection φ : U → φ(U ) is a homeomorphism onto the open set φ(U ) ⊂ Rm. This topology is called the intrinsic topology of M and is described in the following deﬁnition. Deﬁnition 2.8.2. Let M be a smooth m-manifold. The intrinsic topology on the set M is the topology induced by the charts, i.e. a subset W ⊂ M is open in the intrinsic topology iﬀ φ(U ∩ W ) is an open subset of Rm for every chart (φ, U ) on M.1 Remark 2.8.3. Let M ⊂ Rk be smooth m-dimensional submanifold of Rk as in Deﬁnition 2.1.3. Then the set of all diﬀeomorphisms (φ, U ∩ M ) as in Deﬁnition 2.1.3 form a smooth atlas as in Deﬁnition 2.8.1. The intrinsic topology on the resulting smooth manifold is the same as the relative topology deﬁned in §1.3. Remark 2.8.4. A topological manifold is a topological space such that each point has a neighborhood U homeomorphic to an open subset of
Rm. Thus a smooth manifold (with the intrinsic topology) is a topological manifold and its maximal smooth atlas A is a subset of the set A0 of all pairs (φ, U ) where U ⊂ M is an open set and φ is a homeomorphism from U to an open subset of Rm. One says that the maximal smooth atlas A is a smooth structure on the topological manifold M iﬀ the topology of M is the intrinsic topology of the smooth structure and every chart of the smooth structure is a homeomorphism. As explained in §1.4 a topological manifold can have many distinct smooth structures (see Remark 1.4.6). However, it is a deep theorem beyond the scope of this book that there are topological manifolds which do not admit any smooth structure. 1At this point we do not assume that the intrinsic topology on the manifold M is Hausdorﬀ or second countable. These hypotheses will be imposed after the end of the present chapter. For explanations see the comments at the end of §2.8.1 and of §2.9.5. 2.8. THE INTRINSIC DEFINITION OF A MANIFOLD* 89 Example 2.8.5. The complex projective space CPn is the set CPn = ⊂ Cn+1 \| is a 1-dimensional complex subspace of complex lines in Cn+1. It can be identiﬁed with the quotient space CPn = Cn+1 \ {0} /C∗ of nonzero vectors in Cn+1 modulo the action of the multiplicative group C∗ = C \ {0} of nonzero complex numbers. The equivalence class of a nonzero vector z = (z0,..., zn) ∈ Cn+1 will be denoted by [z] = [z0 : z1 : · · · : zn] := {λz \| λ ∈ C∗} and the associated line is = Cz. An atlas on CPn is given by the open cover Ui := {[z0 : · · · : zn] \| zi = 0} for i = 0, 1,..., n and the coordinate charts φi : Ui → Cn are φi([z0 : · · · :
zn]) := z0 zi,..., zi−1 zi, zi+1 zi,...,. zn zi (2.8.2) Exercise: Prove that each φi is a homeomorphism and the transition maps are holomorphic. Prove that the manifold topology is the quotient topology, i.e. if π : Cn+1 \ {0} → CPn denotes the obvious projection, then a subset U ⊂ CPn is open if and only if π−1(U ) is an open subset of Cn+1 \ {0}. Example 2.8.6. The real projective space RPn is the set RPn = ⊂ Rn+1 \| is a 1-dimensional linear subspace of real lines in Rn+1. It can again be identiﬁed with the quotient space RPn = Rn+1 \ {0} /R∗ of nonzero vectors in Rn+1 modulo the action of the multiplicative group R∗ = R \ {0} of nonzero real numbers, and the equivalence class of a nonzero vector x = (x0,..., xn) ∈ Rn+1 will be denoted by [x] = [x0 : x1 : · · · : xn] := {λx \| λ ∈ R∗}. An atlas on RPn is given by the open cover Ui := {[x0 : · · · : xn] \| xi = 0} and the coordinate charts φi : Ui → Rn are again given by (2.8.2), with zj replaced by xj. The arguments in Example 2.8.5 show that these coordinate charts form an atlas and the manifold topology is the quotient topology. The transition maps are real analytic diﬀeomorphisms. 90 CHAPTER 2. FOUNDATIONS Example 2.8.7. The real n-torus is the topological space Tn := Rn/Zn equipped with the quotient topology. Thus two vectors x, y ∈ Rn are equivalent iﬀ their diﬀerence x − y ∈ Zn is an integer vector and we denote by π : Rn → Tn the
obvious projection which assigns to each vector x ∈ Rn its equivalence class π(x) := [x] := x + Zn. Then a set U ⊂ Tn is open if and only if the set π−1(U ) is an open subset of Rn. An atlas on Tn is given by the open cover Uα := {[x] \| x ∈ Rn, \|x − α\| < 1/2}, parametrized by vectors α ∈ Rn, and the coordinate charts φα : Uα → Rn deﬁned by φα([x]) := x for x ∈ Rn with \|x − α\| < 1/2. Exercise: Show that each transition map for this atlas is a translation by an integer vector. Example 2.8.8. Consider the complex Grassmannian Gk(Cn) := {V ⊂ Cn \| v is a k-dimensional complex linear subspace}. This set can again be described as a quotient space Gk(Cn) ∼= Fk(Cn)/U(k). Here Fk(Cn) := D ∈ Cn×k \| D∗D = 1l denotes the set of unitary k-frames in Cn and the group U(k) acts on Fk(Cn) contravariantly by D → Dg for g ∈ U(k). The projection π : Fk(Cn) → Gk(Cn) sends a matrix D ∈ Fk(Cn) to its image V := π(D) := im D. A subset U ⊂ Gk(Cn) is open if and only if π−1(U ) is an open subset of Fk(Cn). Given a k-dimensional subspace V ⊂ Cn we can deﬁne an open set UV ⊂ Gk(Cn) as the set of all k-dimensional subspaces of Cn that can be represented as graphs of linear maps from V to V ⊥. This set of graphs can be identiﬁed with the complex vector space HomC(V, V ⊥) of complex linear maps from V to V ⊥ and hence with C(n−k)×k. This leads to an atlas
on Gk(Cn) with holomorphic transition maps and shows that Gk(Cn) is a manifold of complex dimension kn − k2. Exercise: Verify the details of this construction. Find explicit formulas for the coordinate charts and their transition maps. Carry this over to the real setting. Show that CPn and RPn are special cases. 2.8. THE INTRINSIC DEFINITION OF A MANIFOLD* 91 Example 2.8.9 (The real line with two zeros). A topological space M is called Hausdorﬀ iﬀ any two points in M can be separated by disjoint open neighborhoods. This example shows that a manifold need not be a Hausdorﬀ space. Consider the quotient space M := R × {0, 1}/ ≡ where [x, 0] ≡ [x, 1] for x = 0. An atlas on M consists of two coordinate charts φ0 : U0 → R and φ1 : U1 → R where Ui := {[x, i] \| x ∈ R}, φi([x, i]) := x for i = 0, 1. Thus M is a 1-manifold. But the topology on M is not Hausdorﬀ, because the points [0, 0] and [0, 1] cannot be separated by disjoint open neighborhoods. Example 2.8.10 (A 2-manifold without a countable atlas). Consider the vector space X = R × R2 with the equivalence relation [t1, x1, y1] ≡ [t2, x2, y2] ⇐⇒ either y1 = y2 = 0, t1 + x1y1 = t2 + x2y2 or y1 = y2 = 0, t1 = t2, x1 = x2. For y = 0 we have [0, x, y] ≡ [t, x − t/y, y], however, each point (x, 0) on the x-axis gets replaced by the uncountable set R × {(x, 0)}. Our manifold is the quotient space M := X/ ≡. This time we do not use the quotient topology but the topology induced by our atlas (see Deﬁnition 2.8.2). The coordinate charts
are parametrized by the reals: for t ∈ R the set Ut ⊂ M and the coordinate chart φt : Ut → R2 are given by Ut := {[t, x, y] \| x, y ∈ R}, φt([t, x, y]) := (x, y). A subset U ⊂ M is open, by deﬁnition, iﬀ φt(U ∩ Ut) is an open subset of R2 for every t ∈ R. With this topology each φt is a homeomorphism from Ut onto R2 and M admits a countable dense subset S := {[0, x, y] \| x, y ∈ Q}. However, there is no atlas on M consisting of countably many charts. (Each coordinate chart can contain at most countably many of the points [t, 0, 0].) The function f : M → R given by f ([t, x, y]) := t + xy is smooth and each point [t, 0, 0] is a critical point of f with value t. Thus f has no regular value. Exercise: Show that M is a path-connected Hausdorﬀ space. In Theorem 2.9.12 we will show that smooth manifolds whose topology is Hausdorﬀ and second countable are precisely those that can be embedded in Euclidean space. Most authors tacitly assume that manifolds are Hausdorﬀ and second countable and so will we after the end of the present chapter. However before §2.9.1 there is no need to impose these hypotheses. 92 CHAPTER 2. FOUNDATIONS 2.8.2 Smooth Maps and Diﬀeomorphisms Our next goal is to carry over all the deﬁnitions from embedded manifolds in Euclidean space to the intrinsic setting. Deﬁnition 2.8.11 (Smooth map). Let (M, {(φα, Uα)}α∈A), (N, {(ψβ, Vβ)}β∈B) be smooth manifolds. A map f : M → N is called smooth iﬀ it is continuous and the map fβα := ψβ ◦ f ◦ φ−1 α : φα(Uα �
� f −1(Vβ)) → ψβ(Vβ) (2.8.3) is smooth for every α ∈ A and every β ∈ B. It is called a diﬀeomorphism iﬀ it is bijective and f and f −1 are smooth. The manifolds M and N are called diﬀeomorphic iﬀ there exists a diﬀeomorphism f : M → N. The reader may check that the notion of a smooth map is independent of the atlas used in the deﬁnition, that compositions of smooth maps are smooth, and that sums and products of smooth maps from M to R are smooth. Exercise 2.8.12. Let M be a smooth m-dimensional manifold with an atlas A = {(φα, Uα)}α∈A. Consider the quotient space M := α∈A {α} × φα(Uα) ∼, where (α, x) ∼ (β, y) def⇐⇒ α (x) = φ−1 φ−1 β (y). for α, β ∈ A, x ∈ φα(Uα), and y ∈ φβ(Uβ). Deﬁne an atlas on M by Uα := [α, x] x ∈ φα(Uα), φα([α, x]) := x. Prove that M is a smooth m-manifold and that it is diﬀeomorphic to M. Exercise 2.8.13. Prove that CP1 is diﬀeomorphic to S2. Hint: Stereographic projection. 2.8. THE INTRINSIC DEFINITION OF A MANIFOLD* 93 2.8.3 Tangent Spaces and Derivatives In the situation where M is a submanifold of Euclidean space and p ∈ M we have deﬁned the tangent space of M at p as the set of all derivatives ˙γ(0) of smooth curves γ : R → M that pass through p = γ(0). We cannot do this for manifolds in the intrinsic sense, as the derivative of a curve has yet to be deﬁned. In fact, the purpose of introducing a tangent space
of M is precisely to allow us to deﬁne what we mean by the derivative of a smooth map. There are two approaches. One is to introduce an appropriate equivalence relation on the set of curves through p and the other is to use local coordinates. Deﬁnition 2.8.14. Let M be a manifold with an atlas A = {(φα, Uα)}α∈A and let p ∈ M. Two smooth curves γ0, γ1 : R → M with γ0(0) = γ1(0) = p are called p-equivalent iﬀ for some (and hence every) α ∈ A with p ∈ Uα we have t=0 p We write γ0 ∼ γ1 iﬀ γ0 is p-equivalent to γ1 and denote the equivalence class of a smooth curve γ : R → M with γ(0) = p by [γ]p. Every such equivalence class is called a tangent vector of M at p. The tangent space of M at p is the set of equivalence classes TpM := [γ]p (2.8.4) Deﬁnition 2.8.15. Let M be a manifold with an atlas A = {(φα, Uα)}α∈A and let p ∈ M. The A -tangent space of M at p is the quotient space γ : R → M is smooth and γ(0) = p. φα(γ0(t)) = t=0 φα(γ1(t)). d dt d dt T A p M := {α} × Rm p ∼, (2.8.5) p∈Uα where the union runs over all α ∈ A with p ∈ Uα and (α, ξ) p ∼ (β, η) ⇐⇒ d φβ ◦ φ−1 α (x)ξ = η, x := φα(p). Each equivalence class [α, ξ]p is called a tangent vector of M at p. In Deﬁnition 2.8.14 it is not immediately obvious that the set TpM p M in (2.8.5) is
in (2.8.4) is a vector space. However, the quotient space T A obviously a vector space of dimension m and there is a natural bijection TpM → T A p M : [γ]p → α, d dt t=0 φα(γ(t)). p (2.8.6) This bijection induces a vector space structure on the set TpM. In other words, the set TpM in (2.8.4) admits a unique vector space structure such that the map TpM → T A p M in (2.8.6) is a vector space isomorphism. 94 CHAPTER 2. FOUNDATIONS Exercise 2.8.16. Verify the phrase “and hence every” in Deﬁnition 2.8.14 and deduce that the map TpM → T A p M in (2.8.6) is well deﬁned. Show that it is bijective. From now on we will use either Deﬁnition 2.8.14 or Deﬁnition 2.8.15 or both, whichever way is most convenient, and drop the superscript A. Deﬁnition 2.8.17 (Derivative of a smooth curve). For each smooth curve γ : R → M with γ(0) = p we deﬁne the derivative ˙γ(0) ∈ TpM as the equivalence class ∼= ˙γ(0) := [γ]p t=0 Deﬁnition 2.8.18 (Derivative of a smooth map). Let f : M → N be a smooth map between two smooth manifolds (M, {(φα, Uα)}α∈A) and (N, {(ψβ, Vβ)}β∈B) and let p ∈ M. The derivative of f at p is the map φα(γ(t)) ∈ TpM. d dt α, p deﬁned by the formula df (p) : TpM → Tf (p)N df (p)[γ]p := [f ◦ γ]f (p) (2.8.7) for each smooth curve γ : R → M with γ(0
) = p. Here we use (2.8.4). Under the isomorphism (2.8.6) this corresponds to the linear map df (p)[α, ξ]p := [β, dfβα(x)ξ]f (p), x := φα(p), (2.8.8) for α ∈ A with p ∈ Uα and β ∈ B with f (p) ∈ Vβ, where fβα is given by (2.8.3). Remark 2.8.19. Think of N = Rn as a manifold with a single coordinate chart, namely the identity map ψβ = id : Rn → Rn. For every q ∈ N = Rn the tangent space TqN is then canonically isomorphic to Rn via (2.8.5). Thus for every smooth map f : M → Rn the derivative of f at p ∈ M is a linear map df (p) : TpM → Rn, and the formula (2.8.8) reads x := φα(p). df (p)[α, ξ]p = d(f ◦ φ−1 α )(x)ξ, This formula also applies to maps deﬁned on some open subset of M. In particular, with f = φα : Uα → Rm we have dφα(p)[α, ξ]p = ξ. Thus the map dφα(p) : TpM → Rm is the canonical vector space isomorphism determined by α. With these deﬁnitions the derivative of f at p is a linear map and we have the chain rule for the composition of two smooth maps as in Theorem 2.2.14. 2.8. THE INTRINSIC DEFINITION OF A MANIFOLD* 95 2.8.4 Submanifolds and Embeddings Deﬁnition 2.8.20 (Submanifold). Let M be a smooth m-manifold and let n ∈ {0, 1,..., m}. A subset N ⊂ M is called an n-dimensional submanifold of M iﬀ, for every p ∈ N, there exists a local coordinate chart φ : U →
Ω for M, deﬁned on an an open neighborhood U ⊂ M of p and with values in an open set Ω ⊂ Rn × Rm−n, such that φ(U ∩ N ) = Ω ∩ (Rn × {0}). By Theorem 2.1.10 an m-manifold M ⊂ Rk in the sense of Deﬁnition 2.1.3 is a submanifold of Rk in the sense of Deﬁnition 2.8.20. By Theorem 2.3.4 the notion of a submanifold N ⊂ M of a manifold M ⊂ Rk in Deﬁnition 2.3.1 agrees with the notion of a submanifold in Deﬁnition 2.8.20. Exercise 2.8.21. Let N be a submanifold of M. Show that if M is Hausdorﬀ, so is N, and if M is paracompact, so is N. Exercise 2.8.22. Let N be a submanifold of M and let P be a submanifold of N. Prove that P is a submanifold of M. Hint: Use Theorem 2.1.10. Exercise 2.8.23. Let N be a submanifold of M. Prove that there exists an open set U ⊂ M such that N ⊂ U and N is closed in the relative topology of U. All the theorems we have proved for embedded manifolds and their proofs carry over almost word for word to the present setting. For example we have the inverse function theorem, the notion of a regular value, the notions of a submersion and of an immersion, the notion of an embedding as a proper injective immersion, and the fact from Theorem 2.3.4 that a subset P ⊂ M is a submanifold if and only if it is the image of an embedding. Exercise 2.8.24 (Lines in Euclidean space). The tangent bundle of the 2-sphere is the 4-manifold T S2 = (x, y) ∈ R3 × R3 \|x\| = 1, x, y = 0 (see Example
2.2.6). Deﬁne an equivalence relation on T S2 by y = y x = ±x, (x, y) ∼ (x, y) def⇐⇒ for (x, y), (x, y) ∈ T S2. Show that the quotient space T S2/∼ can be identiﬁed with the set L of all lines in R3, by assigning to each pair (x, y) ∈ T S2 the line x,y := y + tx t ∈ R ⊂ R3. Show that the space L of lines in R3 admits the unique structure of a smooth manifold such that the canonical projection T S2 → L : (x, y) → x,y is a submersion. Show that the manifold topology on L agrees with the quotient topology on T S2/∼. Show that the map L → RP2 × R3 : x,y → ([x], y) is an embedding. 96 CHAPTER 2. FOUNDATIONS Example 2.8.25 (Veronese embedding). The map CP2 → CP5 : [z0 : z1 : z2] → [z2 0 : z2 1 : z2 2 : z1z2 : z2z0 : z0z1] is an embedding. (Exercise: Prove this.) It restricts to an embedding of the real projective plane into RP5 and also gives rise to embeddings of RP2 into R4 as well as to the Roman surface: an immersion of RP2 into R3. (See Example 2.1.17.) There are similar embeddings CPn → CPN −1, N := n + d d, for all n and d, deﬁned in terms of monomials of degree d in n + 1 variables. These are the Veronese embeddings. Example 2.8.26 (Pl¨ucker embedding). The Grassmannian G2(R4) of 2-planes in R4 is a smooth 4-manifold and can be expressed as the quotient of the space F2(R4) of orthonormal 2-frames in R4 by the orthogonal group O(2). (See Example 2.8.8.) Write an orthonormal 2-frame in R4 as a
matrix D =     x0 y0 x1 y1 x2 y2 x3 y3    , DTD = 1l. Then the map f : G2(R4) → RP5, deﬁned by f ([D]) := [p01 : p02 : p03 : p23 : p31 : p12], pij := xiyj − xjyi, is an embedding and its image is the quadric X := f (G2(R4)) = p ∈ RP5 \| p01p23 + p02p31 + p03p12 = 0. (Exercise: Prove this.) There are analogous embeddings f : Gk(Rn) → RPN −1, N :=, n k for all k and n, deﬁned in terms of the k × k-minors of the (orthonormal) frames. These are the Pl¨ucker embeddings. 2.8. THE INTRINSIC DEFINITION OF A MANIFOLD* 97 2.8.5 Tangent Bundle and Vector Fields Let M be a m-manifold with an atlas A = {(φα, Uα)}α∈A. The tangent bundle of M is deﬁned as the disjoint union of the tangent spaces, i.e. T M := p∈M {p} × TpM = {(p, v) \| p ∈ M, v ∈ TpM }. Denote by π : T M → M the projection given by π(p, v) := p. Recall the notion of a submersion as a smooth map between smooth manifolds, whose derivative is surjective at each point. Lemma 2.8.27. The tangent bundle of M is a smooth 2m-manifold with coordinate charts φα : Uα := π−1(Uα) → φα(Uα) × Rm, φα(p, v) := (φα(p), dφα(p)v). The projection π : T M → M is a surjective submersion. If M is second countable and Hausdorﬀ, so is T
M. Proof. For each pair α, β ∈ A the set φα( Uα ∩ Uβ) = φα(Uα ∩ Uβ) × Rm is open in Rm × Rm and the transition map φβα := φβ ◦ φ−1 α : φα( Uα ∩ Uβ) → φβ( Uα ∩ Uβ) is given by φβα(x, ξ) = (φβα(x), dφβα(x)ξ) for x ∈ φα(Uα ∩ Uβ) and ξ ∈ Rm where φβα := φβ ◦ φ−1 α. Thus the transition maps are all diﬀeomorphisms and so the coordinate charts φα deﬁne an atlas on T M. The topology on T M is determined by this atlas via Deﬁnition 2.8.2. If M has a countable atlas, so does T M. The remaining assertions are easy exercises. Deﬁnition 2.8.28. Let M be a smooth m-manifold. A (smooth) vector ﬁeld on M is a collection of tangent vectors X(p) ∈ TpM, one for each point p ∈ M, such that the map M → T M : p → (p, X(p)) is smooth. The set of all smooth vector ﬁelds on M will be denoted by Vect(M ). 98 CHAPTER 2. FOUNDATIONS Associated to a vector ﬁeld is a smooth map M → T M whose composition with the projection π : T M → M is the identity map on M. Strictly speaking this map should be denoted by a symbol other than X, for example by X. However, it is convenient at this point, and common practice, to slightly abuse notation and denote the map from M to T M also by X. Thus a vector ﬁeld can be deﬁned as a smooth map such that X : M → T M π ◦ X = id : M → M. Such a map is also called a section of the tangent bundle. Now suppose A = {(φα, Uα
)}α∈A is an atlas on M and X : M → T M is a vector ﬁeld on M. Then X determines a collection of smooth maps Xα : φα(Uα) → Rm given by Xα(x) := dφα(p)X(p), p := φ−1 α (x), (2.8.9) for x ∈ φα(Uα). We can think of each Xα as a vector ﬁeld on the open set φα(Uα) ⊂ Rm, representing the vector ﬁeld X on the coordinate patch Uα. These local vector ﬁelds Xα satisfy the condition Xβ(φβα(x)) = dφβα(x)Xα(x) (2.8.10) for x ∈ φα(Uα ∩ Uβ). This equation can also be expressed in the form Xα\|φα(Uα∩Uβ ) = φ∗ βαXβ\|φβ (Uα∩Uβ ). (2.8.11) Conversely, any collection of smooth maps Xα : φα(Uα) → Rm satisfying (2.8.10) determines a unique vectorﬁeld X on M via (2.8.9). Thus we can deﬁne the Lie bracket of two vector ﬁelds X, Y ∈ Vect(M ) by [X, Y ]α(x) := [Xα, Yα](x) = dXα(x)Yα(x) − dYα(x)Xα(x) (2.8.12) for α ∈ A and x ∈ φα(Uα). It follows from equation (2.4.18) in Lemma 2.4.21 that the local vector ﬁelds [X, Y ]α : φα(Uα) → Rm satisfy (2.8.11) and hence determine a unique vector ﬁeld [X, Y ] on M via [X, Y ](p) := dφα(p)−1[Xα, Yα](φα(p)), p ∈ Uα.
(2.8.13) 2.8. THE INTRINSIC DEFINITION OF A MANIFOLD* 99 Thus the Lie bracket of X and Y is deﬁned on Uα as the pullback of the Lie bracket of the vector ﬁelds Xα and Yα under the coordinate chart φα. With this understood all the results in §2.4 about vector ﬁelds and ﬂows along with their proofs carry over word for word to the intrinsic setting whenever M is a Hausdorﬀ space. This includes the existence and uniquess result for integral curves in Theorem 2.4.7, the concept of the ﬂow of a vector ﬁeld in Deﬁnition 2.4.8 and its properties in Theorem 2.4.9, the notion of completeness of a vector ﬁeld (that the integral curves exist for all time), and the various properties of the Lie bracket such as the Jacobi identity (2.4.20), the formulas in Lemma 2.4.18, and the fact that the Lie bracket of two vector ﬁelds vanishes if and only if the corresponding ﬂows commute (see Lemma 2.4.26). One can also carry over the notion of a subbundle E ⊂ T M of rank n to the intrinsic setting by the condition that E is a smooth submanifold of T M and intersects each ﬁber TpM in an n-dimensional linear subspace Ep := {v ∈ TpM \| (p, v) ∈ E}. Then the characterization of subbundles in Theorem 2.6.10 and the theorem of Frobenius 2.7.2 including their proofs also carry over to the intrinsic setting whenever M is a Hausdorﬀ space. 2.8.6 Coordinate Notation Fix a coordinate chart φα : Uα → Rm on an m-manifold M. The components of φα are smooth real valued functions on the open subset Uα of M and it is customary to denote them by x1,..., xm : Uα → R. The derivatives of these functions at p ∈ Uα are linear functionals dxi(p) : TpM → R, i = 1,.
.., m. (2.8.14) They form a basis of the dual space T ∗ p M := Hom(TpM, R). (A coordinate chart on M can in fact be characterized as an m-tuple of real valued functions on an open subset of M whose derivatives are everywhere linearly independent and which, taken together, form an injective map.) The dual basis of TpM will be denoted by ∂ ∂x1 (p),..., ∂ ∂xm (p) ∈ TpM. (2.8.15) 100 Thus CHAPTER 2. FOUNDATIONS dxi(p) ∂ ∂xj (p) = δi j := 1, 0, if i = j, if i = j, for i, j = 1,..., m and ∂/∂xi is a vector ﬁeld on the coordinate patch Uα. For each p ∈ Uα it is the canonical basis of TpM determined by φα. In the notation of (2.8.5) and Remark 2.8.19 we have ∂ ∂xi (p) = [α, ei]p = dφα(p)−1ei, where ei = (0,..., 0, 1, 0,..., 0) (with 1 in the ith place) denotes the standard basis vector of Rm. In other words, for all ξ = (ξ1,..., ξm) ∈ Rm and all p ∈ Uα, the tangent vector is given by v := dφα(p)−1ξ ∈ TpM v = [α, ξ]p = m i=1 ξi ∂ ∂xi (p). (2.8.16) Thus the restriction of a vector ﬁeld X ∈ Vect(M ) to Uα has the form X\|Uα = m i=1 ξi ∂ ∂xi, where ξ1,..., ξm : Uα → R are smooth real valued functions. If the map Xα : φα(Uα) → Rm is deﬁned by (2.8.9), then Xα ◦ �
�−1 α = (ξ1,..., ξm). The above notation is motivated by the observation that the derivative of a smooth function f : M → R in the direction of a vector ﬁeld X on a coordinate patch Uα is given by LX f \|Uα = m i=1 ξi ∂f ∂xi. Here the term ∂f /∂xi is understood as ﬁrst writing f as a function of x1,..., xm, then taking the partial derivative, and afterwards expressing this partial derivative again as a function of p. Thus ∂f /∂xi is the shorthand notation for the function ∂ α ) ◦ φα : Uα → R. ∂xi (f ◦ φ−1 2.9. CONSEQUENCES OF PARACOMPACTNESS* 101 2.9 Consequences of Paracompactness* In geometry it is often necessary to turn a construction in local coordinates into a global geometric object. A key technical tool for such “local to global” constructions is an existence theorem for partitions of unity. 2.9.1 Paracompactness The existence of a countable atlas is of fundamental importance for almost everything we will prove about manifolds. The next two remarks describe several equivalent conditions. Remark 2.9.1. Let M be a smooth manifold and denote by U ⊂ 2M the topology induced by the atlas as in Deﬁnition 2.8.2. Then the following are equivalent. (a) M admits a countable atlas. (b) M is σ-compact, i.e. there is a sequence of compact subsets Ki ⊂ M such that Ki ⊂ int(Ki+1) for every i ∈ N and M = ∞ (c) Every open cover of M has a countable subcover. i=1 Ki. (d) M is second countable, i.e. there is a countable collection of open sets B ⊂ U such that every open set U ∈ U is a union of open sets from the collection B. (B is then called a countable base for the topology of M.) That (a) =⇒ (b) =⇒ (c) =⇒ (a) and
(a) =⇒ (d) follows directly from the deﬁnitions. The proof that (d) implies (a) requires the construction of a countable reﬁnement and the axiom of choice. (A reﬁnement of an open cover {Ui}i∈I is an open cover {Vj}j∈J such that each set Vj is contained in one of the sets Ui.) Remark 2.9.2. Let M and U be as in Remark 2.9.1 and suppose in addition that M is a connected Hausdorﬀ space. Then the existence of a countable atlas is also equivalent to each of the following conditions. (e) M is metrizable, i.e. there is a distance function d : M × M → [0, ∞) such that U is the topology induced by d. (f ) M is paracompact, i.e. every open cover of M has a locally ﬁnite reﬁnement. (An open cover {Vj}j∈J is called locally ﬁnite iﬀ every p ∈ M has a neighborhood that intersects only ﬁnitely many Vj.) 102 CHAPTER 2. FOUNDATIONS That (a) implies (e) follows from the Urysohn Metrization Theorem which asserts (in its original form) that every normal second countable topological space is metrizable [51, Theorem 34.1]. A topological space M is called normal iﬀ points are closed and, for any two disjoint closed sets A, B ⊂ M, there are disjoint open sets U, V ⊂ M such that A ⊂ U and B ⊂ V. It is called regular iﬀ points are closed and, for every closed set A ⊂ M and every b ∈ M \ A, there are disjoint open sets U, V ⊂ M such that A ⊂ U and b ∈ V. It is called locally compact iﬀ, for every open set U ⊂ M and every p ∈ U, there is a compact neighborhood of p contained in U. It is easy to show that every manifold is locally compact and every locally compact Hausdor�
� space is regular. Tychonoﬀ ’s Lemma asserts that a regular topological space with a countable base is normal [51, Theorem 32.1]. Hence it follows from the Urysohn Metrization Theorem that every Hausdorﬀ manifold with a countable base is metrizable. That (e) implies (f) follows from a more general theorem which asserts that every metric space is paracompact (see [51, Theorem 41.4] and [62]). Conversely, the Smirnov Metrization Theorem asserts that a paracompact Hausdorﬀ space is metrizable if and only if it is locally metrizable, i.e. every point has a metrizable neighborhood (see [51, Theorem 42.1]). Since every manifold is locally metrizable this shows that (f) implies (e). Thus we have (a) =⇒ (e) ⇐⇒ (f) for every Hausdorﬀ manifold. The proof that (f) implies (a) does not require the Hausdorﬀ property (A manifold with but we do need the assumption that M is connected. uncountably many connected components, each of which is paracompact, is itself paracompact but does not admit a countable atlas.) Here is a sketch. If M is a paracompact manifold, then there is a locally ﬁnite open cover {Uα}α∈A by coordinate charts. Since each set Uα has a countable dense subset, the set {α ∈ A \| Uα ∩ Uα0 = ∅} is at most countable for each α0 ∈ A. Since M is connected we can reach each point from Uα0 through a ﬁnite sequence of sets Uα1,..., Uα with Uαi−1 ∩ Uαi = ∅. This implies that the index set A is countable and hence M admits a countable atlas. Remark 2.9.3. A Riemann surface is a 1-dimensional complex manifold (i.e. the coordinate charts take values in C and the transition maps are holomorphic) with a Hausdorﬀ topology. It is a deep theorem in the theory of Riemann surfaces that every connected R
iemann surface is necessarily second countable (see [2]). Thus pathological examples of the type discussed in Example 2.8.10 cannot be constructed with holomorphic transition maps. 2.9. CONSEQUENCES OF PARACOMPACTNESS* 103 Exercise 2.9.4. Prove that every manifold is locally compact. Find an example of a manifold M and a point p0 ∈ M such that every closed neighborhood of p0 is non-compact. Hint: The example is necessarly non-Hausdorﬀ. Exercise 2.9.5. Prove that a manifold M admits a countable atlas if and only if it is σ-compact if and only if every open cover of M has a countable subcover if and only if it is second countable. Hint: The topology of Rm is second countable and every open subset of Rm is σ-compact. Exercise 2.9.6. Prove that every submanifold M ⊂ Rk (Deﬁnition 2.1.3) is second countable. Exercise 2.9.7. Prove that every connected component of a manifold M is an open subset of M and is path-connected. 2.9.2 Partitions of Unity Deﬁnition 2.9.8 (Partition of unity). Let M be a smooth manifold. A partition of unity on M is a collection of smooth functions θα : M → [0, 1], α ∈ A, such that each point p ∈ M has an open neighborhood V ⊂ M on which only ﬁnitely many θα do not vanish, i.e. # {α ∈ A \| θα\|V ≡ 0} < ∞, (2.9.1) and, for every p ∈ M, we have α∈A θα(p) = 1. (2.9.2) If {Uα}α∈A is an open cover of M, then a partition of unity {θα}α∈A (indexed by the same set A) is called subordinate to the cover iﬀ each θα is supported in Uα, i.e. supp(θα) := {p ∈ M \| θα(p) = 0} ⊂ U
α. Theorem 2.9.9 (Partitions of unity). Let M be a smooth manifold whose topology is paracompact and Hausdorﬀ. Then, for every open cover of M, there exists a partition of unity subordinate to that cover. The proof requires two preparatory lemmas. 104 CHAPTER 2. FOUNDATIONS Lemma 2.9.10. Let M be a smooth manifold with a Hausdorﬀ topology. Then, for every open set V ⊂ M and every compact set K ⊂ V, there exists a smooth function κ : M → [0, ∞) with compact support such that κ(p) > 0 for every p ∈ K and supp(κ) ⊂ V. Proof. Assume ﬁrst that K = {p0} is a single point. Since M is a manifold it is locally compact. Hence there is a compact neighborhood C ⊂ V of p0. Since M is Hausdorﬀ C is closed and hence the set U := int(C) is a neighborhood of p0 whose closure U ⊂ C is compact and contained in V. Shrinking U, if necessary, we may assume that there is a coordinate chart φ : U → Ω with values in some open neighborhood Ω ⊂ Rm of the origin such that φ(p0) = 0. (Here m is the dimension of M.) Now choose a smooth function κ0 : Ω → [0, ∞) with compact support such that κ0(0) > 0. Then the function κ : M → [0, 1], deﬁned by κ\|U := κ0 ◦ φ and κ(p) := 0 for p ∈ M \ U is supported in V and satisﬁes κ(p0) > 0. This proves the lemma in the case where K is a point. Now let K be any compact subset of V. Then, by the ﬁrst part of the proof, there is a collection of smooth functions κp : M → [0, ∞), one for every p ∈ K, such that κp(p) > 0 and supp(κp) ⊂ V. Since K is compact there are ﬁnitely
many points p1,..., pk ∈ K such that the sets p ∈ M \| κpj (p) > 0 cover K. Hence the function κ := j κpj is supported in V and is everywhere positive on K. This proves Lemma 2.9.10. Lemma 2.9.11. Let M be a topological space. If {Vi}i∈I is a locally ﬁnite collection of open sets in M, then Vi = i∈I0 V i i∈I0 for every subset I0 ⊂ I. i∈I0 V i is obviously contained in the closure of Proof. The set Vi. To prove the converse choose a point p0 ∈ M \ V i. Since the collection {Vi}i∈I is locally ﬁnite, there exists an open neighborhood U of p0 such that the set I1 := {i ∈ I \| Vi ∩ U = ∅} is ﬁnite. Hence the set i∈I0 i∈I0 U0 := U \ V i i∈I0∩I1 is an open neighborhood of p0 and we have U0 ∩ Vi = ∅ for every i ∈ I0. Hence p0 /∈ Vi. This proves Lemma 2.9.11. i∈I0 2.9. CONSEQUENCES OF PARACOMPACTNESS* 105 Proof of Theorem 2.9.9. Let {Uα}α∈A be an open cover of M. We prove in four steps that there is a partition of unity subordinate to this cover. The proofs of steps one and two are taken from [51, Lemma 41.6]. Step 1. There is a locally ﬁnite open cover {Vi}i∈I of M such that, for every i ∈ I, the closure V i is compact and contained in one of the sets Uα. Denote by V ⊂ 2M the set of all open sets V ⊂ M such that V is compact and V ⊂ Uα for some α ∈ A. Since M is a locally compact Hausdorﬀ space the collection V is an open cover of M. (If p ∈ M, then there is an α ∈ A such that p �
� Uα; since M is locally compact there is a compact neighborhood K ⊂ Uα of p; since M is Hausdorﬀ K is closed and thus V := int(K) is an open neighborhood of p with V ⊂ K ⊂ Uα.) Since M is paracompact the open cover V has a locally ﬁnite reﬁnement {Vi}i∈I. This cover satisﬁes the requirements of Step 1. Step 2. There is a collection of compact sets Ki ⊂ Vi, one for each i ∈ I, such that M = i∈I Ki. Denote by W ⊂ 2M the set of all open sets W ⊂ M such that W ⊂ Vi for some i. Since M is a locally compact Hausdorﬀ space, the collection W is an open cover of M. Since M is paracompact W has a locally ﬁnite reﬁnement {Wj}j∈J. By the axiom of choice there is a map such that J → I : j → ij W j ⊂ Vij ∀ j ∈ J. Since the collection {Wj}j∈J is locally ﬁnite, we have Ki := Wj = ij =i ij =i W j ⊂ Vi by Lemma 2.9.11. Since V i is compact so is Ki. Step 3. There is a partition of unity subordinate to the cover {Vi}i∈I. Choose a collection of compact sets Ki ⊂ Vi for i ∈ I as in Step 2. Then, by Lemma 2.9.10 and the axiom of choice, there is a collection of smooth functions κi : M → [0, ∞) with compact support such that supp(κi) ⊂ Vi, κi\|Ki > 0 ∀ i ∈ I. 106 CHAPTER 2. FOUNDATIONS Since the cover {Vi}i∈I is locally ﬁnite the sum κ := i∈I κi : M → R is locally ﬁnite (i.e. each point in M has a neighborhood in which only ﬁnitely many terms do not vanish) and thus deﬁnes a smooth
function on M. This function is everywhere positive, because each summand is nonnegative and, for each p ∈ M, there is an i ∈ I with p ∈ Ki so that κi(p) > 0. Thus the funtions χi := κi/κ deﬁne a partition of unity satisfying supp(χi) ⊂ Vi for every i ∈ I as required. Step 4. There is a partition of unity subordinate to the cover {Uα}α∈A. Let {χi}i∈I be the partition of unity constructed in Step 3. By the axiom of choice there is a map I → A : i → αi such that Vi ⊂ Uαi for every i ∈ I. For α ∈ A deﬁne θα : M → [0, 1] by θα := αi=α χi. Here the sum runs over all indices i ∈ I with αi = α. This sum is locally ﬁnite and hence is a smooth function on M. Moreover, each point in M has an open neighborhood in which only ﬁnitely many of the θα do not vanish. Hence the sum of the θα is a well deﬁned function on M and α∈A θα = α∈A αi=α χi = i∈I χi ≡ 1. This shows that the functions θα form a partition of unity. To prove the inclusion supp(θα) ⊂ Uα we consider the open sets Wi := {p ∈ M \| χi(p) > 0} for i ∈ I. Since Wi ⊂ Vi this collection is locally ﬁnite. Hence, by Lemma 2.9.11, we have supp(θα) = Wi = W i = supp(χi) ⊂ αi=α αi=α αi=α αi=α Vi ⊂ Uα. This proves Theorem 2.9.9. 2.9. CONSEQUENCES OF PARACOMPACTNESS* 107 2.9.3 Embedding in Euclidean Space Theorem 2.9.12. Let M be a second countable smooth m-manifold with a Hausdor
ﬀ topology. Then there exists an embedding f : M → R2m+1 with a closed image. Proof. The proof has ﬁve steps. Step 1. Let U ⊂ M be an open set and let K ⊂ U be a compact set. Then there exists an integer k ∈ N, a smooth map f : M → Rk, and an open set V ⊂ M, such that K ⊂ V ⊂ U, the restriction f \|V : V → Rk is an injective immersion, and f (p) = 0 for all p ∈ M \ U. Choose a smooth atlas A = {(φα, Uα)}α∈A on M such that, for each α ∈ A, either Uα ⊂ U or Uα ∩ K = ∅. Since M is a paracompact Hausdorﬀ manifold, Theorem 2.9.9 asserts that there exists a partition of unity {θα}α∈A subordinate to the open cover {Uα}α∈A of M. Since the sets {p ∈ Uα \| θα(p) > 0} with Uα ⊂ U form an open cover of K and K is a compact subset of M, there exist ﬁnitely many indices α1,..., α ∈ A such that K ⊂ p ∈ M θα1(p) + · · · + θα(p) > 0 =: V ⊂ U. Let k := (m + 1) and, for i = 1,...,, abbreviate φi := φαi, θi := θαi. Deﬁne the smooth map f : M → Rk by f (p) :=        θ1(p) θ1(p)φ1(p)... θ(p) θ(p)φ(p)        for p ∈ M. Then the restriction f \|V : V → Rk is injective. Namely, if p0, p1 ∈ V satisfy f
(p0) = f (p1), then I := i θi(p0) > 0 = i θi(p1) > 0 = ∅ and, for i ∈ I, we have θi(p0) = θi(p1), hence φi(p0) = φi(p1), and so p0 = p1. Moreover, for every p ∈ V the derivative df (p) : TpM → Rk is injective, and this proves Step 1. 108 CHAPTER 2. FOUNDATIONS Step 2. Let f : M → Rk be an injective immersion and let A ⊂ R(2m+1)×k be a nonempty open set. Then there exists a matrix A ∈ A such that the map Af : M → R2m+1 is an injective immersion. The proof of Step 2 uses the Theorem of Sard (see [1, 50]). The sets W0 := (p, q) ∈ M × M W1 := (p, v are open subsets of smooth second countable Hausdorﬀ 2m-manifolds and the maps F0 : A × W0 → R2m+1, F1 : A × W1 → R2m+1, deﬁned by F0(A, p, q) := A(f (p) − f (q)), F1(A, p, v) := Adf (p)v for A ∈ A, (p, q) ∈ W0, and (p, v) ∈ W1, are smooth. Moreover, the zero vector in R2m+1 is a regular value of F0 because f is injective and of F1 because f is an immersion. Hence it follows from the intrinsic analogue of Theorem 2.2.19 that the sets M0 := F −1 M1 := F −1 0 1 (0) = (A, p, q) ∈ A × W0 (0) = (A, p, v) ∈ A × W1 Af (p) = Af (q), Adf (p)v = 0 are smooth manifolds of dimension dim M0 = dim M1 = (2m + 1)k − 1. Since M is a second countable Hausdorﬀ manifold,
so are M0 and M1. Hence the Theorem of Sard asserts that the canonical projections M0 → A : (A, p, q) → A =: π0(A, p, q), M1 → A : (A, p, v) → A =: π1(A, p, v), have a common regular value A ∈ A. Since dim M0 = dim M1 < dim A, this implies A ∈ A \ (π0(M0) ∪ π1(M1)). Hence Af : M → R2m+1 is an injective immersion and this proves Step 2. 2.9. CONSEQUENCES OF PARACOMPACTNESS* 109 If M is compact, the result follows from Steps 1 and 2 with K = U = M. In the noncompact case the proof requires two more steps to construct an embedding into R4m+4 and a further step to reduce the dimension to 2m + 1. Step 3. Assume M is not compact. Then there exists a sequence of open sets Ui ⊂ M, a sequence of smooth functions ρi : M → [0, 1], and a sequence of compact sets Ki ⊂ Ui such that supp(ρi) ⊂ Ui, Ki = ρ−1 i (1) ⊂ Ui, Ui ∩ Uj = ∅ for all i, j ∈ N with \|i − j\| ≥ 2 and M = ∞ i=1 Ki. Since M is second countable, there exists a sequence of compact sets Ci ⊂ M such that Ci ⊂ int(Ci+1) for all i ∈ N and M = i∈N Ci (Remark 2.9.1). Deﬁne the compact sets Bi ⊂ M by C0 := ∅ and Bi := Ci \ Ci−1 for i ∈ N. Then M = i∈N Bi and, for all i, j ∈ N with j ≥ i + 2, we have Bi ⊂ Ci ⊂ int(Cj−1), Bj ⊂ Cj \ int(Cj−1) and so Bi ∩ Bj = ∅. Since M is metrizable by Remark 2.9.2, there exists a distance function d : M × M → [
0, ∞) that induces the intrinsic topology on M. Deﬁne Ai := j∈N\{i−1,i,i+1} Bj, εi := d(Ai, Bi) = inf p∈Ai,q∈Bi d(p, q). Then Ai is a closed subset of M, because any convergent sequence in M must belong to a ﬁnite union of the Bj. Since Ai ∩ Bi = ∅, this implies εi > 0. For i ∈ N deﬁne the set Ui ⊂ M by Ui := p ∈ M there exists a q ∈ Bi with d(p, q) < εi/3. Then {Ui}i∈N is a sequence of open subsets of M such that Bi ⊂ Ui ⊂ Ci+1 for all i ∈ N and Ui ∩ Uj = ∅ for \|i − j\| ≥ 2. In particular, each set Ui has a compact closure. For each i there exists of a partition of unity subordinate to the open cover M = Ui ∪ (M \ Bi) and hence a smooth function ρi : M → [0, 1] such that supp(ρi) ⊂ Ui and ρi\|Bi ≡ 1. Deﬁne Ki := ρ−1 i (1) = {p ∈ Ui \| ρi(p) = 1} for i ∈ N. Then Ki is a compact set and Bi ⊂ Ki ⊂ Ui for each i ∈ N. Hence M = i∈N Ki and this proves Step 3. 110 CHAPTER 2. FOUNDATIONS Step 4. Assume M is not compact. Then there exists an embedding f : M → R4m+4 with a closed image and a pair of orthonormal vectors x, y ∈ R4m+4 such that, for every ε > 0, there exists a compact set K ⊂ M with sup p∈M \K inf s,t∈R f (p) \|f (p)\| − sx − ty < ε. (2.9.3) Assume M is not compact and let Ki, Ui, ρi be as in Step 3. Then, by Steps
1 and 2, there exists a sequence of smooth maps gi : M → R2m+1 such that gi\|M \Ui ≡ 0, the restriction gi\|Ki : Ki → R2m+1 is injective, and the derivative dgi(p) : TpM → R2m+1 is injective for all p ∈ Ki and all i ∈ N. Let ξ ∈ R2m+1 be a unit vector and deﬁne the maps fi : M → R2m+1 by  fi(p) := ρi(p) iξ +   gi(p) 1 + \|gi(p)\|2 (2.9.4) for p ∈ M and i ∈ N. Then the restriction fi\|Ki : Ki → R2m+1 is injective, the derivative dfi(p) : TpM → R2m+1 is injective for all p ∈ Ki, and supp(fi) ⊂ Ui, fi(Ki) ⊂ B1(iξ), fi(M ) ⊂ Bi+1(0). Deﬁne the maps f odd, f ev : M → R2m+1 and ρodd, ρev : M → R by ρodd(p) := f odd(p) := ρev(p) := f ev(p) := ρ2i−1(p), 0, f2i−1(p), 0, ρ2i(p), 0, f2i(p), 0, if i ∈ N and p ∈ U2i−1, if p ∈ M \ i∈N U2i−1, if i ∈ N and p ∈ U2i−1, if p ∈ M \ i∈N U2i−1, if i ∈ N and p ∈ U2i, if p ∈ M \ i∈N U2i, if i ∈ N and p ∈ U2i, if p ∈ M \ i∈N U2i, and deﬁne the map f : M → R4m+4 by f (p) := ρodd(p), f odd(p), ρev
(p), f ev(p) for p ∈ M. 2.9. CONSEQUENCES OF PARACOMPACTNESS* 111 We prove that f is injective. To see this, note that p ∈ K2i−1 =⇒ 2i − 2 < f odd(p) \|f ev(p)\| < 2i + 1, < 2i, 2i − 1 < \|f ev(p)\| < 2i + 1, (2.9.5) =⇒ p ∈ K2i < 2i + 2, f odd(p) Now let p0, p1 ∈ M such that f (p0) = f (p1). Assume ﬁrst that p0 ∈ K2i−1. Then ρodd(p1) = ρodd(p0) = 1 and hence p1 ∈ j∈N K2j−1. By (2.9.5), we also have 2i − 2 < \|f odd(p1)\| = \|f odd(p0)\| < 2i and hence p1 ∈ K2i−1. This implies f2i−1(p1) = f odd(p1) = f odd(p0) = f2i−1(p0) and so p0 = p1. Now assume p0 ∈ K2i. Then ρev(p1) = ρev(p0) = 1 and hence p1 ∈ j∈N K2j. By (2.9.5), we also have 2i − 1 < \|f ev(p1)\| = \|f ev(p0)\| < 2i + 1, so p1 ∈ K2i, which implies f2i(p1) = f ev(p1) = f ev(p0) = f2i(p0), and so again p0 = p1. This shows that f is injective. That f is an immersion follows from the fact that the derivative dfi(p) is injective for all p ∈ Ki and all i ∈ N. We prove that f is proper and has a closed image. Let (pν)ν∈N be a sequence in M such that the sequence (f (pν))ν∈N in R4m+4 is bounded.
Choose i ∈ N such that \|f odd(pν)\| < 2i and \|f ev(pν)\| < 2i + 1 for all ν ∈ N. Then pν ∈ 2i j=1 Kj for all ν ∈ N by (2.9.5). Hence (pν)ν∈N has a convergent subsequence. Thus f : M → R4m+4 is an embedding with a closed image. Next consider the pair of orthonormal vectors x := (0, ξ, 0, 0), y := (0, 0, 0, ξ) in R4m+4 = R × R2m+1 × R × R2m+1. Let (pν)ν∈N be a sequence in M that does not have a convergent subsequence and choose a sequence iν ∈ N such that pν ∈ K2iν −1 ∪ K2iν for all ν ∈ N. Then iν tends to inﬁnity. If pν ∈ K2iν −1 for all ν, then we have lim supν→∞\|f odd(pν)\|−1\|f ev(pν)\| ≤ 1 by (2.9.5). Passing to a subsequence, still denoted by (pν)ν∈N, we may assume that the limit λ := limν→∞\|f odd(fν)\|−1\|f ev(pν)\| exists. Then 0 ≤ λ ≤ 1, f odd(pν) \|f (pν)\| lim ν→∞ = √ 1 1 + λ2, lim ν→∞ \|f ev(pν)\| \|f (pν)\| = √ λ 1 + λ2, and it follows from (2.9.4) that lim ν→∞ f odd(pν) \|f odd(pν)\| = ξ, lim ν→∞ f ev(pν) \|f odd(pν)\| = λξ. This implies lim ν→∞ f (pν) \|f (pν)\| = 0, √ ξ 1 + λ2, 0, √ λξ 1 + λ2 = √ 1 1 + λ2 x + �
� λ 1 + λ2 y. 112 CHAPTER 2. FOUNDATIONS Similarly, if pν ∈ K2iν for all ν, there exists a subsequence such that the limit λ := limν→∞\|f ev(pν)\|−1\|f odd(pν)\| exists and, by (2.9.4), this implies lim ν→∞ f (pν) \|f (pν)\| = 0, √ λξ 1 + λ2, 0, √ ξ 1 + λ2 = √ λ 1 + λ2 x + √ 1 1 + λ2 y. This shows that the vectors x and y satisfy the requirements of Step 4. Step 5. There exists an embedding f : M → R2m+1 with a closed image. For compact manifolds the result was proved in Steps 1 and 2 and for m = 0 the assertion is obvious, because then M is a ﬁnite or countable set with the discrete topology. Thus assume that M is not compact and m ≥ 1. Choose f : M → R4m+4 and x, y ∈ R4m+4 as in Step 4 and deﬁne A := A ∈ R(2m+1)×(4m+4) the vectors Ax and Ay are linearly independent. Since m ≥ 1, this is a nonempty open subset of R(2m+1)×(4m+4). We prove that the map Af : M → R2m+1 is proper and has a closed image for every A ∈ A. To see this, ﬁx a matrix A ∈ A. Let (pν)ν∈N be a sequence in M that does not have a convergent subsequence. Then by Step 4 there exists a subsequence, still denoted by (pν)ν∈N, and real numbers s, t ∈ R such that s2 + t2 = 1, This implies lim ν→∞ f (pν) \|f (pν)\| lim ν→∞ Af (pν) \|f (pν)\| = sx + ty, lim ν→∞ \|f (pν)\| = ∞. = sAx + tAy = 0 and hence limν→∞ \|Af (pν
)\| = ∞. Thus the preimage of every compact subset of R2m+1 under the map Af : M → R2m+1 is a compact subset of M, and hence Af is proper and has a closed image (Remark 2.3.3). Now it follows from Step 2 that there exists a matrix A ∈ A such that the map Af : M → R2m+1 is an injective immersion. Hence it is an embedding with a closed image. This proves Step 5 and Theorem 2.9.12. The Whitney Embedding Theorem asserts that every second countable Hausdorﬀ m-manifold M admits an embedding f : M → R2m. The proof is based on the Whitney Trick and goes beyond the scope of this book. The next exercise shows that Whitney’s theorem is sharp. Remark 2.9.13. The manifold RP2 cannot be embedded into R3. The same is true for the Klein bottle K := R2/ ≡ where the equivalence relation is given by [x, y] ≡ [x + k, − y] for x, y ∈ R and k, ∈ Z. 2.9. CONSEQUENCES OF PARACOMPACTNESS* 113 2.9.4 Leaves of a Foliation Let M be an m-dimensional paracompact Hausdorﬀ manifold and E ⊂ T M be an integrable subbundle of rank n. Let L ⊂ M be a closed leaf of the foliation determined by E. Then L is a smooth n-dimensional submanifold of M. Here is a sketch of David Epstein’s proof of this fact in [19]. (a) The space L with the intrinsic topology admits the structure of a manifold such that the obvious inclusion ι : L → M is an injective immersion. This is an easy exercise. For the deﬁnition of the intrinsic topology see Exercise 2.7.10. The dimension of L is n. (b) If f : X → Y is a continuous map between topological spaces such that Y is paracompact and there is an open cover {Vj}j∈J of Y such that f −1(Vj) is paracompact for each j, then X is paracompact. To see this, we may assume that the
cover {Vj}j∈J is locally ﬁnite. Now let {Uα}α∈A be an open cover of X. Then the sets Uα ∩ f −1(Vj) deﬁne an open cover of f −1(Vj). Choose a locally ﬁnite reﬁnement {Wij}i∈Ij of this cover. Then the open cover {Wij}j∈J, i∈Ij of M is a locally ﬁnite reﬁnement of {Uα}α∈A. (c) The intrinsic topology of L is paracompact. This follows from (b) and the fact that the intersection of L with every foliation box is paracompact in the intrinsic topology. (d) The intrinsic topology of L is second countable. This follows from (a) and (c) and the fact that every connected paracompact manifold is second countable (see Remark 2.9.2). (e) The intersection of L with a foliation box consists of at most countably many connected components. This follows immediately from (d). (f ) If L is a closed subset of M, then the intersection of L with a foliation box has only ﬁnitely many connected components. To see this, we choose a transverse slice of the foliation at p0 ∈ L, i.e. a connected submanifold T ⊂ M through p0, diﬀeomorphic to an open ball in Rm−n, whose tangent space at each point p ∈ T is a complement of Ep. By (d) we have that T ∩ L is at most countable. If this set is not ﬁnite, even after shrinking T, there must be a sequence pi ∈ (T ∩L)\{p0} converging to p0. Using the holonomy of the leaf (obtained by transporting transverse slices along a curve via a lifting argument) we ﬁnd that every point p ∈ T ∩ L is the limit point of a sequence in (T ∩ L) \ {p}. Hence the one-point set {p} has empty interior in the relative topology of T ∩ L for each p ∈ T ∩ L. Thus T
∩ L is a countable union of closed subsets with empty interior. Since T ∩L admits the structure of a complete metric space, this contradicts the Baire category theorem. (g) It follows immediately from (f) that L is a submanifold of M. 114 CHAPTER 2. FOUNDATIONS 2.9.5 Principal Bundles An interesting class of foliations arises from smooth Lie group actions. Let G ⊂ GL(N, R) be a compact Lie group and let P be a smooth mmanifold whose topology is Hausdorﬀ and second countable. A smooth (contravariant) G-action on P is a smooth map that satisﬁes the conditions P × G → P : (p, g) → pg (pg)h = p(gh), p1l = p (2.9.6) (2.9.7) for all p ∈ P and all g, h ∈ G. Fix any such group action. Then every group element g ∈ G determines a diﬀeomorphism P → P : p → pg, whose derivative at p ∈ P is denoted by TpP → TpgP : v → vg. Every Lie algebra element ξ ∈ g := Lie(G) = T1lG determines a vector ﬁeld Xξ ∈ Vect(P ) which assigns to each p ∈ P the tangent vector d dt Xξ(p) := pξ := t=0 The linear map g → Vect(P ) : ξ → Xξ is called the inﬁnitesimal action. It is a Lie algebra anti-homomorphism because the group action is contravariant. (Exercise: Prove that [Xξ, Xη] = −X[ξ,η] for ξ, η ∈ g.) The group action (2.9.6) is said to be with ﬁnite isotropy iﬀ the isotropy subgroup p exp(tξ) ∈ TpP. (2.9.8) Gp := {g ∈ G \| pg = p} is ﬁnite for all p ∈ P. The isotropy subgroup Gp is a Lie subgroup of G with Lie algebra
gp := Lie(Gp) = {ξ ∈ g \| Xξ(p) = 0}. Since G is compact, this shows that Gp is a ﬁnite subgroup of G if and only if gp = {0} or, equivalently, the map g → TpP : ξ → Xξ(p) = pξ is injective. Thus, in the ﬁnite isotropy case, the group action determines an involutive subbundle E ⊂ T P with the ﬁbers Ep := pg = {Xξ(p) \| ξ ∈ g} for p ∈ P. When G is connected, the leaves of the corresponding foliation are the group orbits pG := {pg \| g ∈ G}. These are the elements of the orbit space P/G := {pG \| p ∈ P }. There is a natural projection π : P → P/G deﬁned by π(p) := pG for p ∈ P and the orbit space P/G is equipped with the quotient topology (a subset U ⊂ P/G is open if and only if π−1(U ) is an open subset of P ). The group action is called free iﬀ Gp = {1l} for all p ∈ P. The next theorem shows that, in the case of a free action, the quotient space admits a unique smooth structure such that the projection π : P → P/G is a submersion. 2.9. CONSEQUENCES OF PARACOMPACTNESS* 115 Theorem 2.9.14 (Principal Bundle). Let P be a smooth m-manifold whose topology is Hausdorﬀ and second countable. Suppose P is equipped with a smooth contravariant action of a compact Lie group G and assume the group action is free. Then dim(G) ≤ m and B := P/G admits a unique smooth structure such that the projection π : P → B is a submersion. The intrinsic topology of B, induced by the smooth structure, agrees with the quotient topology, and it is Hausdorﬀ and second countable. Proof. For each p ∈ P the map G → P : g → pg is an embedding and this implies k := dim
(G) ≤ dim(P ) = m. Deﬁne n := m − k. A local slice of the group action is a smooth map ι : Ω → P, deﬁned on an open set Ω ⊂ Rn, such that the map Ω × G → P : (x, g) → ι(x)g is an embedding. With this understood, we prove the assertions in ﬁve steps. Step 1. For every p0 ∈ P there exists a local slice ι0 : Ω0 → P, deﬁned on an open neighborhood Ω0 ⊂ Rn of the origin, such that ι0(0) = p0. Choose a coordinate chart φ : V → Rm on an open neighborhood V ⊂ P of p0 such that φ(p0) = 0 and φ(V ) = Rm. Deﬁne v1,..., vm ∈ Tp0P by dφ(p0)vi := ei for i = 1,..., m, where e1,..., em is the standard basis of Rm. Reorder the coordinates on Rm, if necessary, such that the vectors v1,..., vn project to a basis of the quotient space Tp0P/p0g. Deﬁne ι : Rn → P by ι(x1,..., xn) := φ−1(x1,..., xn, 0,..., 0) and deﬁne the map ψ : Rn × G → P by ψ(x, g) := ι(x)g for x ∈ Rn and g ∈ G. Then ψ is smooth and its derivative dψ(0, 1l) : Rn × g → Tp0P is given by dψ(0, 1l)(x, ξ) = n i=1 xivi + p0ξ for x = (x1,..., xn) ∈ Rn and ξ ∈ g. Hence dψ(0, 1l) is bijective and so it follows from the Inverse Function Theorem 2
.2.17 that there exist open neighborhoods Ω ⊂ Rn of 0, Ω1 ⊂ G of 1l, and W ⊂ P of p0 such that the restricted map ψ1 := ψ\|Ω×Ω1 : Ω × Ω1 → W is a diﬀeomorphism. 116 CHAPTER 2. FOUNDATIONS Next we prove that there exists an open neigborhood Ω0 ⊂ Ω of the origin such that the restricted map ψ0 := ψ\|Ω0×G : Ω0 × G → P is injective. Suppose otherwise that no such neighborhood Ω0 exists. Then there exist sequences (xi, gi), (x i, g i) and ψ(xi, gi) = ψ(x i)i∈N in Ω converge to the origin. Since G is compact we may assume, by passing to a subsequence if necessary, that the sequences (gi)i∈N and (g i)i∈N converge. Denote the limits by i) for all i and the sequences (xi)i∈N and (x i) ∈ Ω × G such that (xi, gi) = (x i, g i, g g := lim i→∞ gi ∈ G, g := lim i→∞ g i ∈ G. Then p0g = lim i→∞ ι(xi)gi = lim i→∞ ι(x i)g i = p0g and so g = g because the group action is free. Thus the sequence (g )i∈N in G converges to 1l and hence belongs to the set Ω1 for i suﬃciently large. Since ig−1 i ) i, g ig−1 ig−1 i i = ψ1(x ψ1(xi, 1l) = ι(xi) = ι(x i)g for all i, this contradicts the injectivity of ψ1. Thus we have proved that the map ψ0 : Ω0 × G → P is injective for a suitable neighborhood Ω0 ⊂ Ω of the origin. That it is an immersion is a direct consequence of the
formula dψ0(x, g)(x, g) = dι(x)x + ι(x)(gg−1)g = dψ0(x, 1l)(x, gg−1)g for all x ∈ Ω0, x ∈ Rn, g ∈ G, and g ∈ TgG, and the fact that the derivative dψ0(x, 1l) is bijective for all x ∈ Ω0 (even for all x ∈ Ω). Thus we have proved that ψ0 : Ω0 × G → P is an injective immersion. Shrinking Ω0 further, if necessary, we may assume that Ω0 has a compact closure and that ψ is injective on Ω0 × G. This implies that ψ0 is proper. Namely, if (xi, gi)i∈N is a sequence in Ω0 × G and (x, g) ∈ Ω0 × G such that ψ0(x, g) = limi→∞ ψ0(xi, gi), then there is a subsequence (xiν, giν )ν∈N that converges to a pair (x, g) ∈ Ω0 × G. This subsequence satisﬁes ψ(x, g) = lim ν→∞ ψ0(xiν, giν ) = ψ(x, g). Since ψ is injective on Ω0 × G, this implies x = x and g = g. Thus every subsequence of (xi, gi)i∈N has a further subsequence that converges to (x, g) and so the sequence (xi, gi)i∈N itself converges to (x, g). Thus the map ψ0 : Ω0 × G → P is a proper injective immersion and this proves Step 1. 2.9. CONSEQUENCES OF PARACOMPACTNESS* 117 Step 2. Let ι : Ω → P be a local slice. Then the set U := π(ι(Ω)) ⊂ B is open in the quotient topology and the map π ◦ ι : �
� → U is a homeomorphism with respect to the quotient topology on U. The map ψ : Ω × G → P, deﬁned by ψ(x, g) := ι(x)g for x ∈ Ω and g ∈ G, is an embedding. Hence W := ψ(Ω × G) is an open G-invariant subset of P and ψ : Ω × G → W is a G-equivariant homeomorphism. Moreover, for every element p ∈ P, we have π(p) ∈ U if and only if there exists an element x ∈ Ω and an element g ∈ G such that p = ι(x)g = ψ(x, g). Thus π−1(U ) = ψ(Ω × G) = W is an open subset of P, and so U is an open subset of B = P/G with respect to the quotient topology. The continuity of π ◦ ι : Ω → U follows directly from the deﬁnition. Moreover, if Ω ⊂ Ω is an open set and U := π(ι(Ω)), then π−1(U ) = ψ(Ω × G) is open by the same argument, and so U ⊂ B is open with respect to the quotient topology. Thus π ◦ ι : Ω → U is a homeomorphism and this proves Step 2. Step 3. By Step 1 there exists a collection ια : Ωα → P, α ∈ A, of local slices such that the sets Uα := π(ια(Ωα)) cover the orbit space B = P/G. For α ∈ A deﬁne φα := (π ◦ ια)−1 : Uα → Ωα. Then A = {(φα, Uα)}α∈A is a smooth structure on B which renders the canonical projection π : P → B into a submersion. Moreover, this smooth structure is compatible with the quotient topology on B. For α, β ∈ A deﬁne Ωαβ := φα(
Uα ∩ Uβ) and φβα := φβ ◦ φ−1 α : Ωαβ → Ωβα. We must prove that φβα is smooth. To see this, deﬁne ψα : Ωα × G → P by ψα(x, g) := ια(x)g for α ∈ A, x ∈ Ωα, and g ∈ G. Then ψα is a diﬀeomorphism onto its image and ψα(Ωαβ × G) = ψβ(Ωβα × G) = π−1(Uα ∩ Uβ). For x ∈ Ωαβ the element φβα(x) ∈ Ωβα is the projection of ψ−1 β ◦ ψα(x, 1l) onto the ﬁrst factor. Thus φβα is smooth and so is its inverse φαβ. This shows that {(Uα, φα)}α∈A is a smooth structure on B. Second, π is a submersion with respect to this smooth structure, because φα ◦ π ◦ ψα(x, g) = x for all α ∈ A, all x ∈ Ωα, and all g ∈ G. Third, this smooth structure is compatible with the quotient topology by Step 2. This proves Step 3. Step 4. There is only one smooth structure on B with respect to which the projection π : P → B is a submersion. Fix any smooth structure on B for which the projection π : P → B is a submersion. Then the dimension of B is n = dim(P ) − dim(G), and so the smooth structure consists of bijections φα : Uα → Ωα from subsets Uα ⊂ B onto open sets Ωα ⊂ Rn such that the sets Uα cover B and the transition maps are diﬀeomorphisms between open subsets of Rn. 118 CHAPTER 2. FOUNDATIONS We prove that the intrinsic topology on B agrees with the quotient topol- ogy. To see this, ﬁx a subset
U ⊂ B. Then the following are equivalent. (a) U is open with respect to the intrinsic topology on B. (b) φα(U ∩ Uα) is open in Rn for all α ∈ A. (c) π−1(U ∩ Uα) is open in P for all α ∈ A. (d) π−1(U ) is open in P. (e) U is open with respect to the quotient topology on B. The equivalence of (a) and (b) follows from the deﬁnition of the intrinsic topology. That (b) implies (c) follows from the three observations that the set π−1(Uα) is open in P, the map φα ◦ π : π−1(Uα) → Ωα is continuous, and (φα ◦ π)−1(φα(U ∩ Uα)) = π−1(U ∩ Uα). That (c) implies (b) follows from the fact that the map φα ◦ π : π−1(Uα) → Ωα is a submersion and hence maps the open set π−1(U ∩ Uα) onto an open subset of Ωα (Corollary 2.6.2). The equivalence of (c) and (d) follows from the fact that the map π : P → B is continuous and Uα ⊂ B is open (both with respect to the intrinsic topology on B) and so π−1(Uα) is open in P for all α ∈ A. The equivalence of (d) and (e) follows from the deﬁnition of the quotient topology on B. Now let ι : Ω → P be a local slice and deﬁne the set U := π(ι(Ω)) ⊂ B and the map φ := (π ◦ ι)−1 : U → Ω. Then the composition φα ◦ φ−1 = φα ◦ π ◦ ι : φ(U ∩ Uα) → φα(U ∩ Uα) is a homeomorphism between open
subsets of Rn. Moreover, φα ◦ φ−1 is the composition of the smooth maps ι : {x ∈ Ω \| π(ι(x)) ∈ Uα} → π−1(U ∩ Uα), π : π−1(U ∩ Uα) → U ∩ Uα, and φα : U ∩ Uα → φα(U ∩ Uα). So φα ◦ φ−1 is smooth and its derivative is everywhere bijective because π is a submersion and the kernel of dπ(ι(x)) is transverse to the image of dι(x). Thus φα ◦ φ−1 is a diﬀeomorphism by the Inverse Function Theorem and this proves Step 4. Step 5. The quotient topology on B is a Hausdorﬀ and second countable. Let ια : Ωα → P for α ∈ A be a collection of local slices such that the sets Uα := π(ια(Ωα)) cover B. Then the open sets π−1(Uα) form an open cover of P and so there is a countable subcover. Thus B is second countable. To prove that B is Hausdorﬀ, ﬁx two distinct elements b0, b1 ∈ B and choose p0, p1 ∈ P such that π(p0) = b0 and π(p1) = b1. Then p0G and p1G are disjoint compact subsets of P and hence can be separated by disjoint open subsets U0, U1 ⊂ P, because P is a Hausdorﬀ space. Now for i = 0, 1 the set Vi := {p ∈ P \| pG ⊂ Ui} is open (exercise) and contains the orbit piG. Hence W0 := π(V0) and W1 := π(V1) are disjoint open subsets of B such that b0 ∈ W0 and b1 ∈ W1. This proves Step 5 and Theorem 2.9.14. 2.9. CONSEQUENCES OF PARACOMPACTNESS* 119 Example 2
.9.15. There are many important examples of free group actions and principal bundles. A class of examples arises from orthonormal frame bundles (§3.4). The complex projective space B = CPn arises from the action of the circle G = S1 on the unit sphere P = S2n+1 ⊂ Cn+1 (Example 2.8.5). The real projective space B = RPn arises from the action of the ﬁnite group G = Z/2Z on the unit sphere P = Sn ⊂ Rn+1 (Example 2.8.6). The complex Grassmannian B = Gk(Cn) arises from the action of G = U(k) on the space P = Fk(Cn) of unitary k-frames in Cn (Example 3.7.6). If G is a Lie group and K ⊂ G is a compact subgroup, then by Theorem 2.9.14 the homogeneous space G/K admits a unique smooth structure such that the projection π : G → G/K is a submersion. The example SL(2, C)/SU(2) can be identiﬁed with hyperbolic 3-space (§6.4.3), the example U(n)/O(n) can be identiﬁed with the space of Lagrangian subspaces of a symplectic vector space ([49, Lemma 2.3.2]), the example Sp(2n)/U(n) can be identiﬁed with Siegel upper half space or the space of compatible linear complex structures on a symplectic vector space (Exercise 6.5.24 and [49, Lemma 2.5.12]), and the example G2/SO(4) can be identiﬁed with the associative Grassmannian ([68, Remark 8.4]). Standing Assumption We have seen that all the results in the present chapter carry over to the intrinsic setting, assuming that the topology of M is Hausdorﬀ and paracompact. In fact, in many cases it is enough to assume the Hausdorﬀ property. However, these results mainly deal with introducing the basic concepts such as smooth maps, embeddings, submersions, vector ﬁelds, ﬂows, and
verifying their elementary properties, i.e. with setting up the language for diﬀerential geometry and topology. When it comes to the substance of the subject we shall deal with Riemannian metrics and they only exist on paracompact Hausdorﬀ manifolds. Another central ingredient in diﬀerential topology is the theorem of Sard and that requires second countability. To quote Moe Hirsch [29]: “Manifolds that are not paracompact are amusing, but they never occur naturally and it is diﬃcult to prove anything about them.” Thus we will set the following convention for the remaining chapters. We assume from now on that each intrinsic manifold M is Hausdorﬀ and second countable and hence is also paracompact. For most of this text we will in fact continue to develop the theory for submanifolds of Euclidean space and indicate, wherever necessary, how to extend the deﬁnitions, theorems, and proofs to the intrinsic setting. 120 CHAPTER 2. FOUNDATIONS Chapter 3 The Levi-Civita Connection For a submanifold of Euclidean space the inner product on the ambient space determines an inner product on each tangent space, the ﬁrst fundamental form. The second fundamental form is obtained by diﬀerentiating the map which assigns to each point in M ⊂ Rn the orthogonal projection onto the tangent space (§3.1). The covariant derivative of a vector ﬁeld along a curve is the orthogonal projection of the derivative in the ambient space onto the tangent space (§3.2). We will show how the covariant derivative gives rise to parallel transport (§3.3), examine the frame bundle (§3.4), discuss motions without “sliding, twisting, and wobbling”, and prove the development theorem (§3.5). In §3.6 we will see that the covariant derivative is determined by the Christoﬀel symbols in local coordinates and thus carries over to the intrinsic setting. The intrinsic setting of Riemannian manifolds is explained in §3.7. The covariant derivative takes the form of a family of linear operators ∇ : Vect(γ) → Vect(γ), one for every smooth curve γ : I → M, and these operators are
uniquely characterized by the axioms of Theorem 3.7.8. This family of linear operators is the Levi-Civita connection. 3.1 Second Fundamental Form Let M ⊂ Rn be a smooth m-manifold. Then each tangent space of M is an m-dimensional real vector space and hence is isomorphic to Rm. Thus any two tangent spaces TpM and TqM are of course isomorphic to each other. While there is no canonical isomorphism from TpM to TqM we shall see that every smooth curve γ in M connecting p to q induces an isomorphism between the tangent spaces via parallel transport of tangent vectors along γ. 121 122 CHAPTER 3. THE LEVI-CIVITA CONNECTION Throughout we use the standard inner product on Rn given by v, w = v1w1 + v2w2 + · · · + vnwn for v = (v1,..., vn) ∈ Rn and w = (w1,..., wn) ∈ Rn. The associated Euclidean norm will be denoted by \|v\| = v, v = 1 + v2 v2 2 + · · · + v2 n for v = (v1,..., vn) ∈ Rn. When M ⊂ Rn is a smooth m-dimensional submanifold, a ﬁrst observation is that each tangent space of M inherits an inner product from the ambient space Rn. The resulting ﬁeld of inner products is called the ﬁrst fundamental form. Deﬁnition 3.1.1. Let M ⊂ Rn be a smooth m-dimensional submanifold. The ﬁrst fundamental form on M is the ﬁeld which assigns to each p ∈ M the bilinear map gp : TpM × TpM → R deﬁned by for v, w ∈ TpM. gp(v, w) = v, w (3.1.1) A second observation is that the inner product on the ambient space also determines an orthogonal projection of Rn onto the tangent space TpM for each p ∈ M. This projection can be represented by the matrix Π(p)
∈ Rn×n which is uniquely determined by the conditions Π(p) = Π(p)2 = Π(p)T, and Π(p)v = v ⇐⇒ v ∈ TpM for p ∈ M and v ∈ Rn (see Exercise 2.6.9). (3.1.2) (3.1.3) Lemma 3.1.2. The map Π : M → Rn×n deﬁned by (3.1.2) and (3.1.3) is smooth. Proof. This follows directly from Theorem 2.6.10 and Corollary 2.6.12. More explicitly, if U ⊂ M is an open set and φ : U → Ω is a coordinate chart onto an open subset Ω ⊂ Rm with the inverse ψ := φ−1 : Ω → U, then Π(p) = dψ(φ(p)) dψ(φ(p))Tdψ(φ(p)) −1 dψ(φ(p))T for p ∈ U and this proves Lemma 3.1.2. 3.1. SECOND FUNDAMENTAL FORM 123 Figure 3.1: A unit normal vector ﬁeld. Example 3.1.3 (Gauß map). Let M ⊂ Rm+1 be a submanifold of codimension one. Then T M ⊥ is a vector bundle of rank one (Corollary 2.6.13), and so each ﬁber TpM ⊥ is spanned by a unit vector ν(p) ∈ Rm, determined by TpM up to a sign. By Theorem 2.6.10 each p0 ∈ M has an open neighborhood U ⊂ M on which there exists a smooth map ν : U → Rm+1 satisfying ν(p) ⊥ TpM, \|ν(p)\| = 1 (3.1.4) for all p ∈ U (see Figure 3.1). Such a map ν is called a Gauß map. The function Π : M → Rn×n is in this case given by Π(p) = 1l
− ν(p)ν(p)T (3.1.5) for p ∈ U. Example 3.1.4. Let M = S2 ⊂ R3. Then ν(p) = p and so Π(p) = 1l − ppT =   1 − x2 −xy −yx −zx −xz 1 − y2 −yz 1 − z2 −zy   for p = (x, y, z) ∈ S2. Example 3.1.5 (M¨obius strip). Consider the submanifold   (x, y, z) ∈ R3 M :=  x = (1 + r cos(θ/2)) cos(θ), y = (1 + r cos(θ/2)) sin(θ), z = r sin(θ/2), r, θ ∈ R, \|r\| < ε    for ε > 0 suﬃciently small. Show that there does not exist a global smooth function ν : M → R3 satisfying (3.1.4). T M Mpν( )p 124 CHAPTER 3. THE LEVI-CIVITA CONNECTION Example 3.1.6. Let U ⊂ Rn be an open set and f : U → Rn−m be a smooth function such that 0 ∈ Rn−m is a regular value of f and U ∩ M = f −1(0). Then TpM = ker df (p) and Π(p) = 1l − df (p)T df (p)df (p)T−1 df (p) for every p ∈ U ∩ M. Example 3.1.7. Let Ω ⊂ Rm be an open set and ψ : Ω → M be a smooth embedding. Then Tψ(x)M = im dψ(x) and Π(ψ(x)) = dψ(x) dψ(x)Tdψ(x) −1 dψ(x)T for every x ∈ Ω. Next we diﬀerentiate the map Π : M → Rn
×n in Lemma 3.1.2. The derivative at p ∈ M takes the form of a linear map which, as usual, is deﬁned by dΠ(p) : TpM → Rn×n dΠ(p)v := t=0 for v ∈ TpM, where γ : R → M is chosen such that γ(0) = p and ˙γ(0) = v (see Deﬁnition 2.2.13). We emphasize that the expression dΠ(p)v is a matrix and can therefore be multiplied by a vector in Rn. Π(γ(t)) ∈ Rn×n d dt Lemma 3.1.8. For all p ∈ M and v, w ∈ TpM we have dΠ(p)vw = dΠ(p)wv ∈ TpM ⊥. Proof. Choose a smooth path γ : R → M and a vector ﬁeld X : R → Rn along γ such that γ(0) = p, ˙γ(0) = v, X(0) = w. For example, we can choose X(t) := Π(γ(t))w. Then for every t ∈ R. Diﬀerentiate this equation to obtain X(t) = Π(γ(t))X(t) ˙X(t) = Π(γ(t)) ˙X(t) + dΠ(γ(t)) ˙γ(t)X(t). (3.1.6) Hence dΠ(γ(t)) ˙γ(t)X(t) = 1l − Π(γ(t)) ˙X(t) ∈ Tγ(t)M ⊥ (3.1.7) for every t ∈ R and, with t = 0, we obtain (dΠ(p)v) w ∈ TpM ⊥. 3.1. SECOND FUNDAMENTAL FORM 125 Now choose a smooth map R2 → M : (s, t) → γ(s, t) satisfying γ(0, 0) = p, ∂γ ∂s (0
, 0) = v, ∂γ ∂t (0, 0) = w, (for example by doing this in local coordinates) and denote X(s, t) := ∂γ ∂s (s, t) ∈ Tγ(s,t)M, Y (s, t) := ∂γ ∂t (s, t) ∈ Tγ(s,t)M. Then ∂Y ∂s = ∂2γ ∂s∂t = ∂X ∂t and hence, using (3.1.7), we obtain dΠ(γ) ∂γ ∂t ∂γ ∂s = dΠ(γ) X ∂γ ∂t = 1l − Π(γ) ∂X ∂t = 1l − Π(γ) ∂Y ∂s = dΠ(γ) = dΠ(γ) ∂γ ∂s ∂γ ∂s Y ∂γ ∂t. With s = t = 0 we obtain dΠ(p)wv = dΠ(p)vw ∈ TpM ⊥ and this proves Lemma 3.1.8. Deﬁnition 3.1.9. The collection of symmetric bilinear maps hp : TpM × TpM → TpM ⊥, deﬁned by hp(v, w) := (dΠ(p)v)w = (dΠ(p)w)v (3.1.8) for p ∈ M and v, w ∈ TpM is called the second fundamental form on M. 126 CHAPTER 3. THE LEVI-CIVITA CONNECTION Example 3.1.10. Let M ⊂ Rm+1 be an m-manifold and ν : M → Sm be a Gauß map so that TpM = ν(p)⊥ for every p ∈ M (see Example 3.1.3). Then Π(p) = 1l − ν(p)ν(p)T and hence hp(v, w) = −ν(p)dν(p)v, w for p ∈ M and v, w ∈ TpM. Exercise 3.1
.11. Choose a splitting Rn = Rm × Rn−m and write the elements of Rn as tuples (x, y) = (x1,..., xm, y1,..., yn−m) Let M ⊂ Rn be a smooth m-dimensional submanifold such that p = 0 ∈ M and T0M = Rm × {0}, T0M ⊥ = {0} × Rn−m. By the implicit function theorem, there are open neighborhoods Ω ⊂ Rm and V ⊂ Rn−m of zero and a smooth map f : Ω → V such that M ∩ (Ω × V ) = graph(f ) = {(x, f (x)) \| x ∈ Ω}. Thus f (0) = 0 and df (0) = 0. Prove that the second fundamental form hp : TpM × TpM → TpM ⊥ is given by the second derivatives of f, i.e.  hp(v, w) = 0, m i,j=1 ∂2f ∂xi∂xj  (0)viwj  for v, w ∈ TpM = Rm × {0}. Exercise 3.1.12. Let M ⊂ Rn be an m-manifold. Fix a point p ∈ M and a unit tangent vector v ∈ TpM so that \|v\| = 1 and deﬁne L := {p + tv + w \| t ∈ R, w ⊥ TpM }. Let γ : (−ε, ε) → M ∩ L be a smooth curve such that γ(0) = p, ˙γ(0) = v, and \| ˙γ(t)\| = 1 for all t. Prove that ¨γ(0) = hp(v, v). Draw a picture of M and L in the case n = 3 and m = 2. 3.2. COVARIANT DERIVATIVE 127 Figure 3.2: A vector ﬁeld along a curve. 3.2 Covariant Derivative Deﬁnition 3.2.1. Let I �
�� R be an open interval and let γ : I → M be a smooth curve. A vector ﬁeld along γ is a smooth map X : I → Rn such that X(t) ∈ Tγ(t)M for every t ∈ I (see Figure 3.2). The set of smooth vector ﬁelds along γ is a real vector space and will be denoted by Vect(γ) := X : I → Rn \| X is smooth and X(t) ∈ Tγ(t)M ∀ t ∈ I. The ﬁrst derivative ˙X(t) of a vector ﬁeld along γ at t ∈ I will, in general, not be tangent to M. We may decompose it as a sum of a tangent vector and a normal vector in the form ˙X(t) = Π(γ(t)) ˙X(t) + 1l − Π(γ(t)) ˙X(t), where Π : M → Rn×n is deﬁned by (3.1.2) and (3.1.3). The tangential component of this decomposition plays an important geometric role. It is called the covariant derivative of X at t. Deﬁnition 3.2.2 (Covariant derivative). Let I ⊂ R be an open interval, let γ : I → M be a smooth curve, and let X ∈ Vect(γ). The covariant derivative of X is the vector ﬁeld ∇X ∈ Vect(γ), deﬁned by ∇X(t) := Π(γ(t)) ˙X(t) ∈ Tγ(t)M (3.2.1) for t ∈ I. Lemma 3.2.3 (Gauß–Weingarten formula). The derivative of a vector ﬁeld X along a curve γ is given by ˙X(t) = ∇X(t) + hγ(t)( ˙γ(t), X(t)). (3.2.2) Here the ﬁrst summand is tangent to M and the second summand is orthogonal to the tang
ent space of M at γ(t). Proof. This is equation (3.1.6) in the proof of Lemma 3.1.8. X(t)tγ( ) M 128 CHAPTER 3. THE LEVI-CIVITA CONNECTION It follows directly from the deﬁnition that the covariant derivative along a curve γ : I → M is a linear operator ∇ : Vect(γ) → Vect(γ). The following lemma summarizes the basic properties of this operator. Lemma 3.2.4 (Covariant derivative). The covariant derivative satisﬁes the following axioms for any two open intervals I, J ⊂ R. (i) Let γ : I → M be a smooth curve, let λ : I → R be a smooth function, and let X ∈ Vect(γ). Then ∇(λX) = ˙λX + λ∇X. (3.2.3) (ii) Let γ : I → M be a smooth curve, let σ : J → I be a smooth function and let X ∈ Vect(γ). Then ∇(X ◦ σ) = ˙σ(∇X ◦ σ). (3.2.4) (iii) Let γ : I → M be a smooth curve and let X, Y ∈ Vect(γ). Then d dt X, Y = ∇X, Y + X, ∇Y. (3.2.5) (iv) Let γ : I × J → M be a smooth map, denote by ∇s the covariant derivative along the curve s → γ(s, t) (with t ﬁxed), and denote by ∇t the covariant derivative along the curve t → γ(s, t) (with s ﬁxed). Then ∇s∂tγ = ∇t∂sγ. Proof. Part (i) follows from the Leibniz rule d follows from the chain rule d the orthogonal projections Π(γ(t)) : Rn → Tγ(t)M to obtain (3.2.6) dt (λX) = ˙λX + λ ˙X and (ii) d
t (X ◦ σ) = ˙σ( ˙X ◦ σ). To prove part (iii), use d dt X, Y = ˙X, Y + X, ˙Y = ˙X, Π(γ)Y + Π(γ)X, ˙Y = Π(γ) ˙X, Y + X, Π(γ) ˙Y = ∇X, Y + X, ∇Y. Part (iv) holds because the second derivatives commute and this proves Lemma 3.2.4. Part (i) in Lemma 3.2.4 asserts that the operator ∇ is what is called a connection, part (iii) asserts that it is compatible with the ﬁrst fundamental form, and part (iv) asserts that it is torsion-free. Theorem 3.7.8 below asserts that these conditions (together with an extended chain rule) determine the covariant derivative uniquely. 3.3. PARALLEL TRANSPORT 129 3.3 Parallel Transport Deﬁnition 3.3.1 (Parallel vector ﬁeld). Let I ⊂ R be an interval and let γ : I → M be a smooth curve. A vector ﬁeld X along γ is called parallel iﬀ ∇X(t) = 0 for all t ∈ I. Example 3.3.2. Assume m = n so that M ⊂ Rm is an open set. Then a vector ﬁeld along a smooth curve γ : I → M is a smooth map X : I → Rm. Its covariant derivative is equal to the ordinary derivative ∇X(t) = ˙X(t) and hence X is is parallel if and only if it is constant. Remark 3.3.3. For every X ∈ Vect(γ) and every t ∈ I we have ∇X(t) = 0 ⇐⇒ ˙X(t) ⊥ Tγ(t)M. In particular, ˙γ is a vector ﬁeld along γ and ∇ ˙γ(t) = Π(γ(t))¨γ(t). Hence ˙γ is a parallel vector
ﬁeld along γ if and only if ¨γ(t) ⊥ Tγ(t)M for all t ∈ I. We will return to this observation in Chapter 4. In general, a vector ﬁeld X along a smooth curve γ : I → M is parallel ˙X(t) is orthogonal to Tγ(t)M for every t and, by the Gauß– if and only if Weingarten formula (3.2.2), we have ∇X = 0 ⇐⇒ ˙X = hγ( ˙γ, X). The next theorem shows that any given tangent vector v0 ∈ Tγ(t0)M extends uniquely to a parallel vector ﬁeld along γ. Theorem 3.3.4 (Existence and uniqueness). Let I ⊂ R be an interval and γ : I → M be a smooth curve. Let t0 ∈ I and v0 ∈ Tγ(t0)M be given. Then there is a unique parallel vector ﬁeld X ∈ Vect(γ) such that X(t0) = v0. Proof. Choose a basis e1,..., em of the tangent space Tγ(t0)M and let X1,..., Xm ∈ Vect(γ) be vector ﬁelds along γ such that Xi(t0) = ei, i = 1,..., m. (For example choose Xi(t) := Π(γ(t))ei.) Then the vectors Xi(t0) are linearly independent. Since linear independence is an open condition there is a 130 CHAPTER 3. THE LEVI-CIVITA CONNECTION constant ε > 0 such that the vectors X1(t),..., Xm(t) ∈ Tγ(t)M are linearly independent for every t ∈ I0 := (t0 − ε, t0 + ε) ∩ I. Since Tγ(t)M is an m-dimensional real vector space this implies that the vectors Xi(t) form a basis of Tγ(t)M for every t ∈ I0. We express the vector ∇Xi(t) ∈ Tγ(t)M
in this basis and denote the coeﬃcients by ak i (t) so that ∇Xi(t) = m k=1 ak i (t)Xk(t). i : I0 → R are smooth. Likewise, if X : I → Rn is The resulting functions ak any vector ﬁeld along γ, then there are smooth functions ξi : I0 → R such that m X(t) = ξi(t)Xi(t) for all t ∈ I0. The derivative of X is given by i=1 ˙X(t) = m i=1 ˙ξi(t)Xi(t) + ξi(t) ˙Xi(t) and the covariant derivative by ∇X(t) = = = m i=1 m i=1 m k=1 ˙ξi(t)Xi(t) + ξi(t)∇Xi(t) ˙ξi(t)Xi(t) + ˙ξk(t) + m i=1 m i=1 ξi(t) ak i (t)Xk(t) m k=1 i (t)ξi(t) ak Xk(t) for t ∈ I0. Hence ∇X(t) = 0 if and only if ˙ξ(t) + A(t)ξ(t) = 0, A(t) :=    a1 1(t)... am 1 (t) · · · · · ·   . a1 m(t)... am m(t) Thus we have translated the equation ∇X = 0 over the interval I0 into a time dependent linear ordinary diﬀerential equation. By a theorem in Analysis II (see [64, Lemma 4.4.3]), this equation has a unique solution for any initial condition at any point in I0. Thus we have proved that every t0 ∈ I is 3.3. PARALLEL TRANSPORT 131 contained in an interval I0 ⊂ I, open in the relative topology of I, such that, for every t1 ∈ I0 and every v1 ∈ Tγ(t1)
M, there exists a unique parallel vector ﬁeld X : I0 → Rn along γ\|I0 satisfying X(t1) = v1. We formulate this condition on the interval I0 as a logical formula: ∀ t1 ∈ I0 ∀ v1 ∈ Tγ(t1)M ∃! X ∈ Vect(γ\|I0) such that ∇X = 0 and X(t1) = v1. (3.3.1) If two I-open intervals I0, I1 ⊂ I satisfy this condition and have nonempty intersection, then their union I0 ∪ I1 also satisﬁes (3.3.1). (Prove this!) Now deﬁne J := {I0 ⊂ I \| I0 is an I-open interval, I0 satisﬁes (3.3.1), t0 ∈ I0}. This interval J satisﬁes (3.3.1). Moreover, it is nonempty and, by deﬁnition, it is open in the relative topology of I. We prove that it is also closed in the relative topology of I. Thus let (ti)i∈N be a sequence in J converging to a point t∗ ∈ I. By what we have proved above, there exists a constant ε > 0 such that the interval I ∗ := (t∗ − ε, t∗ + ε) ∩ I satisﬁes (3.3.1). Since the sequence (ti)i∈N converges to t∗, there exists an i ∈ N such that ti ∈ I ∗. Since ti ∈ J there exists an interval I0 ⊂ I, open in the relative topology of I, that contains t0 and ti and satisﬁes (3.3.1). Hence the interval I0 ∪ I ∗ is open in the relative topology of I, contains t0 and t∗, and satisﬁes (3.3.1). This shows that t∗ ∈J. Thus we have proved that the interval J is nonempty, and open and closed in the relative topology of I. Hence J = I and this proves Theorem 3
.3.4. Deﬁnition 3.3.5 (Parallel transport). Let I ⊂ R be an interval and let γ : I → M be a smooth curve. For t0, t ∈ I we deﬁne the map Φγ(t, t0) : Tγ(t0)M → Tγ(t)M by Φγ(t, t0)v0 := X(t) where X ∈ Vect(γ) is the unique parallel vector ﬁeld along γ satisfying X(t0) = v0. The collection of maps Φγ(t, t0) for t, t0 ∈ I is called parallel transport along γ. Recall the notation γ∗T M = (s, v) \| s ∈ I, v ∈ Tγ(s)M for the pullback tangent bundle. This set is a smooth submanifold of I ×Rn. (See Theorem 2.6.10 and Corollary 2.6.13.) The next theorem summarizes the properties of parallel transport. In particular, the last assertion shows that the covariant derivative can be recovered from the parallel transport maps. 132 CHAPTER 3. THE LEVI-CIVITA CONNECTION Theorem 3.3.6 (Parallel transport). Let γ : I → M be a smooth curve on an interval I ⊂ R. (i) The map Φγ(t, s) : Tγ(s)M → Tγ(t)M is linear for all s, t ∈ I. (ii) For all r, s, t ∈ I we have Φγ(t, s) ◦ Φγ(s, r) = Φγ(t, r), Φγ(t, t) = id. (iii) For all s, t ∈ I and all v, w ∈ Tγ(s)M we have Φγ(t, s)v, Φγ(t, s)w = v, w. Thus Φγ(t, s) : Tγ(s)M → Tγ(t)M is an orthogonal transformation. (iv) If J ⊂ R is an interval and σ : J → I is a smooth map, then Φγ
◦σ(t, s) = Φγ(σ(t), σ(s)). for all s, t ∈ J. (v) The map I × γ∗T M → γ∗T M : (t, (s, v)) → (t, Φγ(t, s)v) is smooth. (vi) For all X ∈ Vect(γ) and t, t0 ∈ I we have d dt Φγ(t0, t)X(t) = Φγ(t0, t)∇X(t). Proof. Assertion (i) holds because the sum of two parallel vector ﬁelds along γ is again parallel and the product of a parallel vector ﬁeld with a constant real number is again parallel. Assertion (ii) follows directly from the uniqueness statement in Theorem 3.3.4. We prove (iii). Fix a number s ∈ I and two tangent vectors Deﬁne the vector ﬁelds X, Y ∈ Vect(γ) along γ by v, w ∈ Tγ(s)M. X(t) := Φγ(t, s)v, Y (t) := Φγ(t, s)w. These vector ﬁelds are parallel. Thus, by equation (3.2.5) in Lemma 3.2.4, we have d dt X, Y = ∇X, Y + X, ∇Y = 0. Hence the function I → R : t → X(t), Y (t) is constant and this proves (iii). 3.3. PARALLEL TRANSPORT 133 We prove (iv). Fix an element s ∈ J and a tangent vector v ∈ Tγ(σ(s))M. Deﬁne the vector ﬁeld X along γ by for t ∈ I. Thus X is the unique parallel vector ﬁeld along γ that satisﬁes X(t) := Φγ(t, σ(s))v X(σ(s)) = v. Denote γ := γ ◦ σ : J → M, X := X ◦ σ : I → Rn Then X is a
vector ﬁeld along γ and, by the chain rule, we have d dt X(σ(t)) = ˙σ(t) ˙X(σ(t)). X(t) = d dt Projecting orthogonally onto the tangent space Tγ(σ(t))M we obtain ∇ X(t) = ˙σ(t)∇X(σ(t)) = 0 for every t ∈ J. Hence X is the unique parallel vector ﬁeld along γ that satisﬁes X(s) = v. Thus Φ γ(t, s)v = X(t) = X(σ(t)) = Φγ(σ(t), σ(s))v. This proves (iv). We prove (v). Fix a point t0 ∈ I, choose an orthonormal basis e1,..., em of Tγ(t0)M, and deﬁne Xi(t) := Φγ(t, t0)ei for t ∈ I and i = 1,..., m. Thus Xi ∈ Vect(γ) is the unique parallel vector ﬁeld along γ such that Xi(t0) = ei. Then by (iii) we have Xi(t), Xj(t) = δij for all i, j ∈ {1,..., m} and all t ∈ I. Hence the vectors X1(t),..., Xm(t) form an orthonormal basis of Tγ(t)M for every t ∈ I. This implies that, for each s ∈ I and each tangent vector v ∈ Tγ(s)M, we have v = m i=1 Xi(s), vXi(s). Since each vector ﬁeld Xi is parallel it satisﬁes Xi(t) = Φγ(t, s)Xi(s). Hence Φγ(t, s)v = m i=1 Xi(s), vXi(t) (3.3.2) for all s, t ∈ I and v ∈ Tγ(s)M. This proves (v). 134 CHAPTER 3. THE LEVI-CIVITA CONN
ECTION We prove (vi). Let X1,..., Xm ∈ Vect(γ) be as in the proof of (v). Thus every vector ﬁeld X along γ can be written in the form X(t) = m i=1 ξi(t)Xi(t), ξi(t) := Xi(t), X(t). Since the vector ﬁelds Xi are parallel we have ∇X(t) = m i=1 ˙ξi(t)Xi(t) for all t ∈ I. Hence Φγ(t0, t)X(t) = m i=1 ξi(t)Xi(t0), Φγ(t0, t)∇X(t) = m i=1 ˙ξi(t)Xi(t0). Evidently, the derivative of the ﬁrst sum with respect to t is equal to the second sum. This proves (vi) and Theorem 3.3.6. Remark 3.3.7. For s, t ∈ I we can think of the linear map Φγ(t, s)Π(γ(s)) : Rn → Tγ(t)M ⊂ Rn as a real n × n matrix. The formula (3.3.2) in the proof of (v) shows that this matrix can be expressed in the form Φγ(t, s)Π(γ(s)) = m i=1 Xi(t)Xi(s)T ∈ Rn×n. The right hand side deﬁnes a smooth matrix valued function on I × I and this is equivalent to the assertion in (v). Remark 3.3.8. It follows from assertions (ii) and (iii) in Theorem 3.3.6 that Φγ(t, s)−1 = Φγ(s, t) = Φγ(t, s)∗ for all s, t ∈ I. Here the linear map Φγ(t, s)∗ : Tγ(t)M → Tγ(s)M is understood as the adjoint operator of Φγ(t, s) : Tγ(s)M → Tγ(t)M with respect to the
inner products on the two subspaces of Rn inherited from the Euclidean inner product on the ambient space. 3.3. PARALLEL TRANSPORT 135 The two theorems in this section carry over verbatim to any smooth vector bundle E ⊂ M × Rn over a manifold. As in the case of the tangent bundle one can deﬁne the covariant derivative of a section of E along γ as the orthogonal projection of the ordinary derivative in the ambient space Rn onto the ﬁber Eγ(t). Instead of parallel vector ﬁelds one then speaks about horizontal sections and one proves as in Theorem 3.3.4 that there is a unique horizontal section along γ through any point in any of the ﬁbers Eγ(t0). This gives parallel transport maps from Eγ(s) to Eγ(t) for any pair s, t ∈ I and Theorem 3.3.6 carries over verbatim to all vector bundles E ⊂ M × Rn. We spell this out in more detail in the case where E = T M ⊥ ⊂ M × Rn is the normal bundle of M. Let γ : I → M be a smooth curve. A normal vector ﬁeld along γ is a smooth map Y : I → Rn such that Y (t) ⊥ Tγ(t)M for every t ∈ I. The set of normal vector ﬁelds along γ will be denoted by Vect⊥(γ) := Y : I → Rn \| Y is smooth and Y (t) ⊥ Tγ(t)M for all t ∈ I. This is again a real vector space. The covariant derivative of a normal vector ﬁeld Y ∈ Vect⊥(γ) at t ∈ I is deﬁned as the orthogonal projection of the ordinary derivative onto the orthogonal complement of Tγ(t)M and will be denoted by ∇⊥Y (t) := 1l − Π(γ(t)) ˙Y (t). (3.3.3) Thus the covariant derivative deﬁnes a linear operator ∇⊥ : Vect⊥(γ) → Vect�
��(γ). There is a version of the Gauß–Weingarten formula for the covariant derivative of a normal vector ﬁeld. This is the content of the next lemma. Lemma 3.3.9. Let M ⊂ Rn be a smooth m-manifold. For p ∈ M and u ∈ TpM deﬁne the linear map hp(u) : TpM → TpM ⊥ by hp(u)v := hp(u, v) = dΠ(p)uv (3.3.4) for v ∈ TpM. Then the following holds. (i) The adjoint operator hp(u)∗ : TpM ⊥ → TpM is given by hp(u)∗w = dΠ(p)uw, w ∈ TpM ⊥. (3.3.5) (ii) If I ⊂ R is an interval, γ : I → M is a smooth curve, and Y ∈ Vect⊥(γ), then the derivative of Y satisﬁes the Gauß–Weingarten formula ˙Y (t) = ∇⊥Y (t) − hγ(t)( ˙γ(t))∗Y (t). (3.3.6) 136 CHAPTER 3. THE LEVI-CIVITA CONNECTION Proof. Since Π(p) ∈ Rn×n is a symmetric matrix for every p ∈ M so is the matrix dΠ(p)u for every p ∈ M and every u ∈ TpM. Hence v, hp(u)∗w = hp(u)v, w = dΠ(p)uv, w = v, dΠ(p)uw for every v ∈ TpM and every w ∈ TpM ⊥. This proves (i). To prove (ii) we observe that, for Y ∈ Vect⊥(γ) and t ∈ I, we have Π(γ(t))Y (t) = 0. Diﬀerentiating this identity we obtain Π(γ(t)) ˙Y (t) + dΠ(γ(t)) �
�γ(t)Y (t) = 0 and hence ˙Y (t) = ˙Y (t) − Π(γ(t)) ˙Y (t) − dΠ(γ(t)) ˙γ(t)Y (t) = ∇⊥Y (t) − hγ(t)( ˙γ(t))∗Y (t) for t ∈ I. Here the last equation follows from (i) and the deﬁnition of ∇⊥. This proves Lemma 3.3.9. Theorem 3.3.4 and its proof carry over to the normal bundle T M ⊥. Thus, if γ : I → M is a smooth curve, then for all s ∈ I and w ∈ Tγ(s)M ⊥ there is a unique normal vector ﬁeld Y ∈ Vect⊥(γ) such that ∇⊥Y ≡ 0, Y (s) = w. This gives rise to parallel transport maps Φ⊥ γ (t, s) : Tγ(s)M ⊥ → Tγ(t)M ⊥ deﬁned by Φ⊥ γ (t, s)w := Y (t) for s, t ∈ I and w ∈ Tγ(s)M ⊥, where Y is the unique normal vector ﬁeld along γ satisfying ∇⊥Y ≡ 0 and Y (s) = w. These parallel transport maps satisfy exactly the same conditions that have been spelled out in Theorem 3.3.6 for the tangent bundle and the proof carries over verbatim to the present setting. 3.4. THE FRAME BUNDLE 137 3.4 The Frame Bundle Each tangent space of an m-manifold M is isomorphic to the Euclidean space Rm, however, in general there is no canonical isomorphism. The space of all pairs consisting of a point p in the manifold M and an isomorphism from Rm to the tangent space of M at p is itself a smooth manifold, called the frame bundle of M. 3.4.1 Frames of a Vector Space Let V be an m-dimensional real vector space. A frame of V is a basis e
1,..., em of V. It determines a vector space isomorphism e : Rm → V via eξ := m i=1 ξiei, ξ = (ξ1,..., ξm) ∈ Rm. Conversely, each isomorphism e : Rm → V determines a basis e1,..., em of V via ei = e(0,..., 0, 1, 0..., 0), where the coordinate 1 appears in the ith place. The set of vector space isomorphisms from Rm to V will be denoted by Liso(Rm, V ) := {e : Rm → V \| e is a vector space isomorphism}. The general linear group GL(m) = GL(m, R) (of nonsingular real m × mmatrices) acts on this space by composition on the right via GL(m) × Liso(Rm, V ) → Liso(Rm, V ) : (a, e) → a∗e := e ◦ a. This is a contravariant group action in that a∗b∗e = (ba)∗e, 1l∗e = e for a, b ∈ GL(m) and e ∈ Liso(Rm, V ). Moreover, the action is free, i.e. for all a ∈ GL(m) and e ∈ Liso(Rm, V ), we have a∗e = e ⇐⇒ a = 1l. It is transitive in that for all e, e ∈ Liso(Rm, V ) there is a group element a ∈ GL(m) such that e = a∗e. Thus we can identify the space Liso(Rm, V ) with the group GL(m) via the bijection GL(m) → Liso(Rm, V ) : a → a∗e0 induced by a ﬁxed element e0 ∈ Liso(Rm, V ). This identiﬁcation is not canonical; it depends on the choice of e0. The space Liso(Rm, V ) admits a bijection to a group but is not itself a group. 138 CHAPTER 3. THE LEVI-CIVITA CON
NECTION 3.4.2 The Frame Bundle Deﬁnition 3.4.1 (Frame bundle). Let M ⊂ Rn be a smooth m-manifold. The frame bundle of M is the set F(M ) := {(p, e) \| p ∈ M, e ∈ F(M )p}, (3.4.1) where F(M )p is the space of frames of the tangent space at p, i.e. F(M )p := Liso(Rm, TpM ). Deﬁne a right action of GL(m) on F(M ) by a∗(p, e) := (p, a∗e) = (p, e ◦ a) (3.4.2) for a ∈ GL(m) and (p, e) ∈ F(M ). One can think of a frame e ∈ Liso(Rm, TpM ) as a linear map from Rm to Rn whose image is TpM and hence as an n × m-matrix of rank m. The basis of TpM associated to this frame is given by the columns of the matrix e ∈ Rn×m. Thus the frame bundle F(M ) of an embedded manifold M ⊂ Rn is a subset of the Euclidean space Rn × Rn×m. Lemma 3.4.2. The frame bundle F(M ) ⊂ Rn × Rn×m is a smooth manifold of dimension m + m2, the group action GL(m) × F(M ) → F(M ) : (a, (p, e)) → a∗(p, e) is smooth, and the projection π : F(M ) → M deﬁned by π(p, e) := p for (p, e) ∈ F(M ) is a surjective submersion. The orbits of the GL(m)-action on F(M ) are the ﬁbers of this projection, i.e. GL(m)∗(p, e) = π−1(p) ∼= F(M )p for (p, e) ∈ F(M ), and the group GL(m) acts freely and transitively on each of these �
�bers. 3.4. THE FRAME BUNDLE 139 Proof. Let U ⊂ M be an M -open set. A moving frame over U is a sequence of m smooth vector ﬁelds E1,..., Em ∈ Vect(U ) on U such that the vectors E1(p),..., Em(p) form a basis of TpM for each p ∈ U. Any such moving frame gives a bijection U × GL(m) → F(U ) : (p, a) → a∗(p, E(p)) = (p, E(p) ◦ a), where E(p) := (E1(p),..., Em(p)) ∈ F(M )p for p ∈ U. This bijection (when composed with a parametrization of U ) gives a parametrization of the open set F(U ) in F(M ). The assertions of the lemma then follow from the fact that the diagram U × GL(m) pr1 U F(U ) π commutes. More precisely, suppose that there exists a coordinate chart with values in an open set Ω ⊂ Rm, and denote its inverse by φ : U → Ω ψ := φ−1 : Ω → U. Then the open set F(U ) = π−1(U ) = {(p, e) ∈ F(M ) \| p ∈ U } = (U × Rn×m) ∩ F(M ) is parametrized by the map Ω × GL(m) → F(U ) : (x, a) → ψ(x), dψ(x) ◦ a. This map is amooth and so is its inverse F(U ) → Ω × GL(m) : (p, e) → φ(p), dφ(p) ◦ e. These are the desired coordinate chart on F(M ). Thus F(M ) is a smooth manifold of dimension m + m2. Moreover, in these coordinates the projection π : F(U ) → U is the map Ω × GL(m) → Ω : (x, a) → x and so π is a submersion. The remaining
assertions follow directly from the deﬁnitions and this proves Lemma 3.4.2. 140 CHAPTER 3. THE LEVI-CIVITA CONNECTION The frame bundle F(M ) is a principal bundle over M with structure group GL(m). More generally, a principal bundle over a manifold B with structure group G is a smooth manifold P equipped with a surjective submersion π : P → B and a smooth contravariant action G × P → P : (g, p) → pg by a Lie group G such that π(pg) = π(p) for all p ∈ P and g ∈ G and such that the group G acts freely and transitively on the ﬁber Pb := π−1(b) for each b ∈ B. In this book we shall mostly be concerned with the frame bundle of a manifold M and the orthonormal frame bundle. Deﬁnition 3.4.3. The orthonormal frame bundle of M is the set O(M ) := (p, e) ∈ Rn × Rn×m p ∈ M, im e = TpM, eTe = 1lm. If we denote by ei := e(0,..., 0, 1, 0,..., 0) (with 1 as the ith argument) the basis of TpM induced by the isomorphism e : Rm → TpM, then we have eTe = 1l ⇐⇒ ei, ej = δij ⇐⇒ e1,..., em is an orthonormal basis. Thus O(M ) is the bundle of orthonormal frames of the tangent spaces TpM or the bundle of orthogonal isomorphisms e : Rm → TpM. It is a principal bundle over M with structure group O(m). Exercise 3.4.4. Prove that O(M ) is a submanifold of F(M ) and that the obvious projection π : O(M ) → M is a submersion. Prove that the action of GL(m) on F(M ) restricts to an action of the orthogonal group O(m) on O(M ) whose orbits are the ﬁbers O(M )p := e ∈
Rn×m (p, e) ∈ O(M ) = e ∈ Liso(Rm, TpM ) eTe = 1l. Hint: If φ : U → Ω is a coordinate chart on M with inverse ψ : Ω → U, then ex := dψ(x)(dψ(x)Tdψ(x))−1/2 : Rm → Tψ(x)M is an orthonormal frame of the tangent space Tψ(x)M for every x ∈ Ω. 3.4. THE FRAME BUNDLE 141 3.4.3 Horizontal Lifts We have seen in Lemma 3.4.2 that the frame bundle F(M ) is a smooth submanifold of Rn × Rn×m. Next we examine the tangent space of F(M ) at a point (p, e) ∈ F(M ). By Deﬁnition 2.2.1, this tangent space is given by T(p,e)F(M ) =   ( ˙γ(0), ˙e(0))  R → F(M ) : t → (γ(t), e(t)) is a smooth curve satisfying γ(0) = p and e(0) = e   . The next lemma gives an explicit formula for this tangent space in terms of the second fundamental form hp : TpM × TpM → TpM ⊥ in Deﬁnition 3.1.9. Compare this formula with Lemma 4.3.1 in the next chapter. Lemma 3.4.5. Let M ⊂ Rn be a smooth m-dimensional submanifold. Then the tangent space of F(M ) at (p, e) is given by T(p,e)F(M ) = (p, e) p ∈ TpM, e ∈ Rn×m, and 1l − Π(p) e = hp(p)e. (3.4.3) Proof. We prove the inclusion “⊂” in (3.4.3). Let (p, e) ∈ T(p,e
)F(M ) and choose a smooth curve R → F(M ) : t → (γ(t), e(t)) such that γ(0) = p, e(0) = e, ˙γ(0) = p, ˙e(0) = e. Fix a vector ξ ∈ Rm and deﬁne the vector ﬁeld X ∈ Vect(γ) by X(t) := e(t)ξ for t ∈ R. Then the Gauß–Weingarten formula (3.2.2) asserts that ˙e(t)ξ = ˙X(t) = ∇X(t) + hγ(t)( ˙γ(t), X(t)) = Π(γ(t)) ˙e(t)ξ + hγ(t)( ˙γ(t), e(t)ξ) for all t ∈ R. Take t = 0 to obtain (1l − Π(p))eξ = hp(p, eξ) = hp(p)eξ for all ξ ∈ Rm. This proves the inclusion “⊂” in (3.4.3). Equality holds because both sides of the equation are (m+m2)-dimensional linear subspaces of Rn × Rn×m. This proves Lemma 3.4.5. It is convenient to consider two kinds of curves in F(M ), namely vertical curves with constant projections to M and horizontal lifts of curves in M. We denote by L(Rm, TpM ) the space of linear maps from Rm to TpM. 142 CHAPTER 3. THE LEVI-CIVITA CONNECTION Deﬁnition 3.4.6 (Horizontal lift). Let γ : R → M be a smooth curve. A smooth curve β : R → F(M ) is called a lift of γ iﬀ π ◦ β = γ. Any such lift has the form β(t) = (γ(t), e(t)) with e(t) ∈ Liso(Rm, Tγ(t)M ). The associated curve of frames e(t) of the tangent spaces Tγ(t)M is called a
moving frame along γ. A curve β(t) = (γ(t), e(t)) ∈ F(M ) is called horizontal or a horizontal lift of γ iﬀ the vector ﬁeld X(t) := e(t)ξ along γ is parallel for every ξ ∈ Rm. Thus a horizontal lift of γ has the form β(t) = (γ(t), Φγ(t, 0)e) (3.4.4) for some e ∈ Liso(Rm, Tγ(0)M ). Lemma 3.4.7. (i) The tangent space of F(M ) at (p, e) ∈ F(M ) is the direct sum T(p,e)F(M ) = H(p,e) ⊕ V(p,e) of the horizontal space H(p,e) := (v, hp(v)e) v ∈ TpM and the vertical space V(p,e) := {0} × L(Rm, TpM ). (3.4.5) (3.4.6) (ii) The vertical space V(p,e) at (p, e) ∈ F(M ) is the kernel of the linear map dπ(p, e) : T(p,e)F(M ) → TpM. (iii) A curve β : R → F(M ) is horizontal if and only if it is tangent to the horizontal spaces, i.e. ˙β(t) ∈ Hβ(t) for every t ∈ R. (iv) If β : R → F(M ) is a horizontal curve, so is a∗β for every a ∈ GL(m). 3.4. THE FRAME BUNDLE 143 Proof. The proof has four steps. Step 1. Let (p, e) ∈ F(M ). Then V(p,e) = ker dπ(p, e) ⊂ T(p,e)F(M ). Since π is a submersion, the ﬁber π−1(p) is a submanifold of F(M ) by Theorem 2.2.19 and T(p,e)π−1(p)
= ker dπ(p, e). Now let (p, e) ∈ ker dπ(p, e). Then there exists a vertical curve β : R → F(M ) with π ◦ β ≡ p such that β(0) = (p, e), ˙β(0) = (p, e). Any such curve has the form β(t) := (p, e(t)) where e(t) ∈ Liso(Rm, TpM ). Hence p = 0 and e = ˙e(0) ∈ L(Rm, TpM ). This shows that ker dπ(p, e) ⊂ V(p,e). Conversely, for every e ∈ L(Rm, TpM ), the curve (3.4.7) R → L(Rm, TpM ) : t → e(t) := e + te takes values in the open set Liso(Rm, TpM ) for t suﬃciently small and hence β(t) := (p, e(t)) is a vertical curve with ˙β(0) = (0, e). Thus V(p,e) ⊂ ker dπ(p, e) ⊂ T(p,e)F(M ). (3.4.8) Combining (3.4.7) and (3.4.8) we obtain Step 1 and part (ii). Step 2. Let (p, e) ∈ F(M ). Then H(p,e) ⊂ T(p,e)F(M ). Moreover, every horizontal curve β : R → F(M ) satisﬁes ˙β(t) ∈ Hβ(t) for all t ∈ R. Fix a tangent vector v ∈ TpM, let γ : R → M be a smooth curve satisfying γ(0) = p and ˙γ(0) = v, and let β : R → F(M ) be the horizontal lift of γ with β(0) = (p, e). Then β(t) = (γ(t), e(t)), e(t) := Φγ(t, 0)e. Fix a vector ξ ∈ Rm and consider the vector �
�eld X(t) := e(t)ξ = Φγ(t, 0)eξ along γ. This vector ﬁeld is parallel and hence, by the Gauß–Weingarten formula, it satisﬁes ˙e(0)ξ = ˙X(0) = hγ(0)( ˙γ(0), X(0)) = hp(v)eξ. Here we have used (3.3.4). Thus (v, hp(v)e) = ( ˙γ(0), ˙e(0)) = ˙β(0) ∈ Tβ(0)F(M ) = T(p,e)F(M ) and so H(p,e) ⊂ T(p,e)F(M ). Moreover, and this proves Step 2. ˙β(0) = (v, hp(v)e) ∈ H(p,e) = Hβ(0) 144 CHAPTER 3. THE LEVI-CIVITA CONNECTION Step 3. We prove part (i). We have V(p,e) ⊂ T(p,e)F(M ) by Step 1 and H(p,e) ⊂ T(p,e)F(M ) by Step 2. Moreover H(p,e) ∩ V(p,e) = {0} and so T(p,e)F(M ) = H(p,e) ⊕ V(p,e) for dimensional reasons. This proves Step 3. Step 4. We prove parts (iii) and (iv). By Step 2 every horizontal curve β : R → F(M ) satisﬁes ˙β(t) ∈ Hβ(t). Conversely, let R → F(M ) : t → β(t) = (γ(t), e(t)) be a smooth curve satisfying ˙β(t) ∈ Hβ(t) for all t. Then ˙e(t) = hγ(t)( ˙γ(t))e(t) for all t. By the Gauß– Weingarten formula (3.2.2) this implies that the vector ﬁeld X(

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