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t) = e(t)ξ along γ is parallel for every ξ ∈ Rm, so β is horizontal. This proves part (iii). Part (iv) follows from (iii) and the fact that the horizontal tangent bundle H ⊂ T F(M ) is invariant under the induced action of the group GL(m) on T F(M ). This proves Lemma 3.4.7. Figure 3.3: The frame bundle. The reason for the terminology introduced in Deﬁnition 3.4.6 is that one draws the extremely crude picture of the frame bundle displayed in Figure 3.3. One thinks of F(M ) as “lying over” M. One would then represent the equation γ = π ◦ β by the following commutative diagram: β γ F(M ) ; π M R hence the word “lift”. The vertical space is tangent to the vertical line in Figure 3.3 while the horizontal space is transverse to the vertical space. This crude imagery can be extremely helpful. p(M)FπMp(p,e)(M)F=π−1(p) 3.4. THE FRAME BUNDLE 145 Exercise 3.4.8. The group GL(m) acts on F(M ) by diﬀeomorphisms. Thus for each a ∈ GL(m) the map F(M ) → F(M ) : (p, e) → a∗(p, e) = (p, e ◦ a) is a diﬀeomorphism of F(M ). The derivative of this diﬀeomorphism is a diﬀeomorphism of the tangent bundle T F(M ) and this is called the induced action of GL(m) on T F(M ). Prove that the horizontal and vertical subbundles are invariant under the induced action of GL(m) on T F(M ). Exercise 3.4.9. Prove that H(p,e) ⊂ T(p,e)O(M ) and that T(p,e)O(M ) = H(p,e) ⊕ V (p,e), V (p,e) := V(p,e) ∩ T(p,e)O(M )
for every (p, e) ∈ O(M ). The following deﬁnition introduces an important class of vector ﬁelds on the frame bundle that will play a central role in Section 3.5. They will be used to prove the Development Theorem 3.5.21 in §3.5.4 below. Deﬁnition 3.4.10 (Basic vector ﬁeld). Every vector ξ ∈ Rm determines a vector ﬁeld Bξ ∈ Vect(F(M )) deﬁned by Bξ(p, e) := eξ, hp(eξ)e (3.4.9) for (p, e) ∈ F(M ). This vector ﬁeld is horizontal, i.e. Bξ(p, e) ∈ H(p,e), and projects to eξ, i.e. dπ(p, e)Bξ(p, e) = eξ for all (p, e) ∈ F(M ). These two conditions determine the vector ﬁeld Bξ uniquely. It is called the basic vector ﬁeld corresponding to ξ. Exercise 3.4.11. (i) Prove that every basic vector ﬁeld Bξ ∈ Vect(F(M )) is tangent to the orthonormal frame bundle O(M ). (ii) Let R → F(M ) : t → (γ(t), e(t)) be an integral curve of the vector ﬁeld Bξ and a ∈ GL(m). Prove that R → F(M ) : t → a∗β(t) = (γ(t), a∗e(t)) is an integral curve of Ba−1ξ. (iii) Prove that the vector ﬁeld Bξ ∈ Vect(F(M )) is complete for all ξ ∈ Rm if and only if the restricted vector ﬁeld Bξ\|O(M ) ∈ Vect(O(M )) on the orthonormal frame bundle is complete for all ξ ∈ Rm. Deﬁnition 3.4.12 (Complete manifold). A smoth m-manifold
M ⊂ Rn is called complete iﬀ, for every smooth curve ξ : R → Rm and every element (p0, e0) ∈ F(M ), there exists a smooth curve β : R → F(M ) such that β(0) = (p0, e0) and ˙β(t) = Bξ(t)(β(t)) for all t ∈ R. 146 CHAPTER 3. THE LEVI-CIVITA CONNECTION 3.5 Motions and Developments Our aim in this sections is to deﬁne motion without sliding, twisting, or wobbling. This is the motion that results when a heavy object is rolled, with a minimum of friction, along the ﬂoor. It is also the motion of the large snowball a child creates as it rolls it into the bottom part of a snowman. We shall eventually justify mathematically the physical intuition that either of the curves of contact in such ideal rolling may be speciﬁed arbitrarily; the other is then determined uniquely. Thus for example the heavy object may be rolled along an arbitrary curve on the ﬂoor; if that curve is marked in wet ink, another curve will be traced in the object. Conversely, if a curve is marked in wet ink on the object, the object may be rolled so as to trace a curve on the ﬂoor. However, if both curves are prescribed, it will be necessary to slide the object as it is being rolled, if one wants to keep the curves in contact. We assume throughout this section that M and M are two m-dimensional submanifolds of Rn. Objects on M will be denoted by the same letter as the corresponding objects on M with primes aﬃxed. Thus for example Π(p) ∈ Rn×n denotes the orthogonal projection of Rn onto the tangent space TpM, ∇ denotes the covariant derivative of a vector ﬁeld along a curve in M, and Φ γ denotes parallel transport along a curve in M. 3.5.1 Motion Deﬁnition 3.5.1. A motion of M along M (on an interval I ⊂ R) is a triple (Ψ, γ, γ) of smooth maps Ψ : I → O(n), γ : I → M
, γ : I → M such that Ψ(t)Tγ(t)M = Tγ(t)M ∀ t ∈ I. Note that a motion also matches normal vectors, i.e. Ψ(t)Tγ(t)M ⊥ = Tγ(t)M ⊥ ∀ t ∈ I. Remark 3.5.2. Associated to a motion (Ψ, γ, γ) of M along M is a family of (aﬃne) isometries ψt : Rn → Rn deﬁned by ψt(p) := γ(t) + Ψ(t)p − γ(t) (3.5.1) for t ∈ I and p ∈ Rn. These isometries satisfy ψt(γ(t)) = γ(t), dψt(γ(t))Tγ(t)M = Tγ(t)M ∀ t ∈ I. 3.5. MOTIONS AND DEVELOPMENTS 147 Remark 3.5.3. There are three operations on motions. If (Ψ, γ, γ) is a motion of M along M on an inReparametrization. terval I ⊂ R and σ : J → I is a smooth map between intervals, then the triple (Ψ ◦ σ, γ ◦ σ, γ ◦ σ) is a motion of M along M on the interval J. Inversion. If (Ψ, γ, γ) is a motion of M along M, then (Ψ−1, γ, γ) is a motion of M along M. Composition. and (Ψ, γ, γ) is a motion of M along M on the same interval, then If (Ψ, γ, γ) is a motion of M along M on an interval I (ΨΨ, γ, γ) is a motion of M along M. We now give the three simplest examples of “bad” motions; i.e. motions which do not satisfy the concepts we are about to deﬁne. In all three of these examples, p is a point of M and M is the aﬃne tangent space
to M at p: M := p + TpM = {p + v \| v ∈ TpM }. Example 3.5.4 (Pure sliding). Take a nonzero tangent vector v ∈ TpM and let γ(t) := p, γ(t) = p + tv, Ψ(t) := 1l. Then ˙γ(t) = 0, ˙γ(t) = v = 0, and so Ψ(t) ˙γ(t) = ˙γ(t). (See Figure 3.4.) Figure 3.4: Pure sliding. MpM’ 148 CHAPTER 3. THE LEVI-CIVITA CONNECTION Example 3.5.5 (Pure twisting). Let γ and γ be the constant curves γ(t) = γ(t) = p and take Ψ(t) to be the identity on TpM ⊥ and any curve of rotations on the tangent space TpM. As a concrete example with m = 2 and n = 3 one can take M to be the sphere of radius one centered at the point (0, 1, 0) and p to be the origin: M := (x, y, z) ∈ R3 \| x2 + (y − 1)2 + z2 = 1, p := (0, 0, 0). Then M is the (x, z)-plane and A(t) is any curve of rotations in the (x, z)plane, i.e. about the y-axis TpM ⊥. (See Figure 3.5.) Figure 3.5: Pure twisting. Example 3.5.6 (Pure wobbling). This is the same as pure twisting except that Ψ(t) is the identity on TpM and any curve of rotations on TpM ⊥. As a concrete example with m = 1 and n = 3 one can take M to be the circle of radius one in the (x, y)-plane centered at the point (0, 1, 0) and p to be the origin: M := (x, y, 0) ∈ R3 \| x2 + (y − 1)2 = 1, p := (0, 0, 0). Then M is the x-axis and Ψ(t) is any curve
of rotations in the (y, z)-plane, i.e. about the axis M. (See Figure 3.6.) Figure 3.6: Pure wobbling. MpM’MpM’ 3.5. MOTIONS AND DEVELOPMENTS 149 3.5.2 Sliding When a train slides on the track (e.g. in the process of stopping suddenly), there is a terriﬁc screech. Since we usually do not hear a screech, this means that the wheel moves along without sliding. In other words the velocity of the point of contact in the train wheel M equals the velocity of the point of contact in the track M. But the track is not moving; hence the point of contact in the wheel is not moving. One may explain the paradox this way: the train is moving forward and the wheel is rotating around the axle. The velocity of a point on the wheel is the sum of these two velocities. When the point is on the bottom of the wheel, the two velocities cancel. Deﬁnition 3.5.7. A motion (Ψ, γ, γ) is said to be without sliding iﬀ it satisﬁes Ψ(t) ˙γ(t) = ˙γ(t) for every t. Here is the geometric picture of the no sliding condition. As explained in Remark 3.5.2 we can view a motion as a smooth family of isometries ψt(p) := γ(t) + Ψ(t)p − γ(t) acting on the manifold M with γ(t) ∈ M being the point of contact with M. Diﬀerentiating the curve t → ψt(p) which describes the motion of the point p ∈ M in the space Rn we obtain d dt ψt(p) = ˙γ(t) − Ψ(t) ˙γ(t) + ˙Ψ(t)p − γ(t). Taking p = γ(t0) we ﬁnd d dt t=t0 ψt(γ(t0)) = ˙γ(t0) − Ψ(t0) ˙γ(t0). This expression vanishes under the no sliding
condition. In general the curve t → ψt(γ(t0)) will be non-constant, but (when the motion is without sliding) its velocity will vanish at the instant t = t0; i.e. at the instant when it becomes the point of contact. In other words the motion is without sliding if and only if the point of contact is motionless. We remark that, if the motion is without sliding, we have: ˙γ(t) so that the curves γ and γ have the same arclength: t1 = \|Ψ(t) ˙γ(t)\| = \| ˙γ(t)\| t1 ˙γ(t) dt = t0 t0 \| ˙γ(t)\| dt on any interval [t0, t1] ⊂ I. Hence any motion with ˙γ = 0 and ˙γ = 0 is not without sliding (such as the example of pure sliding above). 150 CHAPTER 3. THE LEVI-CIVITA CONNECTION Exercise 3.5.8. Give an example of a motion where \| ˙γ(t)\| = \| ˙γ(t)\| for every t but which is not without sliding. Example 3.5.9. We describe mathematically the motion of the train wheel. Let the center of the wheel move right parallel to the x-axis at height one and the wheel have radius one and make one revolution in 2π units of time. Then the track M is the x-axis and we take M := (x, y) ∈ R2 \| x2 + (y − 1)2 = 1. Choose γ(t) := (cos(t − π/2), 1 + sin(t − π/2)) = (sin(t), 1 − cos(t)), γ(t) := (t, 0), and deﬁne Ψ(t) ∈ GL(2) by Ψ(t) := cos(t) − sin(t). sin(t) cos(t) The reader can easily verify that this is a motion without sliding. A ﬁxed point p0 on M, say p0 = (0, 0), sweeps out a cycloid with parametric equations x = t − sin(t), y = 1 − cos(t).
(Check that ( ˙x, ˙y) = (0, 0) when y = 0; i.e. for t = 2nπ.) Remark 3.5.10. These same formulas give a motion of a sphere M rolling without sliding along a straight line in a plane M. Namely in coordinates (x, y, z) the sphere is given by the equation x2 + (y − 1)2 + z2 = 1, the plane is y = 0 and the line is the x-axis. The z-coordinate of a point is unaﬀected by the motion. Note that the curve γ traces out a straight line in the plane M and the curve γ traces out a great circle on the sphere M. Exercise 3.5.11. The operations of reparametrization, inversion, and comi.e. if (Ψ, γ, γ) and (Ψ, γ, γ) position respect motion without sliding; are motions without sliding on an interval I and σ : J → I is a smooth map between intervals, then the motions (Ψ ◦ σ, γ ◦ σ, γ ◦ σ), (Ψ−1, γ, γ), and (ΨΨ, γ, γ) are also without sliding. 3.5. MOTIONS AND DEVELOPMENTS 151 3.5.3 Twisting and Wobbling A motion (Ψ, γ, γ) on an intervall I ⊂ R transforms vector ﬁelds along γ into vector ﬁelds along γ by the formula X (t) = (ΨX)(t) := Ψ(t)X(t) ∈ Tγ(t)M for t ∈ I and X ∈ Vect(γ); so X ∈ Vect(γ). Lemma 3.5.12. Let (Ψ, γ, γ) be a motion of M along M on an interval I ⊂ R. Then the following are equivalent. (i) The instantaneous velocity of each tangent vector is normal, i.e. for t ∈ I ˙Ψ(t)Tγ(t)M ⊂ Tγ(t)M ⊥. (ii) �
� intertwines covariant diﬀerentiation, i.e. for X ∈ Vect(γ) ∇(ΨX) = Ψ∇X. (iii) Ψ transforms parallel vector ﬁelds along γ into parallel vector ﬁelds along γ, i.e. for X ∈ Vect(γ) ∇X = 0 =⇒ ∇(ΨX) = 0. (iv) Ψ intertwines parallel transport, i.e. for s, t ∈ I and v ∈ Tγ(s)M Ψ(t)Φγ(t, s)v = Φ γ(t, s)Ψ(s)v. A motion that satisﬁes these conditions is called without twisting. Proof. We prove that (i) is equivalent to (ii). A motion satisﬁes the equation Ψ(t)Π(γ(t)) = Π(γ(t))Ψ(t) for every t ∈ I. This restates the condition that Ψ(t) maps tangent vectors of M to tangent vectors of M and normal vectors of M to normal vectors of M. Diﬀerentiating the equation X (t) = Ψ(t)X(t) we obtain ˙X (t) = Ψ(t) ˙X(t) + ˙Ψ(t)X(t). Applying Π(γ(t)) this gives ∇X = Ψ∇X + Π(γ) ˙ΨX. Hence (ii) holds if and only if Π(γ(t)) ˙Ψ(t) = 0 for every t ∈ I. Thus we have proved that (i) is equivalent to (ii). That (ii) implies (iii) is obvious. 152 CHAPTER 3. THE LEVI-CIVITA CONNECTION We prove that (iii) implies (iv). Let t0 ∈ I and v0 ∈ Tγ(t0)M. Deﬁne the vector ﬁeld X ∈ Vect(γ) by X(t) := Φγ(t, t0)v0 for t ∈ I and let X := ΨX
. Then ∇X = 0, hence ∇X = 0 by (iii), and hence X (t) = Φ γ(t, t0)X (t0) = Φ γ(t, t0)Ψ(t0)v0 for all t ∈ I. Since X (t) = Ψ(t)X(t) = Ψ(t)Φγ(t, t0)v0, this implies (iv). We prove that (iv) implies (ii). Let X ∈ Vect(γ) and X := ΨX. By (iv) we have Φ γ(t0, t)X (t) = Ψ(t0)Φγ(t0, t)X(t). Diﬀerentiating this equation with respect to t at t = t0 and using Theorem 3.3.6, we obtain ∇X (t0) = Ψ(t0)∇X(t0). This proves the lemma. Lemma 3.5.13. Let (Ψ, γ, γ) be a motion of M along M on an interval I ⊂ R. Then the following are equivalent. (i) The instantaneous velocity of each normal vector is tangent, i.e. for t ∈ I ˙Ψ(t)Tγ(t)M ⊥ ⊂ Tγ(t)M. (ii) Ψ intertwines normal covariant diﬀerentiation, i.e. for Y ∈ Vect⊥(γ) ∇⊥(ΨY ) = Ψ∇⊥Y. (iii) Ψ transforms parallel normal vector ﬁelds along γ into parallel normal vector ﬁelds along γ, i.e. for Y ∈ Vect⊥(γ) ∇⊥Y = 0 =⇒ ∇⊥(ΨY ) = 0. (iv) Ψ intertwines parallel transport of normal vector ﬁelds, i.e. for s, t ∈ I and w ∈ Tγ(s)M ⊥ Ψ(t)Φ⊥ γ (t, s)w = Φ⊥ �
�(t, s)Ψ(s)w. A motion that satisﬁes these conditions is called without wobbling. The proof that the four conditions in Lemma 3.5.13 are equivalent is word for word analogous to the proof of Lemma 3.5.12 and will be omitted. In summary a motion is without twisting iﬀ tangent vectors at the point of contact are rotating towards the normal space and it is without wobbling iﬀ normal vectors at the point of contact are rotating towards the tangent space. In case m = 2 and n = 3 motion without twisting means that the instantaneous axis of rotation is parallel to the tangent plane. 3.5. MOTIONS AND DEVELOPMENTS 153 Remark 3.5.14. The operations of reparametrization, inversion, and composition respect motion without twisting, respectively without wobbling; i.e. if (Ψ, γ, γ) and (Ψ, γ, γ) are motions without twisting, respectively without wobbling, on an interval I and σ : J → I is a smooth map between intervals, then the motions (Ψ ◦ σ, γ ◦ σ, γ ◦ σ), (Ψ−1, γ, γ), and (ΨΨ, γ, γ) are also without twisting, respectively without wobbling. Remark 3.5.15. Let I ⊂ R be an interval and t0 ∈ I. Given curves γ : I → M and γ : I → M and an orthogonal matrix Ψ0 ∈ O(n) such that Ψ0Tγ(t0)M = Tγ(t0)M there is a unique motion (Ψ, γ, γ) of M along M (with the given γ and γ) without twisting or wobbling satisfying the initial condition: Ψ(t0) = Ψ0. Indeed, the path of matrices Ψ : I → O(n) is uniquely determined by the conditions (iv) in Lemma 3.5.12 and Lemma 3.5.13. It is given by the explicit formula Ψ(t)v = Φ γ(t, t0)Ψ0Φγ(t0, t)Π(γ(t))
v + Φ⊥ γ(t, t0)Ψ0Φ⊥ γ (t0, t)v − Π(γ(t))v (3.5.2) for t ∈ I and v ∈ Rn. We prove below a somewhat harder result where the motion is without twisting, wobbling, or sliding. It is in this situation that γ and γ determine one another (up to an initial condition). Remark 3.5.16. We can now give another interpretation of parallel transport. Given γ : R → M and v0 ∈ Tγ(t0)M take M to be an aﬃne subspace of the same dimension as M. Let (Ψ, γ, γ) be a motion of M along M without twisting (and, if you like, without sliding or wobbling). Let X ∈ Vect(γ) be the constant vector ﬁeld along γ (so that ∇X = 0) with value X (t) = Ψ0v0, Ψ0 := Ψ(t0). Let X ∈ Vect(γ) be the corresponding vector ﬁeld along γ so that Ψ(t)X(t) = Ψ0v0 Then X(t) = Φγ(t, t0)v0. To put it another way, imagine that M is a ball. To deﬁne parallel transport along a given curve γ roll the ball (without sliding) along a plane M keeping the curve γ in contact with the plane M. Let γ be the curve traced out in M. If a constant vector ﬁeld in the plane M is drawn in wet ink along the curve γ, it will mark oﬀ a (covariant) parallel vector ﬁeld along γ in M. Exercise 3.5.17. Describe parallel transport along a great circle in a sphere. 154 CHAPTER 3. THE LEVI-CIVITA CONNECTION 3.5.4 Development A development is an intrinsic version of motion without sliding or twisting. Deﬁnition 3.5.18. A development of M along M (on an interval I) is a triple (Φ, γ, γ) where γ : I → M and �
� : I → M are smooth paths and Φ is a family of orthogonal isomorphisms Φ(t) : Tγ(t)M → Tγ(t)M parametrized by t ∈ I, such that Φ(t) ˙γ(t) = ˙γ(t) for all t ∈ I and Φ intertwines parallel transport, i.e. Φ(t)Φγ(t, s) = Φ γ(t, s)Φ(s) (3.5.3) (3.5.4) for all s, t ∈ I. In particular, the family Φ of isomorphisms is smooth, i.e. if X is a smooth vector ﬁeld along γ, then the formula X (t) := Φ(t)X(t) deﬁnes a smooth vector ﬁeld along γ. Lemma 3.5.19. Let I ⊂ R be an interval, γ : I → M and γ : I → M be smooth curves, and Φ(t) : Tγ(t)M → Tγ(t)M be a family of orthogonal isomorphisms parametrized by t ∈ I. Then the following are equivalent. (i) (Φ, γ, γ) is a development. (ii) Φ satisﬁes (3.5.3) and ∇(ΦX) = Φ∇X (3.5.5) for all X ∈ Vect(γ). (iii) There exists a motion (Ψ, γ, γ) without sliding and twisting such that Φ(t) = Ψ(t)\|Tγ(t)M for all t ∈ I. (3.5.6) (iv) There exists a motion (Ψ, γ, γ) of M along M without sliding, twisting, and wobbling that satisﬁes (3.5.6). Proof. That (3.5.4) is equivalent to (3.5.5) was proved in Lemma 3.5.12. This (i) is equivalent to (ii). That (iv) implies (iii) and (iii) implies (i) is obvious
. To prove that (i) implies (iv) choose any t0 ∈ I and any orthogonal matrix Ψ0 ∈ O(n) such that Ψ0\|Tγ(t0)M = Φ(t0) and deﬁne Ψ(t) : Rn → Rn by (3.5.2). This proves Lemma 3.5.19. 3.5. MOTIONS AND DEVELOPMENTS 155 Remark 3.5.20. The operations of reparametrization, inversion, and composition yield developments when applied to developments; i.e. if (Φ, γ, γ) is a development of M along M on an interval I, (Φ, γ, γ) is a development of M along M on the same interval I, and σ : J → I is a smooth map of intervals, then the triples (Φ ◦ σ, γ ◦ σ, γ ◦ σ), (Φ−1, γ, γ)), (ΦΦ, γ, γ) are all developments. Theorem 3.5.21 (Development Theorem). Let p0 ∈ M and t0 ∈ R, let γ : R → M be a smooth curve, and let Φ0 : Tp0M → Tγ(t0)M be an orthogonal isomorphism. Then the following holds. (i) There exists a development (Φ, γ, γ\|I ) on some open interval I ⊂ R containing t0 that satisﬁes the initial condition γ(t0) = p0, Φ(t0) = Φ0. (3.5.7) (ii) Any two developments (Φ1, γ1, γ\|I1) and (Φ2, γ2, γ\|I2) as in (i) on two intervals I1 and I2 agree on the intersection I1 ∩ I2, i.e. γ1(t) = γ2(t), Φ1(t) = Φ2(t) for every t ∈ I1 ∩ I2. (iii) If M is complete, then (i) holds with I = R. Proof. Let γ
: R → M be any smooth curve such that and, for t ∈ R, deﬁne the linear map γ(t0) = p0 Φ(t) : Tγ(t)M → Tγ(t)M by Φ(t) := Φ γ(t, t0)Φ0Φγ(t0, t). (3.5.8) This is an orthogonal transformation for every t and it intertwines parallel transport. However, in general Φ(t) ˙γ(t) will not be equal to ˙γ(t). 156 CHAPTER 3. THE LEVI-CIVITA CONNECTION To construct a development that satisﬁes (3.5.3), we choose an orthonor- mal frame e0 : Rm → Tp0M and, for t ∈ R, deﬁne e(t) : Rm → Tγ(t)M by e(t) := Φγ(t, t0)e0. (3.5.9) We can think of e(t) as a real n × m-matrix and the map R → Rn×m : t → e(t) is smooth. In fact, the map t → (γ(t), e(t)) is a smooth path in the frame bundle F(M ). Deﬁne the smooth map ξ : R → Rm by ˙γ(t) = Φ γ(t, t0)Φ0e0ξ(t). (3.5.10) We prove the following. Claim: The triple (Φ, γ, γ) is a development on an interval I ⊂ R if and only if the path t → (γ(t), e(t)) satisﬁes the diﬀerential equation ( ˙γ(t), ˙e(t)) = Bξ(t)(γ(t), e(t)) (3.5.11) for every t ∈ I, where Bξ(t) ∈ Vect(F(M )) denotes the basic vector ﬁeld associated to ξ(t) ∈ Rm (see equation (3.4.9)). The triple (Φ,
γ, γ) is a development on I if and only if Φ(t) ˙γ(t) = ˙γ(t) for every t ∈ I. By (3.5.8) and (3.5.10) this is equivalent to the condition Φ γ(t, t0)Φ0Φγ(t0, t) ˙γ(t) = ˙γ(t) = Φ γ(t, t0)Φ0e0ξ(t), hence to and hence to Φγ(t0, t) ˙γ(t) = e0ξ(t), ˙γ(t) = Φγ(t, t0)e0ξ(t) = e(t)ξ(t) (3.5.12) for every t ∈ I. By (3.5.9) and the Gauß–Weingarten formula, we have ˙e(t) = hγ(t)( ˙γ(t))e(t) for every t ∈ R. Hence it follows from (3.4.9) that (3.5.12) is equivalent to (3.5.11). This proves the claim. Parts (i) and (ii) follow directly from the claim. Part (iii) follows from the claim and Deﬁnition 3.4.12. This proves Theorem 3.5.21. 3.5. MOTIONS AND DEVELOPMENTS 157 Remark 3.5.22. As any two developments (Φ1, γ1, γ\|I1) and (Φ2, γ2, γ\|I2) on two intervals I1 and I2 that satisfy the initial condition (3.5.7) agree on I1 ∩ I2 there is a development deﬁned on I1 ∪ I2. Hence there is a unique maximally deﬁned development (Φ, γ, γ\|I ), deﬁned on a maximal interval I = I(t0, p0, Φ0), associated to the initial data t0, p0, Φ0. Denote the space of initial data by P := (t, p, Φ) t
∈ R, p ∈ M, Φ : TpM → Tγ(t)M is an orthogonal transformation, (3.5.13) deﬁne the set D ⊂ R × P by D := {(t, t0, p0, Φ0) \| (t0, p0, Φ0) ∈ P, t ∈ I(t0, p0, Φ0)} (3.5.14) and let D → P : (t, t0, p0, Φ0) → (t, γ(t), Φ(t)), (3.5.15) be the map which assigns to each (t0, p0, Φ0) ∈ P and each t ∈ I(t0, p0, Φ0) the value at time t of the unique development (Φ, γ, γ\|I ) associated to the inital condition (t0, p0, Φ0) on the maximal time interval I = I(t0, p0, Φ0). Then the space P has a natural structure of a smooth manifold (in the intrinsic setting), and it follows from Theorem 2.4.9 and the proof of Theorem 3.5.21 that D is an open subset of R × P and the map (3.5.15) is smooth. The smooth structure on P can be understood as follows. The space O(γ) = (t, e) \| (γ(t), e) ∈ O(M ) is the pullback of the orthonormal frame bundle O(M ) → M under the curve γ : R → M or, equivalently, is the orthonormal frame bundle of the pullback tangent bundle (γ)∗T M. Thus O(γ) is a smooth submanifold of R × Rn×m. The group O(m) acts diagonally on O(γ) × O(M ) and the action is free. Hence the quotient (O(γ) × O(M ))/O(m) is a smooth manifold by Theorem 2.9.14, and it can be naturally identiﬁed with P via the bijection [(t, e), (p, e)] → (t, p, e ◦ e−1).
Remark 3.5.23. The statement of Theorem 3.5.21 is essentially symmetric in M and M as the operation of inversion carries developments to developments. Hence given γ : R → M, 0 ∈ M, p t0 ∈ R, Φ0 : Tγ(t0)M → Tp 0 M, we may speak of the development (Φ, γ, γ) corresponding to γ with initial conditions γ(t0) = p 0 and Φ(t0) = Φ0. 158 CHAPTER 3. THE LEVI-CIVITA CONNECTION Corollary 3.5.24 (Motions). Let p0 ∈ M and t0 ∈ R, let γ : R → M be a smooth curve, and let Ψ0 ∈ O(n) be a matrix such that Ψ0Tp0M = Tγ(t0)M. Then the following holds. (i) There exists a motion (Ψ, γ, γ\|I ) without sliding, twisting and wobbling on some open interval I ⊂ R containing t0 that satisﬁes the initial condition γ(t0) = p0 and Ψ(t0) = Ψ0. (ii) Any two motions as in (i) on two intervals I1 and I2 agree on the intersection I1 ∩ I2. (iii) If M is complete, then (i) holds with I = R. Proof. Theorem 3.5.21 and Remark 3.5.15. Corollary 3.5.25 (Completeness). The following are equivalent. (i) M is complete, i.e. for every smooth curve ξ : R → Rm and every element (p0, e0) ∈ F(M ), there exists a smooth curve β : R → F(M ) such that ˙β(t) = Bξ(t)(β(t)) for all t ∈ R and β(0) = (p0, e0) (Deﬁnition 3.4.12). (ii) For every smooth curve ξ : R → Rm and every element (p0, e0) ∈ O(M ), there is a smooth curve α : R → O(M ) such that
˙α(t) = Bξ(t)(α(t)) for every t ∈ R and α(0) = (p0, e0). (iii) For every smooth curve γ : R → Rm, every p0 ∈ M, and every orthogonal isomorphism Φ0 : Tp0M → Rm there exists a development (Φ, γ, γ) of M along M = Rm on all of R that satisﬁes γ(0) = p0 and Φ(0) = Φ0. Proof. We have already noted that the basic vector ﬁelds are all tangent to the orthonormal frame bundle O(M ) ⊂ F(M ). Now note that if a smooth curve I → F(M ) : t → β(t) = (γ(t), e(t)) on an interval I ⊂ R satisﬁes the diﬀerential equation ˙β(t) = Bξ(t)(β(t)) for all t, then so does the curve I → F(M ) : t → a∗β(t) = (γ(t), e(t) ◦ a) for every a ∈ GL(m, R). Since any frame e0 : Rm → Tp0M can be carried to any other (in particular an orthonormal one) by a suitable matrix a ∈ GL(m, R), this shows that (i) is equivalent to (ii). That (i) implies (iii) was proved in Theorem 3.5.21. 3.5. MOTIONS AND DEVELOPMENTS 159 We prove that (iii) implies (ii). Fix a smooth map ξ : R → Rm and an element (p0, e0) ∈ O(M ). Deﬁne and Φ0 := e−1 0 : Tp0M → Rm γ(t) := t 0 ξ(s) ds ∈ Rm for t ∈ R. By (ii) there exists a development (Φ, γ, γ) of M along Rm on all of R that satisﬁes the initial conditions γ(0) = p0, Φ(0) = Φ0. Then �
�(t) = Φ0Φγ(0, t) : Tγ(t)M → Rm, Φ(t) ˙γ(t) = ˙γ(t) = ξ(t) for all t ∈ R by Deﬁnition 3.5.18. Deﬁne e(t) := Φγ(t, 0)e0 = Φ(t)−1 : Rm → Tγ(t)M for t ∈ R. Then (γ, e) : R → F(M ) is a smooth curve that satisﬁes the initial condition (γ(0), e(0)) = (p0, e0) and the diﬀerential equation ˙γ(t) = Φ(t)−1ξ(t) = e(t)ξ(t), ˙e(t) = hγ(t)( ˙γ(t))e(t) = hγ(t)(e(t)ξ(t))e(t) by the Gauß–Weingarten formula. Thus ( ˙γ(t), ˙e(t)) = Bξ(t)(γ(t), e(t)) for all t ∈ R. This proves Corollary 3.5.25. It is of course easy to give an example of a manifold which is not complete; e.g. if (Φ, γ, γ) is any development of M along M, then M \ {γ(t1)} is not complete as the given development is only deﬁned for t = t1. In §4.6 we give equivalent characterizations of completeness. In particular, we will see that any compact submanifold of Rn is complete. Exercise 3.5.26. An aﬃne subspace of Rn is a subset of the form where E ⊂ Rn is a linear subspace and p ∈ Rn. Prove that every aﬃne subspace of Rn is a complete submanifold. 160 CHAPTER 3. THE LEVI-CIVITA CONNECTION 3.6 Christoﬀel Symbols The goal of this subsection is to examine the covariant derivative in local coordinates on an
embedded manifold M ⊂ Rn of dimension m. Let φ : U → Ω be a coordinate chart, deﬁned on an M -open subset U ⊂ M with values in an open set Ω ⊂ Rm, and denote its inverse by ψ := φ−1 : Ω → U ⊂ M. At this point it is convenient to use superscripts for the coordinates of a vector x ∈ Ω. Thus we write x = (x1,..., xm) ∈ Ω. If p = ψ(x) ∈ U is the corresponding element of M, then the tangent space of M at p is the image of the linear map dψ(x) : Rm → Rn (Theorem 2.2.3) and thus two tangent vectors v, w ∈ TpM can be written in the form v = dψ(x)ξ = w = dψ(x)η = m i=1 m j=1 ξi ∂ψ ∂xi (x), ηj ∂ψ ∂xj (x) (3.6.1) for ξ = (ξ1,..., ξm) ∈ Rm and η = (η1,..., ηm) ∈ Rm. Recall that the restriction of the inner product in the ambient space Rn to the tangent space is the ﬁrst fundamental form gp : TpM × TpM → R (Deﬁnition 3.1.1). Thus gp(v, w) = v, w = m i,j=1 ξigij(x)ηj, (3.6.2) where the functions gij : Ω → R are deﬁned by gij(x) := ∂ψ ∂xi (x), ∂ψ ∂xj (x) for x ∈ Ω. (3.6.3) In other words, the ﬁrst fundamental form is in local coordinates represented by the matrix valued function g = (gij)m i,j=1 : Ω → Rm×m. 3.6. CHRISTOFFEL SYMBOLS
161 Figure 3.7: A vector ﬁeld along a curve in local coordinates. Now let c = (c1,..., cm) : I → Ω be a smooth curve in Ω, deﬁned on an interval I ⊂ R, and consider the curve see Figure 3.7). Our goal is to describe the operator X → ∇X on the space of vector ﬁelds along γ in local coordinates. Let X : I → Rn be a vector ﬁeld along γ. Then X(t) ∈ Tγ(t)M = Tψ(c(t))M = im dψ(c(t)) : Rm → Rn for every t ∈ I and hence there exists a unique smooth function ξ = (ξ1,..., ξm) : I → Rm such that X(t) = dψ(c(t))ξ(t) = m i=1 ξi(t) ∂ψ ∂xi (c(t)). Diﬀerentiate this identity to obtain ˙X(t) = m i=1 ˙ξi(t) ∂ψ ∂xi (c(t)) + m i,j=1 ξi(t) ˙cj(t) ∂2ψ ∂xi∂xj (c(t)). (3.6.4) (3.6.5) We examine the projection ∇X(t) = Π(γ(t)) ˙X(t) of this vector onto the tangent space of M at γ(t). The ﬁrst summand on the right in (3.6.5) is already tγ( )X(t)c(t)UΩψφ Mξ( )t 162 CHAPTER 3. THE LEVI-CIVITA CONNECTION tangent to M. For the second summand we simply observe that the vector Π(ψ(x))∂2ψ/∂xi∂xj lies in tangent space Tψ(x)M and can therefore be expressed as a linear combination of the basis vectors ∂ψ/∂x1,.
.., ∂ψ/∂xm. The coeﬃcients will be denoted by Γk ij(x). Thus there exist smooth functions Γk ij : Ω → R for i, j, k = 1,..., m such that Π(ψ(x)) ∂2ψ ∂xi∂xj (x) = m k=1 Γk ij(x) ∂ψ ∂xk (x) (3.6.6) ij : Ω → R are called for all x ∈ Ω and all i, j ∈ {1,..., m}. The coeﬃcients Γk the Christoﬀel symbols associated to the coordinate chart φ : U → Ω. To sum up we have proved the following. Lemma 3.6.1. Let c : I → Ω be a smooth curve and deﬁne γ := ψ ◦ c : I → M. If ξ : I → Rm is a smooth map and X ∈ Vect(γ) is given by (3.6.4), then its covariant derivative at time t ∈ I is given by ∇X(t) =   ˙ξk(t) + m k=1 m i,j=1  ij(c(t))ξi(t) ˙cj(t) Γk  ∂ψ ∂xk (c(t)), (3.6.7) where the Γk ij are the Christoﬀel symbols deﬁned by (3.6.6). Our next goal is to understand how the Christoﬀel symbols are determined by the metric in local coordinates. Recall from equation (3.6.2) that the inner products on the tangent spaces inherited from the standard Euclidean inner product on the ambient space Rn are in local coordinates represented by the matrix valued function g = (gij)m i,j=1 : Ω → Rm×m given by gij := ∂ψ ∂xi, ∂ψ ∂xj. Rn (3.6.8) We
shall see that the Christoﬀel symbols are completely determined by the functions gij : Ω → R. Here are ﬁrst some elementary observations. 3.6. CHRISTOFFEL SYMBOLS 163 Remark 3.6.2. The matrix g(x) ∈ Rm×m is symmetric and positive deﬁnite for every x ∈ Ω. This follows from the fact that the matrix dψ(x) ∈ Rn×m has rank m and the matrix g(x) is given by Thus ξTg(x)ξ = \|dψ(x)ξ\|2 > 0 for all ξ ∈ Rm \ {0}. g(x) = dψ(x)Tdψ(x) Remark 3.6.3. For x ∈ Ω we have det(g(x)) > 0 by Remark 3.6.2 and so the matrix g(x) is invertible. Denote the entries of the inverse matrix g(x)−1 ∈ Rm×m by gk(x). They are determined by the condition m j=1 gij(x)gjk(x) = δk i = 1, 0, if i = k, if i = k. Since g(x) is symmetric and positive deﬁnite, so is its inverse matrix g(x)−1. In particular, we have gk(x) = gk(x) for all x ∈ Ω and all k, ∈ {1,..., m}. Remark 3.6.4. Suppose that X, Y ∈ Vect(γ) are vector ﬁelds along our curve γ = ψ ◦ c : I → M and ξ, η : I → Rm are deﬁned by X(t) = m i=1 ξi(t) ∂ψ ∂xi (c(t)), Y (t) = m j=1 ηj(t) ∂ψ ∂xj (c(t)). Then the inner product of X(t) and Y (t) is given by X(t), Y (t) = m i,j=1 ξi(t)
gij(c(t))ηj(t). Lemma 3.6.5 (Christoﬀel symbols). Let Ω ⊂ Rm be an open set and let gij : Ω → R for i, j = 1,..., m be smooth functions such that each maij : Ω → R be trix (gij(x))m i,j=1 is symmetric and positive deﬁnite. Let Γk smooth functions for i, j, k = 1,..., m. Then the Γk ij satisfy the conditions ij = Γk Γk ji, ∂gij ∂x = m k=1 gikΓk j + gjkΓk i for i, j, k, = 1,..., m if and only if they are given by Γk ij = m =1 gk 1 2 ∂gi ∂xj + ∂gj ∂xi − ∂gij ∂x. (3.6.9) (3.6.10) If the Γk isfy (3.6.9) and hence are given by (3.6.10). ij are deﬁned by (3.6.6) and the gij by (3.6.8), then the Γk ij sat- 164 CHAPTER 3. THE LEVI-CIVITA CONNECTION Proof. Suppose that the Γk ij are given by (3.6.6) and the gij by (3.6.8). Let c : I → Ω, ξ, η : I → Rm be smooth functions and suppose that the vector ﬁelds X, Y along the curve γ := ψ ◦ c : I → M are given by X(t) := m i=1 ξi(t) ∂ψ ∂xi (c(t)), Y (t) := m j=1 ηj(t) ∂ψ ∂xj (c(t)). Dropping the argument t in each term, we obtain from Remark 3.6.4 and Lemma 3.6.1 that X, Y = X, ∇Y = ∇X,
Y = i,j i,k j,k gij(c)ξiηj, gik(c)ξi   ˙ηk + j,  j(c)ηj ˙c Γk ,   ˙ξk + gkj(c)   ηj. i(c)ξi ˙c Γk i, Hence it follows from equation (3.2.5) in Lemma 3.2.4 that 0 = d dt X, Y − X, ∇Y − ∇X, Y = gij(c) ˙ξiηj + gij(c)ξi ˙ηj + ∂gij ∂x (c)ξiηj ˙c i,j − − i,k gik(c)ξi ˙ηk − gkj(c) ˙ξkηj − i,j,k, j(c)ξiηj ˙c gik(c)Γk gkj(c)Γk i(c)ξiηj ˙c j,k i,j, = i,j,k, ∂gij ∂x (c) − k gik(c)Γk j(c) − k gjk(c)Γk i(c) ξiηj ˙c. This holds for all smooth maps c : I → Ω and ξ, η : I → Rm, so the Γk ij satisfy the second equation in (3.6.9). That they are symmetric in i and j is obvious. 3.6. CHRISTOFFEL SYMBOLS 165 To prove that (3.6.9) is equivalent to (3.6.10), deﬁne Γij := m k=1 gkΓk ij. Then (3.6.9) is equivalent to Γij = Γji, ∂gij ∂x = Γij + Γji. and (3.6.10) is equivalent to �
�ij = ∂gi ∂xj + 1 2 ∂gj ∂xi − ∂gij ∂x. If the Γij are given by (3.6.13), then they satisfy Γij = Γji (3.6.11) (3.6.12) (3.6.13) and 2Γij + 2Γji = ∂gij ∂x + ∂gij ∂x = 2 ∂gi ∂xj − ∂gj ∂xi + ∂gji ∂x + ∂gj ∂xi − ∂gi ∂xj for all i, j,. Conversely, if the Γij satisfy (3.6.12), then ∂gij ∂x = Γij + Γji, ∂gi ∂xj = Γij + Γij = Γij + Γij, ∂gj ∂xi = Γji + Γji = Γij + Γji. Take the sum of the last two minus the ﬁrst of these equations to obtain ∂gi ∂xj + ∂gj ∂xi − ∂gij ∂x = 2Γij. Thus (3.6.12) is equivalent to (3.6.13) and so (3.6.9) is equivalent to (3.6.10). This proves Lemma 3.6.5. 166 CHAPTER 3. THE LEVI-CIVITA CONNECTION 3.7 Riemannian Metrics* We wish to carry over the fundamental notions of diﬀerential geometry to the intrinsic setting. First we need an inner product on the tangent spaces to replace the ﬁrst fundamental form in Deﬁnition 3.1.1. This is the content of Deﬁnition 3.7.1 and Lemma 3.7.4 below. Second we must introduce the covariant derivative of a vector ﬁeld along a curve. With this understood all the deﬁnitions, theorems, and proofs in this chapter carry over in an almost word by word fashion to the intrinsic setting. 3.7.1 Existence of Riemannian Metrics We will always consider norms that are induced by inner products.
But in general there is no ambient space that can induce an inner product on each tangent space. This leads to the following deﬁnition. Deﬁnition 3.7.1. Let M be a smooth m-manifold. A Riemannian metric on M is a collection of inner products TpM × TpM → R : (v, w) → gp(v, w), one for every p ∈ M, such that the map M → R : p → gp(X(p), Y (p)) is smooth for every pair of vector ﬁelds X, Y ∈ Vect(M ). We will also denote the inner product by v, wp and drop the subscript p if the base point is understood from the context. A smooth manifold equipped with a Riemannian metric is called a Riemannian manifold. Example 3.7.2. If M ⊂ Rn is a smooth submanifold, then a Riemannian metric on M is given by restricting the standard inner product on Rn to the tangent spaces TpM ⊂ Rn. This is the ﬁrst fundamental form of an embedded manifold (see Deﬁnition 3.1.1). More generally, assume that M is a Riemannian m-manifold in the intrinsic sense of Deﬁnition 3.7.1 with an atlas A = {(φα, Uα)}α∈A. Then the Riemannian metric g determines a collection of smooth functions gα = (gα,ij)m i,j=1 : φα(Uα) → Rm×m, (3.7.1) one for each α ∈ A, deﬁned by ξTgα(x)η := gp(v, w), φα(p) = x, dφα(p)v = ξ, dφα(p)w = η, (3.7.2) for x ∈ φα(Uα) and ξ, η ∈ Rm. 3.7. RIEMANNIAN METRICS* 167 Each matrix gα(x) is symmetrix and positive deﬁnite. Note that the tangent vectors v and w
in (3.7.2) can also be written in the form v = [α, ξ]p, w = [α, η]p. Choosing standard basis vectors in Rm we obtain and hence ξ = ei, η = ej [α, ei]p = dφα(p)−1ei =: ∂ ∂xi (p) gα,ij(x) = ∂ ∂xi (φ−1 α (x)), ∂ ∂xj (φ−1 α (x)). (3.7.3) For diﬀerent coordinate charts the maps gα and gβ are related through the transition map φβα := φβ ◦ φ−1 α : φα(Uα ∩ Uβ) → φβ(Uα ∩ Uβ) via gα(x) = dφβα(x)Tgβ(φβα(x))dφβα(x) (3.7.4) for x ∈ φα(Uα ∩ Uβ). Equation (3.7.4) can also be written in the shorthand notation gα = φ∗ βαgβ for α, β ∈ A. Exercise 3.7.3. Every collection of smooth maps gα : φα(Uα) → Rm×m with values in the set of positive deﬁnite symmetric matrices that satisﬁes (3.7.4) for all α, β ∈ A determines a global Riemannian metric via (3.7.2). In this intrinsic setting there is no canonical metric on M (such as the metric induced by Rn on an embedded manifold). In fact, it is not completely obvious that a manifold admits a Riemannian metric and this is the content of the next lemma. 168 CHAPTER 3. THE LEVI-CIVITA CONNECTION Lemma 3.7.4. Every paracompact Hausdorﬀ manifold admits a Riemannian metric. Proof. Let m be the dimension of M and let A = {(φα, Uα)}α∈A be an atlas on M. By Theorem 2.9.9 there is a partition of unity {
θα}α∈A subordinate to the open cover {Uα}α∈A. Now there are two equivalent ways to construct a Riemannian metric on M. The ﬁrst method is to carry over the standard inner product on Rm to the tangent spaces TpM for p ∈ Uα via the coordinate chart φα, multiply the resulting Riemannian metric on Uα by the compactly supported function θα, extend it by zero to all of M, and then take the sum over all α. This leads to the following formula. The inner product of two tangent vectors v, w ∈ TpM is deﬁned by v, wp := p∈Uα θα(p)dφα(p)v, dφα(p)w, (3.7.5) where the sum runs over all α ∈ A with p ∈ Uα and the inner product is the standard inner product on Rm. Since supp(θα) ⊂ Uα for each α and the sum is locally ﬁnite we ﬁnd that the function M → R : p → X(p), Y (p)p is smooth for every pair of vector ﬁelds X, Y ∈ Vect(M ). Moreover, the right hand side of (3.7.5) is symmetric in v and w and is positive for v = w = 0 because each summand is nonnegative and each summand with θα(p) > 0 is positive. Thus equation (3.7.5) deﬁnes a Riemannian metric on M. The second method is to deﬁne the functions gα : φα(Uα) → Rm×m by gα(x) := γ∈A θγ(φ−1 α (x))dφγα(x)Tdφγα(x) (3.7.6) for x ∈ φα(Uα) where each summand is deﬁned on φα(Uα ∩ Uγ) and is understood to be zero for x /∈ φα(Uα ∩ Uγ). We leave it to the reader to verify that these functions are smooth and satisfy the
condition (3.7.4) for all α, β ∈ A. Moreover, the formulas (3.7.5) and (3.7.6) determine the same Riemannian metric on M. (Prove this!) This proves Lemma 3.7.4. 3.7. RIEMANNIAN METRICS* 169 3.7.2 Two Examples Example 3.7.5 (Fubini–Study metric). The complex projective space carries a natural Riemannian metric, deﬁned as follows. Identify CPn with the quotient of the unit sphere S2n+1 ⊂ Cn+1 by the diagonal action of the circle S1, i.e. CPn = S2n+1/S1. Then the tangent space of CPn at the equivalence class [z] = [z0 : · · · : zn] ∈ CPn of a point z = (z0,..., zn) ∈ S2n+1 can be identiﬁed with the orthogonal complement of Cz in Cn+1. Now choose the inner product on T[z]CPn to be the one inherited from the standard inner product on Cn+1 via this identiﬁcation. The resulting metric on CPn is called the Fubini–Study metric. Exercise: Prove that the action of U(n + 1) on Cn+1 induces a transitive action of the quotient group PSU(n + 1) := U(n + 1)/S1 by isometries. If z ∈ S2n+1, prove that the unitary matrix g := 2zz∗ − 1l descends to an isometry φ on CPn with ﬁxed point p := [z] and dφ(p) = −id. Show that, in the case n = 1, the pullback of the Fubini–Study metric on CP1 under the stereographic projection S2 \ {(0, 0, 1)} → CP1 \ {[0 : 1]} : (x1, x2, x3) → 1 : x1 + ix2 1 − x3 is one quarter of the standard metric on S2. Example 3.7.6. Think of the complex Grassmannian Gk(Cn
) of k-planes in Cn as a quotient of the space Fk(Cn) := D ∈ Cn×k \| D∗D = 1l of unitary k-frames in Cn by the right action of the unitary group U(k). The space Fk(Cn) inherits a Riemannian metric from the ambient Euclidean space Cn×k. Show that the tangent space of Gk(Cn) at a point Λ = imD, with D ∈ Fk(Cn) can be identiﬁed with the space TΛGk(Cn) = D ∈ Cn×k \| D∗ D = 0. Deﬁne the inner product on this tangent space to be the restriction of the standard inner product on Cn×k to this subspace. Exercise: Prove that the unitary group U(n) acts on Gk(Cn) by isometries. 170 CHAPTER 3. THE LEVI-CIVITA CONNECTION 3.7.3 The Levi-Civita Connection A subtle point in this discussion is how to extend the notion of covariant derivative to general Riemannian manifolds. In this case the idea of projecting the derivative in the ambient space orthogonally onto the tangent space has no obvious analogue. Instead we shall see how the covariant derivatives of vector ﬁelds along curves can be characterized by several axioms and these can be used to deﬁne the covariant derivative in the intrinsic setting. An alternative, but somewhat less satisfactory, approach is to carry over the formula for the covariant derivative in local coordinates to the intrinsic setting and show that the result is independent of the choice of the coordinate chart. Of course, these approaches are equivalent and lead to the same result. We formulate them as a series of exercises. The details are straightforward. Assume throughout that M is a Riemannian m-manifold with an atlas A = {(φα, Uα)}α∈A and suppose that the Riemannian metric is in local coordinates given by gα = (gα,ij)m i,j=1 : φα(Uα) → Rm×m for α ∈ A. These functions satisfy (3.7.4) for all α, β ∈
A. Deﬁnition 3.7.7. Let f : N → M be a smooth map between manifolds. A vector ﬁeld along f is a collection of tangent vectors X(q) ∈ Tf (q)M, one for each q ∈ N, such that the map N → T M : q → (f (q), X(q)) is smooth. The space of vector ﬁelds along f will be denoted by Vect(f ). As before we will not distinguish in notation between the collection of tangent vectors X(q) ∈ Tf (q)M and the associated map N → T M and denote them both by X. The following theorem introduces the Levi-Civita connection as a collection of linear operators ∇ : Vect(γ) → Vect(γ), one for each smooth curve γ : I → M. 3.7. RIEMANNIAN METRICS* 171 Theorem 3.7.8 (Levi-Civita connection). There exists a unique collection of linear operators ∇ : Vect(γ) → Vect(γ) (called the covariant derivative), one for every smooth curve γ : I → M on an open interval I ⊂ R, satisfying the following axioms. (Leibniz Rule) For every smooth curve γ : I → M, every smooth function λ : I → R, and every vector ﬁeld X ∈ Vect(γ), we have ∇(λX) = ˙λX + λ∇X. (3.7.7) (Chain Rule) Let Ω ⊂ Rn be an open set, let c : I → Ω be a smooth curve, let γ : Ω → M be a smooth map, and let X be a smooth vector ﬁeld along γ. Denote by ∇iX the covariant derivative of X along the curve xi → γ(x) (with the other coordinates ﬁxed). Then ∇iX is a smooth vector ﬁeld along γ and the covariant derivative of the vector ﬁeld X ◦ c ∈ Vect(γ ◦ c) is ∇(X ◦ c) = n j=1 ˙c
j(t)∇jX(c(t)). (3.7.8) (Riemannian) For any two vector ﬁelds X, Y ∈ Vect(γ) we have d dt X, Y = ∇X, Y + X, ∇Y. (3.7.9) (Torsion-free) Let I, J ⊂ R be open intervals and 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 by ∇t the covariant derivative along the curve t → γ(s, t) (with s ﬁxed). Then ∇s∂tγ = ∇t∂sγ. (3.7.10) Proof. The proof is based on a reformulation of the axioms in local coordinates. The (Leibnitz Rule) and (Chain Rule) axioms assert that the covariant derivative is in local coordinates given by Christoﬀel symbols Γk ij as in equation (3.6.7) in Lemma 3.6.1. The (Riemannian) and (Torsion-free) axioms assert that the Christoﬀel symbols satisfy the equations in (3.6.9) and hence, by Lemma 3.6.5, are given by (3.6.10). (See also Exercise 3.7.10.) This proves Theorem 3.7.8. 172 CHAPTER 3. THE LEVI-CIVITA CONNECTION Exercise 3.7.9. The Christoﬀel symbols of the Riemannian metric are the functions Γk α,ij : φα(Uα) → R. deﬁned by Γk α,ij := m =1 gk α 1 2 ∂gα,i ∂xj + ∂gα,j ∂xi − ∂gα,ij ∂x (3.7.11) (see Lemma 3.6.5). Prove that they are related by the equation k ∂φk βα ∂xk Γk α,ij = ∂2φk β
α ∂xi∂xj + i,j Γk β,ij ◦ φβα ∂φi βα ∂xi ∂φj βα ∂xj. for all α, β ∈ A. Exercise 3.7.10. Denote ψα := φ−1 ant derivative of a vector ﬁeld α : φα(Uα) → M. Prove that the covari- X(t) = m i=1 ξi α(t) ∂ψα ∂xi (cα(t)) along γ = ψα ◦ cα : I → M is given by ∇X(t) =   ˙ξk α(t) + m k=1 m i,j=1  α,ij(c(t))ξi Γk α(t) ˙cj α(t)  ∂ψα ∂xk (cα(t)). (3.7.12) Prove that ∇X is independent of the choice of the coordinate chart. Exercise 3.7.11. Let Ω ⊂ R2 be open and λ : Ω → (0, ∞) be a smooth function. Let g : Ω → R2×2 be given by λ(x) 0 0 λ(x) g(x) =. Compute the Christoﬀel symbols Γk ij via (3.6.10). Exercise 3.7.12. Let φ : S2 \ {(0, 0, 1)} → C be the stereographic projection, given by φ(p) := p1 1 − p3, p2 1 − p3 Prove that the metric g : R2 → R2×2 has the form g(x) = λ(x)1l where λ(x) := 4 (1 + \|x\|2)2 for x = (x1, x2) ∈ R2. 3.7. RIEMANNIAN METRICS* 173 3.7.4 Basic Vector Fields in the Intrinsic Setting Let M be a Riemannian m-manifold with an atlas A = {(φα, U
α)}α∈A. Then the frame bundle (3.4.1) admits the structure of a smooth manifold with the open cover Uα := π−1(Uα) and coordinate charts φα : Uα → φα(Uα) × GL(m) given by φα(p, e) := (φα(p), dφα(p)e). The derivatives of the horizontal curves in Deﬁnition 3.4.6 form a horizontal subbundle H ⊂ T F(M ) of the tangent bundle of the Frame bundle whose ﬁbers H(p,e) over an element (p, e) ∈ F(M ) can in local coordinates be described as follows. Let x := φα(p), a := dφα(p)e ∈ GL(m), (3.7.13) and let (x, a) ∈ Rm × Rm×m. This pair has the form (x, a) = d φα(p, e)(p, e), (p, e) ∈ H(p,e), (3.7.14) if and only if ak = − m i,j=1 α,ij(x)xiaj Γk (3.7.15) α,ij : φα(Uα) → R are the Christoffor k, = 1,..., m, where the functions Γk fel symbols deﬁned by (3.7.11). Thus a tangent vector (p, e) ∈ T(p,e)F(M ) is horizontal if and only if its coordinates (x, a) in (3.7.14) satisfy (3.7.15). Hence, for every vector ξ ∈ Rm, there exists a unique horizontal vector ﬁeld Bξ ∈ Vect(F(M )) (the basic vector ﬁeld associated to ξ) such that dπ(p, e)Bξ(p, e) = eξ for all (p, e) ∈ F(M ). This vector ﬁeld assigns to a pair (p, e) ∈ F(M ) with the coordinates (x, a) ∈ Rm × GL(
m) as in (3.7.13) the horizontal tangent vector (p, e) ∈ H(p,e) ⊂ T(p,e)F(M ) whose coordinates (x, a) ∈ Rm × Rm×m satisfy (3.7.15) and x = aξ. Exercise 3.7.13. Verify the equivalence of (3.7.14) and (3.7.15). Prove that the notion of a horizontal tangent vector of F(M ) is independent of the choice of the coordinate chart. Hint: Use Exercise 3.7.9. Exercise 3.7.14. Examine the orthonormal frame bundle O(M ) in the intrinsic setting. Exercise 3.7.15. Carry over the proofs of Theorem 3.3.4, Theorem 3.3.6, and Theorem 3.5.21 to the intrinsic setting. 174 CHAPTER 3. THE LEVI-CIVITA CONNECTION Chapter 4 Geodesics This chapter introduces geodesics in Riemannian manifolds. It begins in §4.1 by introducing geodesics as extremals of the energy and length functionals and characterizing them as solutions of a second order diﬀerential equation. In §4.2 we show that minimizing the length with ﬁxed endpoints gives rise to an intrinsic distance function d : M × M → R which induces the topology M inherits from the ambient space Rn. §4.3 introduces the exponential map, §4.4 shows that geodesics minimize the length on short time intervals, §4.5 establishes the existence of geodesically convex neighborhoods, and §4.6 shows that the geodesic ﬂow is complete if and only if (M, d) is a complete metric space, and that in the complete case any two points are joined by a minimal geodesic. §4.7 discusses geodesics in the intrinsic setting. 4.1 Length and Energy This section explains the length and energy functionals on the space of paths with ﬁxed endpoints and introduces geodesics as their extremal points. 4.1.1 The Length and Energy Functionals The concept of a geodesic in a manifold generalizes that of a straight line in Euclidean space. A straight line has parametrizations of form
t → p + σ(t)v where σ : R → R is a diﬀeomorphism and p, v ∈ Rn. Diﬀerent choices of σ yield diﬀerent parametrizations of the same line. Certain parametrizations are preferred, for example those parametrizations which are “proportional to the arclength”, i.e. where σ(t) = at + b for constants a, b ∈ R, so that the tangent vector ˙σ(t)v has constant length. The same distinctions can 175 176 CHAPTER 4. GEODESICS be made for geodesics. Some authors use the term geodesic to include all parametrizations of a geodesic while others restrict the term to cover only geodesics parametrized proportional to the arclength. We follow the latter course, referring to the more general concept as a “reparametrized geodesic”. Thus a reparametrized geodesic need not be a geodesic. We assume throughout that M ⊂ Rn is a smooth m-manifold. Deﬁnition 4.1.1 (Length and energy). Let I = [a, b] ⊂ R be a compact interval with a < b and let γ : I → M be a smooth curve in M. The length L(γ) and the energy E(γ) are deﬁned by b L(γ) := \| ˙γ(t)\| dt, a b \| ˙γ(t)\|2 dt. E(γ) := 1 2 (4.1.1) (4.1.2) a A variation of γ is a family of smooth curves γs : I → M, where s ranges over the reals, such that the map R × I → M : (s, t) → γs(t) is smooth and γ0 = γ. The variation {γs}s∈R is said to have ﬁxed endpoints iﬀ γs(a) = γ(a) and γs(b) = γ(b) for all s ∈ R. Remark 4.1.2. The length of a continuous function
γ : [a, b] → Rn can be deﬁned as the supremum of the numbers N i=1 \|γ(ti) − γ(ti−1)\| over all partitions a = t0 < t1 < · · · < tN = b of the interval [a, b]. By a theorem in ﬁrst year analysis [64] this supremum is ﬁnite whenever γ is continuously diﬀerentiable and is given by (4.1.1). We shall sometimes suppress the notation for the endpoints a, b ∈ I. When γ(a) = p and γ(b) = q we say that γ is a curve from p to q. One can always compose γ with an aﬃne reparametrization t = a + (b − a)t to obtain a new curve γ(t) := γ(t) on the unit interval 0 ≤ t ≤ 1. This new curve satisﬁes L(γ) = L(γ) and E(γ) = (b − a)E(γ). More generally, the length L(γ), but not the energy E(γ), is invariant under reparametrization. Remark 4.1.3 (Reparametrization). Let I = [a, b] and I = [a, b] be If γ : I → Rn is a smooth curve and σ : I → I is a compact intervals. smooth function such that σ(a) = a, σ(b) = b, and ˙σ(t) ≥ 0 for all t ∈ I, then the curves γ and γ ◦ σ have the same length. Namely, d dt γ(σ(t)) ˙σ(t) dt = L(γ). ˙γ(σ(t)) L(γ ◦ σ) = (4.1.3) dt = b b a a Here second equality follows from the chain rule and the third equality follows from the change of variables formula for the Riemann integral. 4.1. LENGTH AND ENERGY 177 Theorem 4.1.4 (Characterization of geodesics). Let I = [a, b] ⊂ R be a compact interval and
let γ : I → M be a smooth curve. Then the following are equivalent. (i) γ is an extremal of the energy functional, i.e. every variation {γs}s∈R of γ with ﬁxed endpoints satisﬁes d ds s=0 E(γs) = 0. (ii) γ is parametrized proportional to the arclength, i.e. the velocity \| ˙γ(t)\| ≡ c ≥ 0 is constant, and either γ is constant, i.e. γ(t) = p = q for all t ∈ I, or c > 0 and γ is an extremal of the length functional, i.e. every variation {γs}s∈R of γ with ﬁxed endpoints satisﬁes d ds s=0 L(γs) = 0. (iii) The velocity vector of γ is parallel, i.e. ∇ ˙γ(t) = 0 for all t ∈ I. (iv) The acceleration of γ is normal to M, i.e. ¨γ(t) ⊥ Tγ(t)M for all t ∈ I. (v) If (Φ, γ, γ) is a development of M along M = Rm, then γ : I → Rm is a straight line parametrized proportional to the arclength, i.e. ¨γ ≡ 0. Proof. See §4.1.3. Deﬁnition 4.1.5 (Geodesic). A smooth curve γ : I → M on an interval I is called a geodesic iﬀ its restriction to each compact subinterval satisﬁes the equivalent conditions of Theorem 4.1.4. So γ is a geodesic if and only if ∇ ˙γ(t) = 0 for all t ∈ I. (4.1.4) By the Gauß–Weingarten formula (3.2.2) with X = ˙γ this is equivalent to ¨γ(t) = hγ(t)( ˙γ(t), ˙γ(t)) for all t ∈ I. (4.1.
5) Remark 4.1.6. (i) The conditions (i) and (ii) in Theorem 4.1.4 are meaningless when I is not compact because then the curve has at most one endpoint and the length and energy integrals may be inﬁnite. However, the conditions (iii), (iv), and (v) in Theorem 4.1.4 are equivalent for smooth curves on any interval, compact or not. (ii) The function s → E(γs) associated to a smooth variation is always smooth and so condition (i) in Theorem 4.1.4 is meaningful. However, more care has to be taken in part (ii) because the function s → L(γs) need not be diﬀerentiable. It is diﬀerentiable at s = 0 whenever ˙γ(t) = 0 for all t ∈ I. 178 CHAPTER 4. GEODESICS 4.1.2 The Space of Paths Fix two points p, q ∈ M and a compact interval I = [a, b] and denote by γ is smooth and γ(a) = p, γ(b) = q Ωp,q := Ωp,q(I) := γ : I → M the space of smooth curves in M from p to q, deﬁned on the interval I. Then the length and energy are functionals L, E : Ωp,q → R and their extremal points can be understood as critical points as we now explain. We may think of the space Ωp,q as a kind of “inﬁnite-dimensional manifold”. This is to be understood in a heuristic sense and we use these terms here to emphasize an analogy. Of course, the space Ωp,q is not a manifold in the strict sense of the word. To begin with it is not embedded in any ﬁnite-dimensional Euclidean space. However, it has many features in common with manifolds. The ﬁrst is that we can speak of smooth curves in Ωp,q. Of course Ωp,q is itself a space of curves in M. Thus a smooth curve in Ωp,q would then be a curve of curves, namly a map R →
Ωp,q : s → γs that assigns to each real number s a smooth curve γs : I → M satisfying γs(a) = p and γs(b) = q. We shall call such a curve of curves smooth iﬀ the associated map R × I → M : (s, t) → γs(t) is smooth. Thus smooth curves in Ωp,q are the variations of γ with ﬁxed endpoints introduced in Deﬁnition 4.1.1. Having deﬁned what we mean by a smooth curve in Ωp,q we can also diﬀerentiate such a curve with respect to s. Here we can simply recall that, since M ⊂ Rn, we have a smooth map R × I → Rn and the derivative of the curve s → γs in Ωp,q can simply be understood as the partial derivative of the map (s, t) → γs(t) with respect to s. Thus, in analogy with embedded manifolds, we deﬁne the tangent space of the space of curves Ωp,q at γ as the set of all derivatives of smooth curves R → Ωp,q : s → γs passing through γ, i.e. TγΩp,q := ∂ ∂s s=0 γs R → Ωp,q : s → γs is smooth and γ0 = γ. Let us denote such a partial derivative by X(t) := ∂ s=0 γs(t) ∈ Tγ(t)M. ∂s Thus we obtain a smooth vector ﬁeld along γ. Since γs(a) = p and γs(b) = q for all s, this vector ﬁeld must vanish at t = a, b. This suggests the formula TγΩp,q = {X ∈ Vect(γ) \| X(a) = 0, X(b) = 0}. (4.1.6) That every tangent vector of the path space Ωp,q at γ is a vector ﬁeld along γ vanishing at the endpoints follows from
the above discussion. The converse inclusion is the content of the next lemma. 4.1. LENGTH AND ENERGY 179 Lemma 4.1.7. Let p, q ∈ M, γ ∈ Ωp,q, and X ∈ Vect(γ) with X(a) = 0 and X(b) = 0. Then there exists a smooth map R → Ωp,q : s → γs such that ∂ ∂s (4.1.7) for all t ∈ I. γ0(t) = γ(t), γs(t) = X(t) s=0 Proof. The proof has two steps. Step 1. There exists smooth map M × I → Rn : (r, t) → Yt(r) with compact support such that Yt(r) ∈ TrM for all t ∈ I and r ∈ M and Ya(r) = Yb(r) = 0 for all r ∈ M. Deﬁne Zt(r) := Π(r)X(t) for t ∈ I and r ∈ M. Choose an open set U ⊂ Rn such that γ(I) ⊂ U and U ∩ M is compact (e.g. take U := a≤t≤b Bε(γ(t)) for ε > 0 suﬃciently small). Now let β : Rn → [0, 1] be a smooth cutoﬀ function with support in the unit ball such that β(0) = 1 and deﬁne the vector ﬁelds Yt by Yt(r) := β(ε−1(r − γ(t)))Zt(r) for t ∈ I and r ∈ M. Step 2. We prove the lemma. The vector ﬁeld Yt : M → T M in Step 1 is complete for each t. Thus there exists a unique smooth map R × I → M : (s, t) → γs(t) such that, for each t ∈ I, the curve R → M : s → γs(t) is the unique solution of the diﬀerential equation ∂ ∂s γs(t) = Y
t(γs(t)) with γ0(t) = γ(t). These maps γs satisﬁes (4.1.7) by Step 1. We can now deﬁne the derivative of the energy functional E at γ in the direction of a tangent vector X ∈ TγΩp,q by dE(γ)X := d ds s=0 E(γs), (4.1.8) where s → γs is as in Lemma 4.1.7. Similarly, the derivative of the length functional L at γ in the direction of X ∈ TγΩp,q is deﬁned by dL(γ)X := d ds s=0 L(γs). (4.1.9) To deﬁne (4.1.8) and (4.1.9) the functions s → E(γs) and s → L(γs) must be diﬀerentiable at s = 0. This is true for E but it only holds for L when ˙γ(t) = 0 for all t ∈ I. Second, we must show that the right hand sides of (4.1.8) and (4.1.9) depend only on X and not on the choice of {γs}s∈R. Third, we must verify that dE(γ) : TγΩp,q → R and dL(γ) : Ωp,q → R are linear maps. This is an exercise in ﬁrst year analysis (see also the proof of Theorem 4.1.4). A curve γ ∈ Ωp,q is then an extremal point of E (respectively L when ˙γ(t) = 0 for all t) if and only if dE(γ) = 0 (respectively dL(γ) = 0). Such a curve is also called a critical point of E (respectively L). 180 CHAPTER 4. GEODESICS 4.1.3 Characterization of Geodesics Proof of Theorem 4.1.4. The equivalence of (iii) and (iv) follows directly from the equations ∇ ˙γ(t) = Π(γ(t))
¨γ(t) and ker(Π(γ(t))) = Tγ(t)M ⊥. We prove that (i) is equivalent to (iii) and (iv). Let X ∈ TγΩp,q and choose a smooth curve of curves R → Ωp,q : s → γs satisfying (4.1.7). Then the function (s, t) → \| ˙γs(t)\|2 is smooth and hence dE(γ)X = = = = = d ds d ds s=0 s=0 b ∂ ∂s 1 2 b a E(γs) a b 1 2 s=0 ∂ ∂s ˙γ(t), a b ˙γ(t), s=0 ˙X(t) dt \| ˙γs(t)\|2 dt \| ˙γs(t)\|2 dt ˙γs(t) dt (4.1.10) a b = − a ¨γ(t), X(t) dt. That (iii) implies (i) follows directly from this identity. To prove that (i) implies (iv) we argue indirectly and assume that there exists a point t0 ∈ [0, 1] such that ¨γ(t0) is not orthogonal to Tγ(t0)M. Then there exists a vector v0 ∈ Tγ(t0)M such that ¨γ(t0), v0 > 0. We may assume without loss of generality that a < t0 < b. Then there exists a constant ε > 0 such that a < t0 − ε < t0 + ε < b and t0 − ε < t < t0 + ε =⇒ ¨γ(t), Π(γ(t))v0 > 0. Now choose a smooth cutoﬀ function β : I → [0, 1] such that β(t) = 0 for all t ∈ I with \|t − t0\| ≥ ε and β(t0) = 1. Deﬁne X ∈ TγΩp,q by X(t) := β(t)Π(γ(t))v0 for t ∈ I. Then
¨γ(t), X(t) ≥ 0 for all t and ¨γ(t0), X(t0) > 0. Hence dE(γ)X = − b a ¨γ(t), X(t) dt < 0 and so γ does not satisfy (i). Thus (i) is equivalent to (iii) and (iv). 4.1. LENGTH AND ENERGY 181 We prove that (i) is equivalent to (ii). Assume ﬁrst that γ satisﬁes (i). Then γ also satisﬁes (iv) and hence ¨γ(t) ⊥ Tγ(t)M for all t ∈ I. This implies 0 = ¨γ(t), ˙γ(t) = 1 2 d dt \| ˙γ(t)\|2. Hence the function I → R : t → \| ˙γ(t)\|2 is constant. Choose c ≥ 0 such that \| ˙γ(t)\| ≡ c. If c = 0, then γ(t) is constant and so γ(t) ≡ p = q. If c > 0, then \| ˙γs(t)\| dt \| ˙γs(t)\| dt dL(γ)X = = = = = a b s=0 ∂ ∂s s=0 d ds b a b \| ˙γ(t)\|−1 a b 1 c 1 c a dE(γ)X. ˙γ(t), ∂ ∂s s=0 ˙γs(t) dt ˙γ(t), ˙X(t) dt Thus, in the case c > 0, γ is an extremal point of E if and only if it is an extremal point of L. Hence (i) is equivalent to (ii). We prove that (iii) is equivalent to (v). Let (Φ, γ, γ) be a development of M along M = Rm. Then ˙γ(t) = Φ(t) ˙γ(t), d dt Φ(t)X(t) = Φ(t)∇X(t) for all X ∈ Vect(γ) and all t
∈ I. Take X = ˙γ to obtain ¨γ(t) = Φ(t)∇ ˙γ(t) for all t ∈ I. Thus ∇ ˙γ ≡ 0 if and only if ¨γ ≡ 0. This proves Theorem 4.1.4. Remark 4.1.3 shows that reparametrization by a nondecreasing surjective map σ : I → I gives rise to map Ωp,q(I) → Ωp,q(I ) : γ → γ ◦ σ which preserves the length functional, i.e. L(γ ◦ σ) = L(γ) for all γ ∈ Ωp,q(I). Thus the chain rule in inﬁnite dimensions should assert that if γ◦σ is an extremal (i.e. critical) point of L, then γ is an extremal point 182 CHAPTER 4. GEODESICS of L. Moreover, if σ is a diﬀeomorphism, the map γ → γ ◦ σ is bijective and should give rise to a bijective correspondence between the extremal points of L on Ωp,q(I) and those on Ωp,q(I ). Finally, if the tangent vector ﬁeld ˙γ vanishes nowhere, then γ can be parametrized by the arclength. This is spelled out in more detail in the next exercise. Exercise 4.1.8. Let γ : I = [a, b] → M be a smooth curve such that for all t ∈ I and deﬁne ˙γ(t) = 0 T := L(γ) = b a \| ˙γ(t)\| dt. (i) Prove that there exists a unique diﬀeomorphism σ : [0, T ] → I such that σ(t) = t ⇐⇒ t = t a \| ˙γ(s)\| ds for all t ∈ [0, T ] and all t ∈ [a, b]. Prove that γ := γ ◦ σ : [0, T ] → M is parametrized by
the arclength, i.e. \| ˙γ(t)\| = 1 for all t ∈ [0, T ]. (ii) Prove that dL(γ)X = − b a ˙V (t), X(t) dt, V (t) := \| ˙γ(t)\|−1 ˙γ(t). (4.1.11) Hint: See the relevant formula in the proof of Theorem 4.1.4. (iii) Prove that γ is an extremal point of L if and only if the curve γ in part (i) is a geodesic. (iv) Prove that γ is an extremal point of L if and only if there exists a geodesic γ : I → M and a diﬀeomorphism σ : I → I such that γ = γ ◦ σ. Next we generalize this exercise to cover the case where ˙γ is allowed to vanish. Recall from Remark 4.1.6 that the function s → L(γs) need not be diﬀerentiable. As an example consider the case where γ = γ0 is constant (see also Exercise 4.4.12 below). Exercise 4.1.9. Let γ : I → M be a smooth curve and let X ∈ TγΩp,q(I). Choose a smooth curve of curves R → Ωp,q(I) : s → γs that satisﬁes (4.1.7). Prove that the one-sided derivatives of the function s → L(γs) exist at s = 0 and satisfy the inequalities ˙X(t) dt ≤ ˙X(t) dt. − ≤ d ds L(γs) s=0 I I Exercise 4.1.10. Let (Φ, γ, γ) be a development of M along M. Show that γ is a geodesic in M if and only if γ is a geodesic in M. 4.2. DISTANCE 4.2 Distance 183 Assume that M ⊂ Rn is a connected smooth m-dimensional submanifold. Two points p, q ∈ M are of distance \|p − q\| apart in the ambient
Euclidean space Rn. In this section we deﬁne a distance function which is more intimately tied to M by minimizing the length functional over the space of curves in M with ﬁxed endpoints. Thus it may happen that two points in M have a very short distance in Rn but can only be joined by very long curves in M (see Figure 4.1). This leads to the intrinsic distance in M. Throughout we denote by I = [0, 1] the unit interval and, for p, q ∈ M, by Ωp,q := {γ : [0, 1] → M \| γ is smooth and γ(0) = p, γ(1) = q} (4.2.1) the space of smooth paths on the unit interval joining p to q. Since M is connected the set Ωp,q is nonempty for all p, q ∈ M. (Prove this!) Figure 4.1: Curves in M. Deﬁnition 4.2.1. The intrinsic distance between two points p, q ∈ M is the real number d(p, q) ≥ 0 deﬁned by d(p, q) := inf γ∈Ωp,q L(γ). (4.2.2) The inequality d(p, q) ≥ 0 holds because each curve has nonnegative length and the inequality d(p, q) < ∞ holds because Ωp,q = ∅. Remark 4.2.2. Every smooth curve γ : [0, 1] → Rn with endpoints γ(0) = p and γ(1) = q satisﬁes the inequality L(γ) = 1 0 \| ˙γ(t)\| dt ≥ 1 0 ˙γ(t) dt = \|p − q\|. Thus d(p, q) ≥ \|p − q\|. For γ(t) := p + t(q − p) we have equality and hence the straight lines minimize the length among all curves from p to q. Mpqpq 184 CHAPTER 4. GEODESICS Lemma 4.2.3. The function d : M × M → [0, ∞) deﬁnes a metric on M : (i) If p
, q ∈ M satisfy d(p, q) = 0, then p = q. (ii) For all p, q ∈ M we have d(p, q) = d(q, p). (iii) For all p, q, r ∈ M we have d(p, r) ≤ d(p, q) + d(q, r). Proof. By Remark 4.2.2 we have d(p, q) ≥ \|p − q\| for all p, q ∈ M and this proves part (i). Part (ii) follows from the fact that the curve γ(t) := γ(1 − t) has the same length as γ and belongs to Ωq,p whenever γ ∈ Ωp,q. To prove part (iii) ﬁx a constant ε > 0 and choose curves γ0 ∈ Ωp,q and γ1 ∈ Ωq,r such that L(γ0) < d(p, q) + ε and L(γ1) < d(q, r) + ε. By Remark 4.1.3 we may assume without loss of generality that γ0(1 − t) = γ1(t) = q for t > 0 suﬃciently small. Under this assumption the curve γ(t) := γ0(2t), γ1(2t − 1), for 0 ≤ t ≤ 1/2, for 1/2 < t ≤ 1 is smooth. Moreover, γ(0) = p and γ(1) = r and so γ ∈ Ωp,r. Thus d(p, r) ≤ L(γ) = L(γ0) + L(γ1) < d(p, q) + d(q, r) + 2ε. Hence d(p, r) < d(p, q) + d(q, r) + 2ε for every ε > 0. This proves part (iii) and Lemma 4.2.3. Remark 4.2.4. It is natural to ask if the inﬁmum in (4.2.2) is always attained. This is easily seen not to be the case in general. For example, let M result from
the Euclidean space Rm by removing a point p0. Then the distance d(p, q) = \|p − q\| is equal to the length of the line segment from p to q and any other curve from p to q is longer. Hence if p0 is in the interior of this line segment, the inﬁmum is not attained. We shall prove below that the inﬁmum is attained whenever M is complete. Figure 4.2: A geodesic on the 2-sphere. qp 4.2. DISTANCE Example 4.2.5. Let 185 M := S2 = p ∈ R3 \| \|p\| = 1 be the unit sphere in R3 and ﬁx two points p, q ∈ S2. Then d(p, q) is the length of the shortest curve on the 2-sphere connecting p and q. Such a curve is a segment on a great circle through p and q (see Figure 4.2) and its length is d(p, q) = cos−1(p, q), (4.2.3) where p, q denotes the standard inner product, and we have 0 ≤ d(p, q) ≤ π. (See Example 4.3.11 below for details.) By Lemma 4.2.3 this deﬁnes a metric on S2. Exercise: Prove directly that (4.2.3) is a distance function on S2. We now have two topologies on our manifold M ⊂ Rn, namely the topology determined by the metric d in Lemma 4.2.3 and the relative topology inherited from Rn. The latter is also determined by a distance function, namely the extrinsic distance function deﬁned as the restriction of the Euclidean distance function on Rn to the subset M. We denote it by d0 : M × M → [0, ∞), d0(p, q) := \|p − q\|. (4.2.4) A natural question is if these two metrics d and d0 induce the same topology on M. In other words is a subset U ⊂ M open with respect to d0 if and only if it is open with respect to d? Or, equivalently, does a sequence pν ∈ M converge to p0 ∈ M with respect to
d if and only if it converges to p0 with respect to d0? Lemma 4.2.7 answers this question in the aﬃrmative. Exercise 4.2.6. Prove that every translation of Rn and every orthogonal transformation preserves the lengths of curves. Lemma 4.2.7. For every p0 ∈ M we have lim p,q→p0 d(p, q) \|p − q\| = 1. Lemma 4.2.8. Let p0 ∈ M and let φ0 : U0 → Ω0 be a coordinate chart onto an open subset of Rm such that its derivative dφ0(p0) : Tp0M → Rm is an orthogonal transformation. Then lim p,q→p0 d(p, q) \|φ0(p) − φ0(q)\| = 1. 186 CHAPTER 4. GEODESICS The proofs will be given below. The lemmas imply that the topology M inherits as a subset of Rm, the topology on M determined by the metric d, and the topology on M induced by the local coordinate systems on M are all the same. Corollary 4.2.9. For every subset U ⊂ M the following are equivalent. (i) U is open with respect to the metric d in (4.2.2). (ii) U is open with respect to the metric d0 in (4.2.4). (iii) For every coordinate chart φ0 : U0 → Ω0 of M onto an open subset Ω0 ⊂ Rm the set φ0(U0 ∩ U ) is an open subset of Rm. Proof. By Remark 4.2.2 we have \|p − q\| ≤ d(p, q) (4.2.5) for all p, q ∈ M. Thus the identity idM : (M, d) → (M, d0) is Lipschitz continuous with Lipschitz constant one and so every d0-open subset of M is d-open. Conversely, let U ⊂ M be a d-open subset of M and let p0 ∈ U and ε > 0. Then, by Lemma 4.2.7, there exists a constant δ >
0 such that all p, q ∈ M with \|p − p0\| < δ and \|q − p0\| < δ satisfy d(p, q) ≤ (1 + ε)\|p − q\|. Since U is d-open, there exists a constant ρ > 0 such that With Bρ(p0, d) ⊂ U. ρ0 := min δ, ρ 1 + ε this implies Bρ0(p0, d0) ⊂ U. Namely, if p ∈ M satisﬁes \|p − p0\| < ρ0 ≤ δ, then d(p, p0) ≤ (1 + ε)\|p − p0\| < (1 + ε)ρ0 ≤ ρ and so p ∈ U. Thus U is d0-open and this proves that (i) is equivalent to (ii). That (ii) implies (iii) follows from the fact that each coordinate chart φ0 is a homeomorphism. To prove that (iii) implies (i), we argue indirectly and assume that U is not d-open. Then there exists a sequence pν ∈ M \ U that converges to an element p0 ∈ U. Let φ0 : U0 → Ω0 be a coordinate chart with p0 ∈ U0. Then limν→∞\|φ0(pν) − φ0(p0)\| = 0 by Lemma 4.2.8. Thus φ0(U0 ∩ U ) is not open and so U does not satisfy (iii). This proves Corollary 4.2.9. 4.2. DISTANCE 187 Figure 4.3: Locally, M is the graph of f. Proof of Lemma 4.2.7. By Remark 4.2.2 the estimate \|p − q\| ≤ d(p, q) holds for all p, q ∈ M. The lemma asserts that, for all p0 ∈ M and all ε > 0, there exists a d0-open neighborhood U0 ⊂ M of p0 such that all p, q ∈ U0 satisfy \|p − q\| ≤ d(p, q) ≤ (1 + ε)\|p − q\|. (4.2.6) Let p0 ∈ M
and ε > 0, and deﬁne x : Rn → Tp0M and y : Rn → Tp0M ⊥ by x(p) := Π(p0)(p − p0), y(p) := (1l − Π(p0)) (p − p0), where Π(p0) : Rn → Tp0M denotes the orthogonal projection as usual. Then the derivative of the map x\|M : M → Tp0M at p = p0 is the identity on Tp0M. Hence the Inverse Function Theorem 2.2.17 asserts that the map x\|M : M → Tp0M is locally invertible near p0. Extending this inverse to a smooth map from Tp0M to Rn and composing it with the map y : M → Tp0M ⊥, we obtain a smooth map f : Tp0M → Tp0M ⊥ and an open neighborhood W ⊂ Rn of p0 such that p ∈ M ⇐⇒ y(p) = f (x(p)) for all p ∈ W (see Figure 4.3). Moreover, by deﬁnition the map f satisﬁes f (0) = 0 ∈ Tp0M ⊥, df (0) = 0 : Tp0M → Tp0M ⊥. Hence there exists a constant δ > 0 such that, for every x ∈ Tp0M, we have \|x\| < δ =⇒ x + f (x) ∈ W and df (x) = sup 0=x∈Tp0 M \|df (x)x\| \|x\| < ε. T Mp0⊥MT Mp0p0 188 Deﬁne CHAPTER 4. GEODESICS U0 := {p ∈ M ∩ W \| \|x(p)\| < δ}. Given p, q ∈ U0 let γ : [0, 1] → M be the curve whose projection to the x-axis is the straight line joining x(p) to x(q), i.e. x(γ(t)) = x(p) + t(x(q) − x(p)) =: x
(t), y(γ(t)) = f (x(γ(t))) = f (x(t)) =: y(t). Then γ(t) ∈ U0 for all t ∈ [0, 1] and L(γx(t) + ˙y(t)\| dt \| ˙x(t) + df (x(t)) ˙x(t)\| dt 1 + df (x(t)) \| ˙x(t)\| dt ≤ (1 + ε) t \| ˙x(t)\| dt 0 = (1 + ε) \|x(p) − x(q)\| = (1 + ε) \|Π(p0)(p − q)\| ≤ (1 + ε) \|p − q\|. Hence d(p, q) ≤ L(γ) ≤ (1 + ε) \|p − q\| and this proves Lemma 4.2.7. Proof of Lemma 4.2.8. By assumption we have \|dφ0(p0)v\| = \|v\| for all v ∈ Tp0M. Fix a constant ε > 0. Then, by continuity of the derivative, there exists a d0-open neighborhood M0 ⊂ M of p0 such that for all p ∈ M0 and all v ∈ TpM we have (1 − ε) \|dφ0(p)v\| ≤ \|v\| ≤ (1 + ε) \|dφ0(p)v\|. Thus for every curve γ : [0, 1] → M0 we have (1 − ε)L(φ0 ◦ γ)) ≤ L(γ) ≤ (1 + ε)L(φ0 ◦ γ). One is tempted to take the inﬁmum over all curves γ : [0, 1] → M0 joining two pints p, q ∈ M0 to obtain the inequality (1 − ε) \|φ0(p) − φ0(q)\| ≤ d(p, q) ≤ (1 + ε) \|φ0(p) − φ0(q)\|. (4.2.7) However, we must justify these inequalities by showing that the inﬁmum
over all curves in M0 agrees with the inﬁmum over all curves in M joining the points p and q. 4.2. DISTANCE 189 It suﬃces to show that the inequalities hold on a smaller heighborhood M1 ⊂ M0 of p0. Choose such a smaller neighborhood M1 such that the open set φ0(M1) is a convex subset of Ω0. Then the right inequality in (4.2.7) follows by taking the curve γ : [0, 1] → M1 from γ(0) = p to γ(1) = q such that φ0 ◦ γ : [0, 1] → φ0(M1) is a straight line. To prove the left inequality in (4.2.7) we use the fact that M0 is d-open by Lemma 4.2.7. Hence, after shrinking M1 if necessary, there exists a constant r > 0 such that p0 ∈ M1 ⊂ Br(p0, d) ⊂ B3r(p0, d) ⊂ M0. Then, for p, q ∈ M1 we have d(p, q) ≤ 2r while L(γ) ≥ 4r for any curve γ from p to q which leaves M0. Hence the distance d(p, q) of p, q ∈ M1 is the inﬁmum of the lengths L(γ) over all curves γ : [0, 1] → M0 that join γ(0) = p to γ(1) = q. This proves the left inequality in (4.2.7) and Lemma 4.2.8. A next question one might ask is the following. Can we choose a coordinate chart φ : U → Ω on M with values in an open set Ω ⊂ Rm so that the length of each smooth curve γ : [0, 1] → U is equal to the length of the curve c := φ ◦ γ : [0, 1] → Ω? We examine this question by considering the inverse map ψ := φ−1 : Ω → U. Denote the components of x and ψ(x) by x = (x1,...
, xm) ∈ Ω, ψ(x) = (ψ1(x),..., ψn(x)) ∈ U. Given a smooth curve [0, 1] → Ω : t → c(t) = (c1(t),..., cm(t)) we can write the length of the composition γ = ψ ◦ c : [0, 1] → M in the form L(ψ ◦ c(c(t)) dt n 2 ψν(c(t)) dt d dt d dt ν=1 n m 2 ∂ψν ∂xi (c(t)) ˙ci(t) dt ν=1 i=1 n m ν=1 i,j=1 ∂ψν ∂xi (c(t)) ∂ψν ∂xj (c(t)) ˙ci(t) ˙cj(t) dt m i,j=1 ˙ci(t)gij(c(t)) ˙cj(t) dt. 190 CHAPTER 4. GEODESICS Here the functions gij : Ω → R are deﬁned by gij(x) := n ν=1 ∂ψν ∂xi (x) ∂ψν ∂xj (x) = ∂ψ ∂xi (x),. ∂ψ ∂xj (x) (4.2.8) Thus we have a smooth function g = (gij) : Ω → Rm×m with values in the positive deﬁnite matrices given by g(x) = dψ(x)Tdψ(x) such that L(ψ ◦ c) = 1 0 ˙c(t)Tg(c(t)) ˙c(t) dt (4.2.9) for every smooth curve c : [0, 1] → Ω. Thus the condition L(ψ ◦ c) = L(c) for every such curve is equivalent to for all x ∈ Ω or, equivalently, gij(x) = δij dψ(x)Td�
�(x) = 1l. (4.2.10) This means that ψ preserves angles and areas. The next example shows that for M = S2 it is impossible to ﬁnd such coordinates. Figure 4.4: A spherical triangle. Example 4.2.10. Consider the manifold M = S2. If there is a diﬀeomorphism ψ : Ω → U from an open set Ω ⊂ R2 onto an open set U ⊂ S2 that satisﬁes (4.2.10), it has to map straight lines onto arcs of great circles and it preserves the area. However, the area A of a spherical triangle bounded by three arcs on great circles satisﬁes the angle sum formula α + β + γ = π + A. (See Figure 4.4.) Hence there can be no such map ψ. Aβαγ 4.3. THE EXPONENTIAL MAP 191 4.3 The Exponential Map Geodesics give rise to a ﬂow on the tangent bundle, the geodesic ﬂow. It is generated by a vector ﬁeld on the tangent bundle, called the geodesic spray. The time-1-map of the geodesic ﬂow gives rise to the exponential map. 4.3.1 Geodesic Spray The tangent bundle T M is a smooth 2m-dimensional manifold in Rn × Rn by Corollary 2.6.12. The next lemma characterizes the tangent bundle of the tangent bundle. Compare this with Lemma 3.4.5. Lemma 4.3.1. The tangent space of T M at (p, v) ∈ T M is given by T(p,v)T M = (p, v) ∈ Rn × Rn p ∈ TpM and 1l − Π(p) v = hp(p, v). (4.3.1) Proof. We prove the inclusion “⊂” in (4.3.1). Let (p, v) ∈ T(p,v)T M and choose a smooth curve R → T M : t → (γ(t), X(t)) such that γ(0) = p, X(0) = v
, ˙γ(0) = p, ˙X(0) = v. Then ˙X = ∇X + hγ( ˙γ, X) by the Gauß–Weingarten formula (3.2.2) and hence (1l − Π(γ(t))) ˙X(t) = hγ(t)( ˙γ(t), X(t)) for all t ∈ R. Take t = 0 to obtain (1l − Π(p))v = hp(p, v). This proves the inclusion “⊂” in (4.3.1). Equality holds because both sides of the equation are 2m-dimensional linear subspaces of Rn × Rn. By Lemma 4.3.1 a smooth map S = (S1, S2) : T M → Rn × Rn is a vector ﬁeld on T M if and only if S1(p, v) ∈ TpM, (1l − Π(p))S2(p, v) = hp(S1(p, v), v) for all (p, v) ∈ T M. A special case is where S1(p, v) = v. Such vector ﬁelds correspond to second order diﬀerential equations on M. Deﬁnition 4.3.2 (Spray). A vector ﬁeld S ∈ Vect(T M ) is called a spray iﬀ it has the form S(p, v) = (v, S2(p, v)) where S2 : T M → Rn is a smooth map satisfying (1l − Π(p))S2(p, v) = hp(v, v), S2(p, λv) = λ2S2(p, v) (4.3.2) for all (p, v) ∈ T M and λ ∈ R. The vector ﬁeld S ∈ Vect(T M ) deﬁned by S(p, v) := (v, hp(v, v)) ∈ T(p,v)T M (4.3.3) for p ∈ M and v ∈ TpM is called
the geodesic spray. 192 CHAPTER 4. GEODESICS 4.3.2 The Exponential Map Lemma 4.3.3. Let γ : I → M be a smooth curve on an open interval I ⊂ R. Then γ is a geodesic if and only if the curve I → T M : t → (γ(t), ˙γ(t)) is an integral curve of the geodesic spray S in (4.3.3). Proof. A smooth curve I → T M : t → (γ(t), X(t)) is an integral curve of S if and only if ˙γ(t) = X(t) and ˙X(t) = hγ(t)(X(t), X(t)) for all t ∈ I. By equation (4.1.5), this holds if and only if γ is a geodesic and ˙γ = X. Combining Lemma 4.3.3 with Theorem 2.4.7 we obtain the following existence and uniqueness result for geodesics. Lemma 4.3.4. Let M ⊂ Rn be an m-dimensional submanifold. (i) For every p ∈ M and every v ∈ TpM there is an ε > 0 and a smooth curve γ : (−ε, ε) → M such that ∇ ˙γ ≡ 0, γ(0) = p, ˙γ(0) = v. (4.3.4) (iI) If γ1 : I1 → M and γ2 : I2 → M are geodesics and t0 ∈ I1 ∩ I2 with γ1(t0) = γ2(t0), ˙γ1(t0) = ˙γ2(t0), then γ1(t) = γ2(t) for all t ∈ I1 ∩ I2. Proof. Lemma 4.3.3 and Theorem 2.4.7. Deﬁnition 4.3.5 (Exponential map). For p ∈ M and v ∈ TpM the interval I ⊂ R Ip,v := I is an open interval containing 0 and there is a geodesic γ : I
→ M satisfying γ(0) = p, ˙γ(0) = v. is called the maximal existence interval for the geodesic through p in the direction v. For p ∈ M deﬁne the set Vp ⊂ TpM by Vp := {v ∈ TpM \| 1 ∈ Ip,v}. (4.3.5) The exponential map at p is the map expp : Vp → M that assigns to every tangent vector v ∈ Vp the point expp(v) := γ(1), where γ : Ip,v → M is the unique geodesic satisfying γ(0) = p and ˙γ(0) = v. 4.3. THE EXPONENTIAL MAP 193 Figure 4.5: The exponential map. Lemma 4.3.6. (i) The set V := {(p, v) \| p ∈ M, v ∈ Vp} ⊂ T M is open and the map V → M : (p, v) → expp(v) is smooth. (ii) If p ∈ M and v ∈ Vp, then Ip,v = {t ∈ R \| tv ∈ Vp} and the geodesic γ : Ip,v → M with γ(0) = p and ˙γ(0) = v is given by γ(t) = expp(tv), t ∈ Ip,v. Proof. Part (i) follows directly from Lemma 4.3.3 and Theorem 2.4.9. To prove part (ii), ﬁx an element p ∈ M and a tangent vector v ∈ Vp, and let γ : Ip,v → M be the unique geodesic with γ(0) = p and ˙γ(0) = v. Fix a nonzero real number λ and deﬁne the map γλ : λ−1Ip,v → M by γλ(t) := γ(λt) for t ∈ λ−1Ip,v. Then ˙γλ(t) = λ ˙γ(λt) ans ¨γλ(t) = λ2¨γ(λ
t) and hence ∇ ˙γλ(t) = Π(γλ(t))¨γλ(t) = λ2Π(γ(λt))¨γ(λt) = λ2∇ ˙γ(λt) = 0 for every t ∈ λ−1Ip,v. This shows that γλ is a geodesic with γλ(0) = p, ˙γλ(0) = λv. In particular, we have λ−1Ip,v ⊂ Ip,λv. Interchanging the roles of v and λv we obtain λ−1Ip,v = Ip,λv. Thus λ ∈ Ip,v ⇐⇒ 1 ∈ Ip,λv ⇐⇒ λv ∈ Vp and for λ ∈ Ip,v. This proves Lemma 4.3.6. γ(λ) = γλ(1) = expp(λv) pM 194 CHAPTER 4. GEODESICS Since expp(0) = p by deﬁnition, the derivative of the exponential map at v = 0 is a linear map from TpM to itself. This derivative is the identity map as illustrated in Figure 4.5 and proved in the following corollary. Corollary 4.3.7. The map expp : Vp → M is smooth and its derivative at the origin is d expp(0) = id : TpM → TpM. Proof. The set Vp is an open subset of the linear subspace TpM ⊂ Rn, with respect to the relative topology, and hence is a manifold. The tangent space of Vp at each point is TpM. By Lemma 4.3.6 the exponential map expp : Vp → M is smooth and its derivative at the origin is given by d expp(0)v = d dt t=0 expp(tv) = ˙γ(0) = v, where γ : Ip,v → M is once again the unique geodesic through p in the direction v. This proves Corollary 4.3.7. Corollary 4.3.8. Let p ∈ M and
, for r > 0, denote Br(p) := {v ∈ TpM \| \|v\| < r}. If r > 0 is suﬃciently small, then Br(p) ⊂ Vp, the set Ur(p) := expp(Br(p)) is an open subset of M, and the restriction of the exponential map to Br(p) is a diﬀeomorphism from Br(p) to Ur(p). Proof. This follows directly from Corollary 4.3.7 and Theorem 2.2.17. Deﬁnition 4.3.9 (Injectivity radius). Let M ⊂ Rn be a smooth mmanifold. The injectivity radius of M at p is the supremum of all real numbers r > 0 such that Br(p) ⊂ Vp and the restriction of the exponential map expp to Br(p) is a diﬀeomorphism onto its image It will be denoted by Ur(p) := expp(Br(p)). inj(p) := inj(p; M ) := sup   r > 0  Br(p) ⊂ Vp and expp : Br(p) → Ur(p) is a diﬀeomorphism   . The injectivity radius of M is the inﬁmum of the injectivity radii of M at p over all p ∈ M. It will be denoted by inj(M ) := inf p∈M inj(p; M ). 4.3. THE EXPONENTIAL MAP 195 4.3.3 Examples and Exercises Example 4.3.10. The exponential map on Rm is given by expp(v) = p + v for p, v ∈ Rm. For every p ∈ Rm this map is a diﬀeomorphism from TpRm = Rm to Rm and hence the injectivity radius of Rm is inﬁnity. Example 4.3.11. The exponential map on Sm is given by expp(v) = cos(\|v\|)p + sin(\|v\|) \|v\| v for every p ∈ Sm and every nonzero tangent vector v ∈ TpSm =
p⊥. The restriction of this map to the open ball of radius r in TpM is a diﬀeomorphism onto its image if and only if r ≤ π. Hence the injectivity radius of Sm at every point is π. Exercise: Given p ∈ Sm and 0 = v ∈ TpSm = p⊥, prove that the geodesic γ : R → Sm with γ(0) = p and ˙γ(0) = v is given by γ(t) = cos(t \|v\|)p + sin(t\|v\|) v for t ∈ R. Show that in the case 0 ≤ \|v\| ≤ π there is no shorter curve in Sm connecting p and q := γ(1) and deduce that the intrinsic distance on Sm is given by d(p, q) = cos−1(p, q) for p, q ∈ Sm (see Example 4.2.5 for m = 2). \|v\| Example 4.3.12. Consider the orthogonal group O(n) ⊂ Rn×n with the standard inner product v, w := trace(vTw) on Rn×n. The orthogonal projection Π(g) : Rn×n → TgO(n) is given by Π(g)v := v − gvTg 1 2 and the second fundamental form by hg(v, v) = −gvTv. Hence a curve γ : R → O(n) is a geodesic if and only if γT¨γ + ˙γT ˙γ = 0 or, equivalently, γT ˙γ is constant. This shows that geodesics in O(n) have the form γ(t) = g exp(tξ) for g ∈ O(n) and ξ ∈ o(n). It follows that expg(v) = g exp(g−1v) = exp(vg−1)g for g ∈ O(n) and v ∈ TgO(n). map exp1l : o(n) → O(n) agrees with the exponential matrix. Exercise 4.3.13. What is the injectivity radius of the 2-torus T2 = S1 × S1
, the punctured 2-plane R2 \ {(0, 0)}, and the orthogonal group O(n)? In particular, for g = 1l the exponential 196 CHAPTER 4. GEODESICS Geodesics in Local Coordinates Lemma 4.3.14. Let M ⊂ Rn be an m-dimensional manifold and choose a coordinate chart φ : U → Ω with inverse ψ := φ−1 : Ω → U. ij : Ω → R be the Christoﬀel symbols deﬁned by (3.6.6) and let c : I → Ω Let Γk be a smooth curve. Then the curve γ := ψ ◦ c : I → M is a geodesic if and only if c satisﬁes the 2nd order diﬀerential equation ¨ck + m i,j=1 ij(c) ˙ci ˙cj = 0 Γk (4.3.6) for k = 1,..., m. Proof. This follows immediately from the deﬁnition of geodesics and equation (3.6.7) in Lemma 3.6.1 with X = ˙γ and ξ = ˙c. We remark that Lemma 4.3.14 gives rise to another proof of Lemma 4.3.4 that is based on the existence and uniqueness of solutions of second order diﬀerential equations in local coordinates. Exercise 4.3.15. Let Ω ⊂ Rm be an open set and g = (gij) : Ω → Rm×m be a smooth map with values in the space of positive deﬁnite symmetric matrices. Consider the energy functional E(c) := 1 0 L(c(t), ˙c(t)) dt on the space of paths c : [0, 1] → Ω, where L : Ω × Rm → R is deﬁned by L(x, ξ) := 1 2 m i,j=1 ξigij(x)ξj. (4.3.7) The Euler–Lagrange equations of this variational problem have the form d dt ∂L �
�ξk (c(t), ˙c(t)) = ∂L ∂xk (c(t), ˙c(t)), k = 1,..., m. (4.3.8) Prove that the Euler–Lagrange equations (4.3.8) are equivalent to the geodesic equations (4.3.6), where the Γk ij : Ω → R are given by (3.6.10). 4.4. MINIMAL GEODESICS 197 4.4 Minimal Geodesics Any straight line segment in Euclidean space is the shortest curve joining its endpoints. The analogous assertion for geodesics in a manifold M is false; consider for example an arc which is more than half of a great circle on a sphere. In this section we consider curves which realize the shortest distance between their endpoints. 4.4.1 Characterization of Minimal Geodesics Lemma 4.4.1. Let I = [a, b] be a compact interval, let γ : I → M be a smooth curve, and deﬁne p := γ(a) and q := γ(b). Then the following are equivalent. (i) γ is parametrized proportional to the arclength, i.e. \| ˙γ(t)\| = c is constant, and γ minimizes the length, i.e. L(γ) ≤ L(γ) for every smooth curve γ in M joining p and q. (ii) γ minimizes the energy, i.e. E(γ) ≤ E(γ) for every smooth curve γ in M joining p and q. Deﬁnition 4.4.2 (Minimal geodesic). A smooth curve γ : I → M on a compact interval I ⊂ R is called a minimal geodesic iﬀ it satisﬁes the equivalent conditions of Lemma 4.4.1. Remark 4.4.3. (i) Condition (i) says that (the velocity \| ˙γ\| is constant and) L(γ) = d(p, q), i.e. that γ is a shortest curve from p to q. It is not precluded that there be more than one such γ; consider
for example the case where M is a sphere and p and q are antipodal. (ii) Condition (ii) implies that d ds s=0 E(γs) = 0 for every smooth variation R × I → M : s → γs(t) of γ with ﬁxed endpoints. Hence a minimal geodesic is a geodesic. (iii) Finally, we remark that L(γ) (but not E(γ)) is independent of the parametrization of γ. Hence, if γ is a minimal geodesic, then L(γ) ≤ L(γ) for every γ (from p to q) whereas E(γ) ≤ E(γ) for those γ deﬁned on (an interval the same length as) I. 198 CHAPTER 4. GEODESICS Proof of Lemma 4.4.1. We prove that (i) implies (ii). Let c be the (constant) value of \| ˙γ(t)\|. Then L(γ) = (b − a)c, E(γ) = (b − a)c2 2. Then, for every smooth curve γ : I → M with γ(a) = p and γ(b) = q, we have 4E(γ)2 = c2L(γ)2 ≤ c2L(γ)2 b = c2 2 dt ˙γ(t) b a ≤ c2(b − a) ˙γ(t) 2 dt a = 2(b − a)c2E(γ) = 4E(γ)E(γ). Here the fourth step follows from the Cauchy–Schwarz inequality. Now divide by 4E(γ) to obtain E(γ) ≤ E(γ). We prove that (ii) implies (i). We have already shown in Remark 4.4.3 that (ii) implies that γ is a geodesic. It is easy to dispose of the case where M is one-dimensional. In that case any γ minimizing E(γ) or L(γ) must be monotonic onto a subarc; otherwise it could be altered so as to make the integral smaller. Hence suppose M is of dimension at least two. Suppose, by contradiction, that L(γ) < L(γ)
for some curve γ from p to q. Since the dimension of M is bigger than one, we may approximate γ by a curve whose tangent vector nowhere vanishes, i.e. we may assume without loss of generality that ˙γ(t) = 0 for all t. Then we can reparametrize γ proportional to arclength without changing its length, and by a further transformation we can make its domain equal to I. Thus we may assume without loss of generality that γ : I → M is a smooth curve with γ(a) = p and γ(b) = q such that \|γ(t)\| = c and (b − a)c = L(γ) < L(γ) = (b − a)c. This implies c < c and hence E(γ) = (b − a)c2 2 < (b − a)c2 2 = E(γ). This contradicts (ii) and proves Lemma 4.4.1. 4.4. MINIMAL GEODESICS 199 4.4.2 Local Existence of Minimal Geodesics The next theorem asserts the existence of minimal geodesics joining two points that are suﬃciently close to each other. It also shows that the set Ur(p) = expp(Br(p)) that was introduced in Deﬁnition 4.3.9 is actually the open ball Ur(p) = {q ∈ M \| d(p, q) < r} whenever r ≤ inj(p; M ). Theorem 4.4.4 (Existence of minimal geodesics). Let M ⊂ Rn be a smooth m-manifold, ﬁx a point p ∈ M, and let r > 0 be smaller than the injectivity radius of M at p. Let v ∈ TpM such that \|v\| < r. Then d(p, q) = \|v\|, q := expp(v), and a curve γ ∈ Ωp,q has minimal length L(γ) = \|v\| if and only if there is a smooth map β : [0, 1] → [0, 1] satisfying β(0) = 0, β(1) = 1, ˙β ≥ 0 such that γ(t) = expp(β(t)
v) for 0 ≤ t ≤ 1. The proof is based on the following lemma. Figure 4.6: The Gauß Lemma. Lemma 4.4.5 (Gauß Lemma). Let M, p, r be as in Theorem 4.4.4, let I ⊂ R be an open interval, and let w : I → Vp be a smooth curve whose norm is constant. Deﬁne \|w(t)\| =: r α(s, t) := expp(sw(t)) for (s, t) ∈ R × I with sw(t) ∈ Vp. Then ∂α ∂s, ∂α ∂t ≡ 0. Thus the geodesics through p are orthogonal to the boundaries of the embedded balls Ur(p) in Corollary 4.3.8 (see Figure 4.6). Upr 200 CHAPTER 4. GEODESICS Proof of Lemma 4.4.5. For every t ∈ I we have α(0, t) = expp(0) = p and so the assertion holds for s = 0, i.e. ∂α ∂s (0, t), ∂α ∂t (0, t) = 0. Moreover, each curve s → α(s, t) is a geodesic, i.e. ∇s ∂α ∂s = Π(α) ∂2α ∂s2 ≡ 0. By Theorem 4.1.4, the function s → ∂α ∂s (s, t) is constant for every t, so that ∂α ∂s (s, t) = ∂α ∂s (0, t) This implies = \|w(t)\| = r for (s, t) ∈ R × I. ∂ ∂s ∂α ∂s, ∂α ∂t = ∇s ∂α ∂s, ∂α ∂t + ∂α ∂s, ∇s ∂α ∂t ∂2α ∂s∂t ∂2α ∂s∂t, Π(α) = = = ∂α ∂s Π(α) ∂α ∂s, ∂ ∂t = 1 2 = 0., �
�α ∂s ∂2α ∂s∂t 2 ∂α ∂s Since the function ∂α ∂s, ∂α ∂t vanishes for s = 0 we obtain ∂α ∂s ∂α ∂t (s, t), (s, t) = 0 for all s and t. This proves Lemma 4.4.5. 4.4. MINIMAL GEODESICS 201 Proof of Theorem 4.4.4. Let r > 0 be as in Corollary 4.3.8 and let v ∈ TpM such that 0 < \|v\| =: ε < r. Denote q := expp(v) and let γ ∈ Ωp,q. Assume ﬁrst that γ(t) ∈ expp Bε(p) = U ε ∀ t ∈ [0, 1]. Then there is a unique smooth function [0, 1] → TpM : t → v(t) such that \|v(t)\| ≤ ε and γ(t) = expp(v(t)) for every t. The set I := {t ∈ [0, 1] \| γ(t) = p} = {t ∈ [0, 1] \| v(t) = 0} ⊂ (0, 1] is open in the relative topology of (0, 1]. Thus I is a union of open intervals in (0, 1) and one half open interval containing 1. Deﬁne β : [0, 1] → [0, 1] and w : I → TpM by β(t) := \|v(t)\| ε, w(t) := ε v(t) \|v(t)\|. Then β is continuous, both β and w are smooth on I, β(0) = 0, β(1) = 1, w(1) = v, and \|w(t)\| = ε, γ(t) = expp(β(t)w(t)) for all t ∈ I. We prove that L(γ) ≥ ε. To see this let α : [0, 1] × I → M be the map of Lemma 4.4.5, i.e. α(s, t)
:= expp(sw(t)). Then γ(t) = α(β(t), t) and hence ˙γ(t) = ˙β(t) ∂α ∂s (β(t), t) + ∂α ∂t (β(t), t) for every t ∈ I. Hence it follows from Lemma 4.4.5 that \| ˙γ(t)\|2 = ˙β(t)2 ∂α ∂s (β(t), t) 2 + ∂α ∂t 2 (β(t), t) ≥ ˙β(t)2ε2 for every t ∈ I. Hence L(γ) = 1 0 \| ˙γ(t)\| dt = I \| ˙γ(t)\| dt ≥ ε I ˙β(t) dt ≥ ε I ˙β(t) dt = ε. 202 CHAPTER 4. GEODESICS Here the last equality follows by applying the fundamental theorem of calculus to each interval in I and using the fact that β(0) = 0 and β(1) = 1. If L(γ) = ε, we must have ∂α ∂t (β(t), t) = 0, ˙β(t) ≥ 0 for all t ∈ I. Thus I is a single half open interval containing 1 and on this interval the condition ∂α ∂t (β(t), t) = 0 implies ˙w(t) = 0. Since w(1) = v we have w(t) = v for every t ∈ I. Hence γ(t) = expp(β(t)v) for every t ∈ [0, 1]. It follows that β is smooth on the closed interval [0, 1] (and not just on I). Thus we have proved that every γ ∈ Ωp,q with values in U ε has length L(γ) ≥ ε with equality if and only if γ is a reparametrized geodesic. But if γ ∈ Ωp,q does not take values only in U ε, there must be a T ∈ (0, 1) such that γ([0, T ]) ⊂ U �
� and γ(T ) ∈ ∂Uε. Then L(γ\|[0,T ]) ≥ ε, by what we have just proved, and L(γ\|[T,1]) > 0 because the restriction of γ to [T, 1] cannot be constant; so in this case we have L(γ) > ε. This proves Theorem 4.4.4. The next corollary gives a partial answer to our problem of ﬁnding length minimizing curves. It asserts that geodesics minimize the length locally. Corollary 4.4.6. Let M ⊂ Rn be a smooth m-manifold, let I ⊂ R be an open interval, and let γ : I → M be a geodesic. Fix a point t0 ∈ I. Then there exists a constant ε > 0 such that t0 − ε < s < t < t0 + ε =⇒ L(γ\|[s,t]) = d(γ(s), γ(t)). Proof. Since γ is a geodesic its derivative has constant norm \| ˙γ(t)\| ≡ c (see Theorem 4.1.4). Choose δ > 0 so small that the interval [t0 − δ, t0 + δ] is contained in I. Then there is a constant r > 0 such that r ≤ inj(γ(t)) whenever \|t − t0\| ≤ δ. Choose ε > 0 such that ε < δ, 2εc < r. If t0 − ε < s < t < t0 + ε, then γ(t) = expγ(s) ((t − s) ˙γ(s)) and \|(t − s) ˙γ(s)\| = \|t − s\| c < 2εc < r ≤ inj(γ(s)). Hence it follows from Theorem 4.4.4 that L(γ\|[s,t]) = \|t − s\| c = d(γ(s), γ(t)). This proves Corollary 4.4.6. 4.4. MINIMAL GEODESICS 203 4.4.3 Examples and Exercises Exercise 4.4.7. How large can the constant ε in Corollary 4.4.
6 be chosen in the case M = S2? Compare this with the injectivity radius. Remark 4.4.8. We conclude from Theorem 4.4.4 that Sr(p) := q ∈ M d(p, q) = r = expp v ∈ TpM \| \|v\| = r (4.4.1) for 0 < r < inj(p; M ). The Gauß Lemma 4.4.5 shows that the geodesic rays [0, 1] → M : s → expp(sv) emanating from p are the orthogonal trajectories to the concentric spheres Sr(p). Exercise 4.4.9. Let M ⊂ R3 be of dimension two and suppose that M is invariant under the (orthogonal) reﬂection about some plane E ⊂ R3. Show that E intersects M in a geodesic. (Hint: Otherwise there would be points p, q ∈ M very close to one another joined by two distinct minimal geodesics.) Conclude for example that the coordinate planes intersect the ellipsoid (x/a)2 + (y/b)2 + (z/c)2 = 1 in geodesics. Exercise 4.4.10. Choose geodesic normal coordinates near p ∈ M via q = expp xi(q)ei, m i=1 where e1,..., em is an orthonormal basis of TpM (see Corollary 4.5.4 below). Then we have xi(p) = 0 and Br(p) = {q ∈ M \| d(p, q) < r} = q ∈ M xi(q) 2 < r2 m i=1 (4.4.2) for 0 < r < inj(p; M ). Hence Theorem 4.5.3 below asserts that Br(p) is convex for r > 0 suﬃciently small. (i) Show that it can happen that a geodesic in Br(p) is not minimal. Hint: Take M to be the hemisphere {(x, y, z) ∈ R3 \| x2 + y2 + z2 = 1, z > 0} together with the disc {(x, y, z) ∈ R3 \| x2 + y2
≤ 1, z = 0}, but smooth the corners along the circle x2 + y2 = 1, z = 0. Take p = (0, 0, 1) and r = π/2. (ii) Show that, if r > 0 is suﬃciently small, then the unique geodesic γ in Br(p) joining two points q, q ∈ Br(p) is minimal and that in fact any curve γ from q to q which is not a reparametrization of γ is strictly longer, i.e. L(γ) > L(γ) = d(q, q). 204 CHAPTER 4. GEODESICS Exercise 4.4.11. Let γ : I = [a, b] → M be a smooth curve with endpoints γ(a) = p and γ(b) = q and nowhere vanishing derivative, i.e. ˙γ(t) = 0 for all t ∈ I. Prove that the following are equivalent. (i) The curve γ is an extremal of the length functional, i.e. every smooth map R × I → M : (s, t) → γs(t) with γs(a) = p and γs(b) = q for all s satisﬁes s=0 L(γs) d ds = 0. (ii) The curve γ is a reparametrized geodesic, i.e. there exists a smooth map σ : [a, b] → [0, 1] with σ(a) = 0, σ(b) = 1, ˙σ(t) ≥ 0 for all t ∈ I, and a vector v ∈ TpM such that q = expp(v), γ(t) = expp(σ(t)v) (We remark that the hypothesis ˙γ(t) = 0 implies that σ is for all t ∈ I. actually a diﬀeomorphism, i.e. ˙σ(t) > 0 for all t ∈ I.) (iii) The curve γ minimizes the length functional locally, i.e. there exists an ε > 0 such that L(γ\|[s,t]) = d(γ(s),
γ(t)) for every closed subinterval [s, t] ⊂ I of length t − s < ε. It is often convenient to consider curves γ where ˙γ(t) is allowed to vanish for some values of t; then γ cannot (in general) be parametrized by arclength. Such a curve γ : I → M can be smooth (as a map) and yet its image may have corners (where ˙γ necessarily vanishes). Note that a curve with corners can never minimize the distance, even locally. Exercise 4.4.12. Show that conditions (ii) and (iii) in Exercise 4.4.11 are equivalent, even without the assumption that ˙γ is nowhere vanishing. Deduce that, if γ : I → M is a shortest curve joining p to q, i.e. L(γ) = d(p, q), then γ is a reparametrized geodesic. Show by example that one can have a variation {γs}s∈R of a reparametrized geodesic γ0 = γ for which the map s → L(γs) is not even diﬀerentiable at s = 0. (Hint: Take γ to be constant. See also Exercise 4.1.9.) Show, however, that conditions (i), (ii) and (iii) in Exercise 4.4.11 remain equivalent if the hypothesis that ˙γ is nowhere vanishing is weakened to the hypothesis that ˙γ(t) = 0 for all but ﬁnitely many t ∈ I. Conclude that a broken geodesic is a reparametrized geodesic if and only if it minimizes arclength locally. (A broken geodesic is a continuous map γ : I = [a, b] → M for which there exist a = t0 < t1 < · · · < tn = b such that γ\|[ti−1,ti] is a geodesic for i = 1,..., n. It is thus a geodesic if and only if ˙γ is continuous at the break points, i.e. ˙γ(t− i ) for i = 1,..., n − 1.) i ) = ˙γ(
t+ 4.5. CONVEX NEIGHBORHOODS 205 4.5 Convex Neighborhoods A subset of an aﬃne space is called convex iﬀ it contains the line segment joining any two of its points. The deﬁnition carries over to a submanifold M of Euclidean space (or indeed more generally to any manifold M equipped with a spray) once we reword the deﬁnition so as to confront the diﬃculty that a geodesic joining two points might not exist nor, if it does, need it be unique. Deﬁnition 4.5.1 (Geodesically convex set). Let M ⊂ Rn be a smooth m-dimensional manifold. A subset U ⊂ M is called geodesically convex iﬀ, for all p0, p1 ∈ U, there exists a unique geodesic γ : [0, 1] → U such that γ(0) = p0 and γ(1) = p1. It is not precluded in Deﬁnition 4.5.1 that there be other geodesics from p to q which leave and then re-enter U, and these may even be shorter than the geodesic in U. Exercise 4.5.2. (a) Find a geodesically convex set U in a manifold M and points p0, p1 ∈ U such that the unique geodesic γ : [0, 1] → U with γ(0) = p0 and γ(1) = p1 has length L(γ) > d(p0, p1). Hint: An interval of length bigger than π in S1. (b) Find a set U in a manifold M such that any two points in U can be joined by a minimal geodesic in U, but U is not geodesically convex. Hint: A closed hemisphere in S2. Theorem 4.5.3 (Convex Neighborhood Theorem). Let M ⊂ Rn be a smooth m-dimensional submanifold and ﬁx a point p0 ∈ M. Let φ : U → Ω be any coordinate chart on an open neighborhood U ⊂ M of p0 with values in an open set �
�� ⊂ Rm. Then the set Ur := {p ∈ U \| \|φ(p) − φ(p0)\| < r} (4.5.1) is geodesically convex for r > 0 suﬃciently small. Before giving the proof of Theorem 4.5.3 we derive a useful corollary. Corollary 4.5.4. Let M ⊂ Rn be a smooth m-manifold and let p0 ∈ M. Then, for r > 0 suﬃciently small, the open ball Ur(p0) := {p ∈ M \| d(p0, p) < r} (4.5.2) is geodesically convex. 206 CHAPTER 4. GEODESICS Proof. Choose an orthonormal basis e1,..., em of Tp0M and deﬁne Ω := {x ∈ Rm \| \|x\| < inj(p0; M )}, U := {p ∈ M \| d(p0, p) < inj(p0; M )}. Deﬁne the map ψ : Ω → U by ψ(x) := expp0 xiei m i=1 (4.5.3) (4.5.4) for x = (x1,..., xm) ∈ Ω. Then ψ is a diﬀeomorphism and d(p0, ψ(x)) = \|x\| for all x ∈ Ω by Theorem 4.4.4. Hence its inverse φ := ψ−1 : U → Ω (4.5.5) satisﬁes φ(p0) = 0 and \|φ(p)\| = d(p0, p) for all p ∈ U. Thus Ur(p0) = {p ∈ U \| \|φ(p) − φ(p0)\| < r} for 0 < r < inj(p0; M ) and so Corollary 4.5.4 follows from Theorem 4.5.3. Deﬁnition 4.5.5 (Geodesically normal coordinates). The coordinate chart φ : U → Ω in (
4.5.4) and (4.5.5) sends geodesics through p0 to straight lines through the origin. Its components x1,..., xm : U → R are called geodesically normal coordinates at p0. Proof of Theorem 4.5.3. Assume without loss of generality that φ(p0) = 0. ij : Ω → R be the Christoﬀel symbols of the coordinate chart and, Let Γk for x ∈ Ω, deﬁne the quadratic function Qx : Rm → R by m m Qx(ξ) := ξk2 − xkΓk ij(x)ξiξj. k=1 i,j,k=1 Shrinking U, if necessary, we may assume that m max i,j=1,...,m 1 2m xkΓk ij(x) ≤ k=1 for all x ∈ Ω. Then, for all x ∈ Ω and all ξ ∈ Rm we have Qx(ξ) ≥ \|ξ\|2 − 1 2m m i=1 2 ξi ≥ 1 2 \|ξ\|2 ≥ 0. Hence Qx is positive deﬁnite for every x ∈ Ω. 4.5. CONVEX NEIGHBORHOODS 207 Now let γ : [0, 1] → U be a geodesic and deﬁne c(t) := φ(γ(t)) for 0 ≤ t ≤ 1. Then, by Lemma 4.3.14, c satisﬁes the diﬀerential equation ¨ck + ij(c) ˙ci ˙cj = 0. Γk i,j Hence d2 dt2 \|c\|2 2 = d dt ˙c, c = \| ˙c\|2 + ¨c, c = Qc( ˙c) ≥ \| ˙c\|2 2 ≥ 0 and so the function t → \|φ(γ(t))\|2 is convex. Thus, if γ(0), γ(1) ∈ Ur for some r > 0, it follows that γ(t
) ∈ Ur for all t ∈ [0, 1]. Consider the exponential map V = {(p, v) ∈ T M \| v ∈ Vp} → M : (p, v) → expp(v) in Lemma 4.3.6. Its domain V is open and the exponential map is smooth. Since it sends the pair (p0, 0) ∈ V to expp0(0) = p0 ∈ U, it follows from continuity that there exist constants ε > 0 and r > 0 such that p ∈ Ur, v ∈ TpM, \|v\| < ε =⇒ v ∈ Vp, expp(v) ∈ U. (4.5.6) Moreover, we have d expp0(0) = id : Tp0M → Tp0M by Corollary 4.3.7. Hence the Implicit Function Theorem 2.6.15 asserts that the constants ε > 0 and r > 0 can be chosen such that (4.5.6) holds and there exists a smooth map h : Ur × Ur → Rn that satisﬁes the conditions h(p, q) ∈ TpM, \|h(p, q)\| < ε (4.5.7) for all p, q ∈ Ur and expp(v) = q ⇐⇒ v = h(p, q) (4.5.8) for all p, q ∈ Ur and all v ∈ TpM with \|v\| < ε. In particular, we have h(p0, p0) = 0 and expp(h(p, q)) = q for all p, q ∈ Ur. 208 CHAPTER 4. GEODESICS Fix two constants ε > 0 and r > 0 and a smooth map h : Ur × Ur → Rn such that (4.5.6), (4.5.7), (4.5.8) are satisﬁed. We show that any two points p, q ∈ Ur are joined by a geodesic in Ur. Let p, q ∈ Ur and deﬁne γ(t) := expp(th(p, q)) for 0 ≤ t ≤ 1. This curve γ : [0, 1] →
M is well deﬁned by (4.5.6) and (4.5.7), it is a geodesic satisfying γ(0) = p ∈ Ur by Lemma 4.3.6, it satisﬁes γ(1) = q ∈ Ur by (4.5.8), it takes values in U by (4.5.6) and (4.5.7), and so γ([0, 1]) ⊂ Ur because the function [0, 1] → R : t → \|φ(γ(t))\|2 is convex. We show that there exists at most one geodesic in Ur joining p and q. Let p, q ∈ Ur and let γ : [0, 1] → Ur be any geodesic such that γ(0) = p and γ(1) = q. Deﬁne v := ˙γ(0) ∈ TpM. Then γ(t) = expp(tv) for 0 ≤ t ≤ 1 by Lemma 4.3.6. We claim that \|v\| < ε. Suppose, by contradiction, that Then \|v\| ≥ ε. T := ε \|v\| ≤ 1 and, for 0 < t < T, we have \|tv\| < ε and expp(tv) = γ(t) ∈ Ur and so h(p, γ(t)) = tv. by (4.5.8). Thus \|h(p, γ(t))\| = t\|v\| for 0 < t < T. Take the limit t T to obtain \|h(p, γ(T ))\| = T \|v\| = ε in contradiction to (4.5.7). This contradiction shows that \|v\| < ε. Since expp(v) = γ(1) = q ∈ Ur it follows from (4.5.8) that v = h(p, q). This proves Theorem 4.5.3. Remark 4.5.6. Theorem 4.5.3 and its proof carry over to general sprays (see Deﬁnition 4.3.2). Exercise 4.5.7. Consider the set Ur(p) = {q ∈ M \| d(p, q
) < r} for p ∈ M and r > 0. Corollary 4.5.4 asserts that this set is geodesically convex for r suﬃciently small. How large can you choose r in the cases M = T2 = S1 × S1, M = R2 \ {0}. M = R2, M = S2, Compare this with the injectivity radius. If the set Ur(p) in these examples is geodesically convex, does it follow that every geodesic in Ur(p) is minimizing? 4.6. COMPLETENESS AND HOPF–RINOW 209 4.6 Completeness and Hopf–Rinow For a Riemannian manifold there are diﬀerent notions of completeness. First, in §3.4 completeness was deﬁned in terms of the completeness of time dependent basic vector ﬁelds on the frame bundle (Deﬁnition 3.4.10). Second, there is a distance function d : M × M → [0, ∞) deﬁned by equation (4.2.2) so that we can speak of completeness of the metric space (M, d) in the sense that every Cauchy sequence converges. Third, there is the question of whether geodesics through any point in any direction exist for all time; if so we call a Riemannian manifold geodesically complete. The remarkable fact is that these three rather diﬀerent notions of completeness are actually equivalent and that, in the complete case, any two points in M can be joined by a shortest geodesic. This is the content of the Hopf–Rinow theorem. We will spell out the details of the proof for embedded manifolds and leave it to the reader (as a straight forward exercise) to extend the proof to the intrinsic setting. Geodesic Completeness Deﬁnition 4.6.1 (Geodesically complete manifold). Let M ⊂ Rn be an m-dimensional manifold. Given a point p ∈ M we say that M is geodesically complete at p iﬀ, for every tangent vector v ∈ TpM, there exists a geodesic γ : R → M (on the entire real axis) satisfying γ(0) = p and
˙γ(0) = v (or equivalently Vp = TpM where Vp ⊂ TpM is deﬁned by (4.3.5)). The manifold M is called geodesically complete iﬀ it is geodesically complete at every point p ∈ M. Deﬁnition 4.6.2. Let (M, d) be a metric space. A subset A ⊂ M is called bounded iﬀ d(p, p0) < ∞ sup p∈A for some (and hence every) point p0 ∈ M. Example 4.6.3. A manifold M ⊂ Rn can be contained in a bounded subset of Rn and still not be bounded with respect to the metric (4.2.2). An example is the 1-manifold M = (x, y) ∈ R2 \| 0 < x < 1, y = sin(1/x). Exercise 4.6.4. Let (M, d) be a metric space. Prove that every compact subset K ⊂ M is closed and bounded. Find an example of a metric space that contains a closed and bounded subset that is not compact. 210 CHAPTER 4. GEODESICS Theorem 4.6.5 (Completeness). Let M ⊂ Rn be a connected m-dimensional manifold and let d : M × M → [0, ∞) be the distance function deﬁned by (4.1.1), (4.2.1), and (4.2.2). Then the following are equivalent. (i) M is geodesically complete. (ii) There exists a point p ∈ M such that M is geodesically complete at p. (iii) Every closed and bounded subset of M is compact. (iv) (M, d) is a complete metric space. (v) M is complete, i.e. for every smooth curve ξ : R → Rm and every element (p0, e0) ∈ F(M ) there exists a smooth curve β : R → F(M ) satisfying (4.6.1) ˙β(t) = Bξ(t)(β(t)), β(0) = (p0, e0). (vi) The basic vector ﬁe
ld Bξ ∈ Vect(F(M )) is complete for every ξ ∈ Rm. (vii) For every smooth curve γ : R → Rm, every p0 ∈ M, and every orthogonal isomorphism Φ0 : Tp0M → Rm there exists a development (Φ, γ, γ) of M along Rm on all of R that satisﬁes γ(0) = p0 and Φ(0) = Φ0. Proof. The proof relies on Theorem 4.6.6 below. Global Existence of Minimal Geodesics Theorem 4.6.6 (Hopf–Rinow). Let M ⊂ Rn be a connected m-manifold and let p ∈ M. Assume M is geodesically complete at p. Then, for every q ∈ M, there exists a geodesic γ : [0, 1] → M such that γ(0) = p, γ(1) = q, L(γ) = d(p, q). Before giving the proof of the Hopf–Rinow Theorem we show that it implies Theorem 4.6.5. Theorem 4.6.6 implies Theorem 4.6.5. That (i) implies (ii) follows directly from the deﬁnitions. We prove that (ii) implies (iii). Thus assume that M is geodesically complete at the point p0 ∈ M and let K ⊂ M be a closed and bounded subset. Then r := supq∈K d(p0, q) < ∞. Hence Theorem 4.6.6 asserts that, for every q ∈ K, there exists a vector v ∈ Tp0M such that \|v\| = d(p0, q) ≤ r and expp0(v) = q. Thus K ⊂ expp0(Br(p0)), Br(p0) = {v ∈ Tp0M \| \|v\| ≤ r}. Then B := {v ∈ Tp0M \| \|v\| ≤ r, expp0(v) ∈ K} is a closed and bounded subset of the Euclidean space Tp0M. Hence B is compact and K = expp0(B).
Since the exponential map expp0 : Tp0M → M is continuous it follows that K is compact. This shows that (ii) implies (iii). 4.6. COMPLETENESS AND HOPF–RINOW 211 We prove that (iii) implies (iv). Thus assume that every closed and bounded subset of M is compact and choose a Cauchy sequence pi ∈ M. Choose i0 ∈ N such that d(pi, pj) ≤ 1 for all i, j ∈ N with i, j ≥ i0. Deﬁne c := max 1≤i≤i0 d(p1, pi) + 1. Then d(p1, pi) ≤ d(p1, pi0) + d(pi0, pi) ≤ d(p1, pi0) + 1 ≤ c for all i ≥ i0 and so d(p1, pi) ≤ c for all i ∈ N. Hence the set {pi \| i ∈ N} is bounded and so is its closure. By (iii) this implies that the sequence pi has a convergent subsequence. Since pi is a Cauchy sequence, this implies that pi converges. Thus we have proved that (iii) implies (iv). We prove that (iv) implies (v). Fix a smooth curve ξ : R → Rm and an element (p0, e0) ∈ F(M ). Assume, by contradiction, that there exists a real number T > 0 such that there exists a solution β : [0, T ) → F(M ) of equation (4.6.1) that cannot be extended to the interval [0, T + ε) for any ε > 0. Write β(t) =: (γ(t), e(t)) so that γ and e satisfy the equations ˙γ(t) = e(t)ξ(t), ˙e(t) = hγ(t)( ˙γ(t))e(t), This implies e(t)η ∈ Tγ(t)M and ˙e(t)η ∈ T ⊥ γ(0) = p0, e(0) = e0. γ(t)M for all η ∈ Rm and therefore d dt η, e(t)
Te(t)ζ = d dt e(t)η, e(t)ζ = ˙e(t)η, e(t)ζ + e(t)η, ˙e(t)ζ = 0 for all η, ζ ∈ Rm and all t ∈ [0, T ). Thus the function t → e(t)Te(t) is constant, hence e(t)Te(t) = eT 0 e0, e(t) = sup 0=η∈Rm \|e(t)η\| \|η\| = e0 (4.6.2) for 0 ≤ t < T, hence \| ˙γ(t)\| = \|e(t)ξ(t)\| ≤ e0 \|ξ(t)\| ≤ e0 sup 0≤s≤T \|ξ(s)\| =: cT and so d(γ(s), γ(t)) ≤ L(γ\|[s,t]) ≤ (t − s)cT for 0 ≤ s < t < T. Since (M, d) is a complete metric space, this shows that the limit p1 := limtT γ(t) ∈ M exists. Thus the set K := γ([0, T )) ∪ {p1} ⊂ M is compact and so is the set K := (p, e) ∈ F(M ) \| p ∈ K, eTe = eT ⊂ F(M ). 0 e0 By equation (4.6.2) the curve [0, T ) → R × F(M ) : t → (t, γ(t), e(t)) takes values in the compact set [0, T ] × K and is the integral curve of a vector ﬁeld on the manifold R × F(M ). Hence Corollary 2.4.15 asserts that [0, T ) cannot be the maximal existence interval of this integral curve, a contradiction. This shows that (iv) implies (v). 212 CHAPTER 4. GEODESICS That (v) implies (vi) follows by taking ξ(t) ≡ ξ in (v). We prove that (vi) implies (i). Fix an element p0 ∈ M and a tangent vector v0
∈ Tp0M. Let e0 ∈ Liso(Rm, Tp0M ) be any isomorphism and choose ξ ∈ Rm such that e0ξ = v0. By (vi) the vector ﬁeld Bξ has a unique integral curve R → F(M ) : t → β(t) = (γ(t), e(t)) with β(0) = (p0, e0). ˙γ(t) = e(t)ξ, ˙e(t) = hγ(t)(e(t)ξ)e(t), Thus and hence ¨γ(t) = ˙e(t)ξ = hγ(t)(e(t)ξ)e(t)ξ = hγ(t)( ˙γ(t), ˙γ(t)). By the Gauß–Weingarten formula, this implies ∇ ˙γ(t) = 0 for every t and hence γ : R → M is a geodesic with γ(0) = p0 and ˙γ(0) = e0ξ = v0. Thus M is geodesically complete and this shows that (vi) implies (i). The equivalence of (v) and (vii) was established in Corollary 3.5.25 and this shows that Theorem 4.6.6 implies Theorem 4.6.5. Proof of the Hopf–Rinow Theorem The proof of Theorem 4.6.6 relies on the next two lemmas. Lemma 4.6.7. Let M ⊂ Rn be a connected m-manifold and p ∈ M. Suppose ε > 0 is smaller than the injectivity radius of M at p and denote Sε(p) := p ∈ M \| d(p, p) = ε. Σ1(p) := {v ∈ TpM \| \|v\| = 1}, Then the map Σ1(p) → Sε(p) : v → expp(εv) is a diﬀeomorphism and, for all q ∈ M, we have d(p, q) > ε =⇒ d(Sε(p), q) =
d(p, q) − ε. Proof. By Theorem 4.4.4, we have d(p, expp(v)) = \|v\| for all v ∈ TpM with \|v\| ≤ ε and d(p, p) > ε for all p ∈ M \ expp(v) \| v ∈ TpM, \|v\| ≤ ε. This shows that Sε(p) = expp(εΣ1(p)) and, since ε is smaller than the injectivity radius, the map Σ1(p) → Sε(p) : v → expp(εv) is a diﬀeomorphism. 4.6. COMPLETENESS AND HOPF–RINOW 213 To prove the second assertion, let q ∈ M such that r := d(p, q) > ε. Fix a constant δ > 0 and choose a smooth curve γ : [0, 1] → M such that γ(0) = p, γ(1) = q, L(γ) ≤ r + δ. Choose t0 > 0 such that γ(t0) is the last point of the curve on Sε(p), i.e. γ(t0) ∈ Sε(p), γ(t) /∈ Sε(p) for t0 < t ≤ 1. Then d(γ(t0), q) ≤ L(γ\|[t0,1]) = L(γ) − L(γ\|[0,t0]) ≤ L(γ) − ε ≤ r + δ − ε. This shows that d(Sε(p), q) ≤ r + δ − ε for every δ > 0 and therefore d(Sε(p), q) ≤ r − ε. Moreover, d(p, q) ≥ d(p, q) − d(p, p) = r − ε for all p ∈ Sε(p). Thus d(Sε(p), q) = r − ε and this proves Lemma 4.6.7. Lemma 4.6.8 (Curve Shortening Lemma). Let M ⊂ Rn be an m-manifold, let p ∈ M, and let ε be a
real number such that 0 < ε < inj(p; M ). Then, for all v, w ∈ TpM, we have \|v\| = \|w\| = ε, d(expp(v), expp(w)) = 2ε =⇒ v + w = 0. 214 CHAPTER 4. GEODESICS Figure 4.7: Two unit tangent vectors. Proof. We will prove that, for all v, w ∈ TpM, we have d(expp(δv), expp(δw)) δ lim δ→0 = \|v − w\|. (4.6.3) Assume this holds and suppose, by contradiction, that there exist two tangent vectors v, w ∈ TpM such that \|v\| = \|w\| = 1, d(expp(εv), expp(εw)) = 2ε, v + w = 0. Then \|v − w\| < 2 (see Figure 4.7). Thus by (4.6.3) there exists a constant 0 < δ < ε such that d(expp(δv), expp(δw)) < 2δ. Then d(expp(εv), expp(εw)) ≤ d(expp(εv), expp(δv)) + d(expp(δv), expp(δw)) + d(expp(δw), expp(εw)) < ε − δ + 2δ + ε − δ = 2ε and this contradicts our assumption. It remains to prove (4.6.3). For this we observe that d(expp(δv), expp(δw)) δ lim δ→0 = lim δ→0 = lim δ→0 = lim δ→0 d(expp(δv), expp(δw)) expp(δv) − expp(δw) expp(δv) − expp(δw) δ expp(δv) − p δ − expp(δw) − p δ expp(δv) − expp(δw) δ = \|v − w\|. Here the second equality follows from Lemma
4.2.7. wv 4.6. COMPLETENESS AND HOPF–RINOW 215 Proof of Theorem 4.6.6. By assumption M ⊂ Rn is a connected submanifold, and p ∈ M is given such that the exponential map expp : TpM → M is deﬁned on the entire tangent space at p. Fix a point q ∈ M \ {p} so that 0 < r := d(p, q) < ∞. Choose a constant ε > 0 smaller than the injectivity radius of M at p and smaller than r. Then, by Lemma 4.6.7, we have d(Sε(p), q) = r − ε. Hence there exists a tangent vector v ∈ TpM such that d(expp(εv), q) = r − ε, \|v\| = 1. Deﬁne the curve γ : [0, r] → M by γ(t) := expp(tv) for 0 ≤ t ≤ r. By Lemma 4.3.6, this is a geodesic and it satisﬁes γ(0) = p. We must prove that γ(r) = q and L(γ) = d(p, q). Instead we will prove the following stronger statement. Claim. For every t ∈ [0, r] we have d(γ(t), q) = r − t. In particular, γ(r) = q and L(γ) = r = d(p, q). Consider the subset I := {t ∈ [0, r] \| d(γ(t), q) = r − t} ⊂ [0, r]. This set is nonempty, because ε ∈ I, it is obviously closed, and t ∈ I =⇒ [0, t] ⊂ I. (4.6.4) Namely, if t ∈ I and 0 ≤ s ≤ t, then d(γ(s), q) ≤ d(γ(s), γ(t)) + d(γ(t), q and d(γ(s), q) ≥ d(p, q) − d(p, γ(s)) ≥ r − s. Hence d(γ(s
), q) = r − s and hence s ∈ I. This proves (4.6.4). 216 CHAPTER 4. GEODESICS We prove that I is open (in the relative topology of [0, r]). Let t ∈ I be given with t < r. Choose a constant ε > 0 smaller than the injectivity radius of M at γ(t) and smaller than r − t. Then, by Lemma 4.6.7 with p replaced by γ(t), we have d(Sε(γ(t)), q) = r − t − ε. Next we choose w ∈ Tγ(t)M such that \|w\| = 1, d(expγ(t)(εw), q) = r − t − ε. Then d(γ(t − ε), expγ(t)(εw)) ≥ d(γ(t − ε), q) − d(expγ(t)(εw), q) = (r − t + ε) − (r − t − ε) = 2ε. The converse inequality is obvious, because both points have distance ε to γ(t) (see Figure 4.8). Figure 4.8: The proof of the Hopf–Rinow theorem. Thus we have proved that Since d(γ(t − ε), expγ(t)(εw)) = 2ε. γ(t − ε) = expγ(t)(−ε ˙γ(t)), it follows from Lemma 4.6.8 that w = ˙γ(t). Hence expγ(t)(sw) = γ(t + s) and this implies that d(γ(t + ε), q) = r − t − ε. Thus t + ε ∈ I and, by (4.6.4), we have [0, t + ε] ∈ I. Thus we have proved that I is open. In other words, I is a nonempty subset of [0, r] which is both open and closed, and hence I = [0, r]. This proves the claim and Theorem 4.6.6. γεεexp ( w)ε(t)pqγγ(t)S ( (t))εεr−t− 4.7. GEODES
ICS IN THE INTRINSIC SETTING* 217 4.7 Geodesics in the Intrinsic Setting* This section examines the distance function on a Riemannian manifold, shows how the results of this chapter extend to the intrinsic setting, and discusses several examples. 4.7.1 Intrinsic Distance Let M be a connected smooth manifold (§2.8) equipped with a Riemannian metric (§3.7). Then we can deﬁne the length of a curve γ : [0, 1] → M by the formula (4.1.1) and it is invariant under reparametrization as in Remark 4.1.3. The distance function d : M × M → R is then given by the same formula (4.2.2). We prove that it still deﬁnes a metric on M and that this metric induces the same topology as the smooth structure. Lemma 4.7.1. Let M be a connected smooth Riemannian manifold and deﬁne the function d : M × M → [0, ∞) by (4.1.1), (4.2.1), and (4.2.2). Then d is a metric and induces the same topology as the smooth structure. Proof. The proof has three steps. Step 1. Fix a point p0 ∈ M and let φ : U → Ω be a coordinate chart of M onto an open subset Ω ⊂ Rm such that p0 ∈ U. Then there exists an open neighborhood V ⊂ U of p0 and constants δ, r > 0 such that δ \|φ(p) − φ(p0)\| ≤ d(p, p0) ≤ δ−1 \|φ(p) − φ(p0)\| (4.7.1) for every p ∈ V and d(p, p0) ≥ δr for every p ∈ M \ V. Denote the inverse of the coordinate chart φ by ψ := φ−1 : Ω → M and deﬁne the map g = (gij)m i,j=1 : Ω → Rm×m by ∂ψ ∂xi (x), ∂ψ ∂xj (x) gij(x
) := ψ(x) for x ∈ Ω. Then a smooth curve γ : [0, 1] → U has the length L(γ) = 1 0 ˙c(t)Tg(c(t)) ˙c(t) dt, c(t) := φ(γ(t)). (4.7.2) Let x0 := φ(p0) ∈ Ω and choose r > 0 such that Br(x0) ⊂ Ω. Then there is a constant δ ∈ (0, 1] such that δ \|ξ\| ≤ ξTg(x)ξ ≤ δ−1 \|ξ\| (4.7.3) for all x ∈ Br(x0) and ξ, η ∈ Rm. Deﬁne V := φ−1(Br(x0)) ⊂ U. 218 CHAPTER 4. GEODESICS Now let p ∈ V and denote x := φ(p) ∈ Br(x0). Then, for every smooth curve γ : [0, 1] → V with γ(0) = p0 and γ(1) = p, the curve c := φ ◦ γ takes values in Br(x0) and satisﬁes c(0) = x0 and c(1) = x. Hence, by (4.7.2) and (4.7.3), we have L(γ) ≥ δ 1 0 \| ˙c(t)\| dt ≥ δ 1 0 ˙c(t) dt = δ \|x − x0\|. If γ : [0, 1] → M is a smooth curve with endpoints γ(0) = p0 and γ(1) = p whose image is not entirely contained in V, then there exists a T ∈ (0, 1] such that γ(t) ∈ V for 0 ≤ t < T and γ(T ) ∈ ∂V, so c(t) = φ(γ(t)) ∈ Br(x0) for 0 ≤ t < T and \|c(T ) − x0\| = r. Hence, by the above argument, we have L(γ) ≥
δr. This shows that d(p0, p) ≥ δr for p ∈ M \ V and d(p0, p) ≥ δ \|φ(p) − φ(p0)\| for p ∈ V. If p ∈ V, x := φ(p), and c(t) := x0 + t(x − x0), then γ := ψ ◦ c is a smooth curve in V with γ(0) = p0 and γ(1) = p and, by (4.7.2) and (4.7.3), L(γ) ≤ δ−1 1 0 \| ˙c(t)\| dt = δ−1 \|x − x0\|. This proves Step 1. Step 2. d is a distance function. Step 1 shows that d(p, p0) > 0 for every p ∈ M \ {p0} and hence d satisﬁes condition (i) in Lemma 4.2.3. The proofs of (ii) and (iii) remain unchanged in the intrinsic setting and this proves Step 2. Step 3. The topology on M induced by d agrees with the topology induced by the smooth structure. Assume ﬁrst that W ⊂ M is open with respect to the manifold topology and let p0 ∈ W. Let φ : U → Ω be a coordinate chart of M onto an open subset Ω ⊂ Rm such that p0 ∈ U, and choose V ⊂ U and δ, r as in Step 1. Then φ(V ∩ W ) is an open subset of Ω containing the point φ(p0). Hence there exists a constant 0 < ε ≤ δr such that Bδ−1ε(φ(p0)) ⊂ φ(V ∩ W ). Thus by Step 1 we have d(p, p0) ≥ δr ≥ ε for all p ∈ M \ V. Hence, if p ∈ M satisﬁes d(p, p0) < ε, then p ∈ V, so \|φ(p) − φ(p0)\| < δ−1d(p, p0) < δ−1ε by (
4.7.1), and therefore φ(p) ∈ φ(V ∩ W ). Thus Bε(p0; d) ⊂ W and this shows that W is open with respect to d. 4.7. GEODESICS IN THE INTRINSIC SETTING* 219 Conversely, assume that W ⊂ M is open with respect to d and choose a coordinate chart φ : U → Ω onto an open set Ω ⊂ Rm. We must prove that φ(W ∩ U ) is an open subset of Ω. To see this, choose x0 ∈ φ(W ∩ U ) and let p0 := φ−1(x0) ∈ W ∩ U. Now choose V ⊂ U and δ, r as in Step 1. Choose ε > 0 such that Bδ−1ε(p0; d) ⊂ W and Bε(x0) ⊂ φ(V ). Let x ∈ Rn such that \|x − x0\| < ε. Then x ∈ φ(V ) and therefore p := φ−1(x) ∈ V. This implies d(p, p0) < δ−1\|φ(p) − φ(p0)\| = δ−1\|x − x0\| < δ−1ε, thus p ∈ W ∩ U, and so x = φ(p) ∈ φ(W ∩ U ). Thus φ(W ∩ U ) is open, and so W is open in the manifold topology of M. This proves Step 3 and Lemma 4.7.1. 4.7.2 Geodesics and the Levi-Civita Connection With the covariant derivative understood (Theorem 3.7.8), we can deﬁne geodesics on M as smooth curves γ : I → M that satisfy the equation ∇ ˙γ = 0, as in Deﬁnition 4.1.5. Then all the above results about geodesics, as well as their proofs, carry over almost verbatim to the intrinsic setting. In particular, geodesics are in local coordinates described by equation (4.3.6) (Lemma 4.3.14) and they
are the critical points of the energy functional E(γ) := 1 2 1 0 \| ˙γ(t)\|2 dt on the space Ωp,q of all paths γ : [0, 1] → M with ﬁxed endpoints γ(0) = p and γ(1) = q. Here we use the fact that Lemma 4.1.7 extends to the intrinsic setting via the Embedding Theorem 2.9.12. So for every vector ﬁeld X ∈ Vect(γ) along γ with X(0) = 0 and X(1) = 0 there exists a curve of curves R → Ωp,q : s → γs with γ0 = γ and ∂sγs\|s=0 = X. Then, by the properties of the Levi-Civita connection, we have dE(γ)X = = 1 1 2 1 ∂s \|∂tγs(t)\|2 dt 0 ˙γ(t), ∇tX(t) dt 0 = − 1 0 ∇t ˙γ(t), X(t) dt. The right hand side vanishes for all X if and only if ∇ ˙γ ≡ 0 (Theorem 4.1.4). With this understood, we ﬁnd that, for all p ∈ M and v ∈ TpM, there exists a unique geodesic γ : Ip,v → M on a maximal open interval Ip,v ⊂ R containing zero that satisﬁes γ(0) = p and ˙γ(0) = v (Lemma 4.3.4). 220 CHAPTER 4. GEODESICS This gives rise to a smooth exponential map expp : Vp = {v ∈ TpM \| 1 ∈ Ip,v} → M as in §4.3 which satisﬁes d expp(0) = id : TpM → TpM as in Corollary 4.3.7. This leads directly to the injectivity radius, the Gauß Lemma 4.4.5, the local length minimizing property of geodesics in Theorem 4.4.4, and the Convex Neighborhood Theorem 4.5
.3. Also the proof of the equivalence of metric and geodesic completeness in Theorem 4.6.5 and of the Hopf–Rinow Theorem 4.6.6 carry over verbatim to the intrinsic setting of general Riemannian manifolds. The only place where some care must be taken is in the proof of the Curve Shortening Lemma 4.6.8 as is spelled out in Exercise 4.7.2 below. 4.7.3 Examples and Exercises Exercise 4.7.2. Choose a coordinate chart φ : U → Ω with φ(p0) = 0 such that the metric in local coordinates satisﬁes gij(0) = δij. Reﬁne the estimate (4.7.1) in the proof of Lemma 4.7.1 and show that lim p,q→p0 d(p, q) \|φ(p) − φ(q)\| = 1. This is the intrinsic analogue of Lemma 4.2.8. Use this to prove that equation (4.6.3) continues to hold for all Riemannian manifolds, i.e. d(expp(δv), expp(δw)) δ lim δ→0 = \|v − w\| for p ∈ M and v, w ∈ TpM. With this understood, the proof of the Curve Shortening Lemma 4.6.8 carries over verbatim to the intrinsic setting. Exercise 4.7.3. The real projective space RPn inherits a Riemannian metric from Sn as it is a quotient of Sn by an isometric involution. Prove that each geodesic in Sn with its standard metric descends to a geodesic in RPn. 4.7. GEODESICS IN THE INTRINSIC SETTING* 221 Exercise 4.7.4. Let f : S3 → S2 be the Hopf ﬁbration deﬁned by f (z, w) = \|z\|2 − \|w\|2, 2Re ¯zw, 2Im ¯zw Prove that the image of a great circle in S3 is a nonconstant geodesic in S2 if and only if it is orthogonal to the ﬁbers of f, which
are also great circles. Here we identify S3 with the unit sphere in C2. (See also Exercise 2.5.22.) Exercise 4.7.5. Prove that a nonconstant geodesic γ : R → S2n+1 descends to a nonconstant geodesic in CPn with the Fubini–Study metric (see Example 3.7.5) if and only if ˙γ(t) ⊥ Cγ(t) for every t ∈ R. Exercise 4.7.6. Consider the manifold Fk(Rn) := D ∈ Rn×k DTD = 1l of orthonormal k-frames in Rn, equipped with the Riemannian metric inherited from the standard inner product X, Y := trace(X TY ) on the space of real n × k-matrices. (a) Prove that TDFk(Rn) = TDFk(Rn)⊥ = DTX + X TD = 0, X ∈ Rn×k DA A = AT ∈ Rk×k. and that the orthogonal projection Π(D) : Rn×k → TDFk(Rn) is given by Π(D)X = X − DDTX + X TD. 1 2 (b) Prove that the second fundamental form of Fk(Rn) is given by hD(X)Y = − DX TY + Y TX 1 2 for D ∈ Fk(Rn) and X, Y ∈ TDFk(Rn). (c) Prove that a smooth map R → Fk(Rn) : t → D(t) is a geodesic if and only if it satisﬁes the diﬀerential equation ¨D = −D ˙DT ˙D. (4.7.4) Prove that the function DT ˙D is constant for every geodesic in Fk(Rn). Compare this with Example 4.3.12. 222 CHAPTER 4. GEODESICS Exercise 4.7.7. Let Gk(Rn) = Fk(Rn)/O(k) be the real Grassmannian of k-dimensional subspaces in Rn, equipped with a Riemannian metric as in Example 3.7.6.
Prove that a geodesics R → Fk(Rn) : t → D(t) descends to a nonconstant geodesic in Gk(Rn) if and only if DT ˙D ≡ 0 and ˙D ≡ 0. Deduce that the exponential map on Gk(Rn) is given by expΛ(Λ) = im D cos DT D 1/2 DT D + D −1/2 sin DT D 1/2 for Λ ∈ Fk(Rn) and Λ ∈ TΛFk(Rn) \ {0}. Here we identify the tangent space TΛFk(Rn) with the space of linear maps from Λ to Λ⊥, and choose the matrices D ∈ Fk(Rn) and D ∈ Rn×k such that Λ = im D, DT D = 0, Λ ◦ D = D : Rk → Λ⊥ = ker DT. Prove that the group O(n) acts on Gk(Rn) by isometries. Which subgroup acts trivially? Exercise 4.7.8. Carry over Exercises 4.7.6 and 4.7.7 to the complex Grassmannian Gk(Cn). Prove that the group U(n) acts on Gk(Cn) by isometries. Which subgroup acts trivially? Chapter 5 Curvature This chapter begins by introducing the notion of an isometry (§5.1). It shows that isometries of embedded manifolds preserve the lengths of curves and can be characterized as diﬀeomorphisms whose derivatives preserve the inner products. The chapter then moves on to the Riemann curvature tensor and establishes its symmetry properties (§5.2). That section also includes a discussion of the covariant derivative of a global vector ﬁeld. The next section is devoted to the generalized Gauß Theorema Egregium which asserts that isometries preserve geodesics, the covariant derivative, and the Riemann curvature tensor (§5.3). The ﬁnal section examines the Riemann curvature tensor in local coordinates and shows how the deﬁnitions and results of the present chapter carry over to the intrinsic setting of Riemannian manifolds (
§5.4). 5.1 Isometries Let M and M be connected submanifolds of Rn. An isometry is an isomorphism of the intrinsic geometries of M and M. Recall the deﬁnition of the intrinsic distance function d : M × M → [0, ∞) in §4.2 by d(p, q) := inf γ∈Ωp,q L(γ), L(γ) = 1 0 \| ˙γ(t)\| dt for p, q ∈ M. Let d denote the intrinisic distance function on M. 223 224 CHAPTER 5. CURVATURE Theorem 5.1.1 (Isometries). Let φ : M → M be a bijective map. Then the following are equivalent. (i) φ intertwines the distance functions on M and M, i.e. d(φ(p), φ(q)) = d(p, q) for all p, q ∈ M. (ii) φ is a diﬀeomorphism and dφ(p) : TpM → Tφ(p)M is an orthogonal isomorphism for every p ∈ M. (iii) φ is a diﬀeomorphism and L(φ ◦ γ) = L(γ) for every smooth curve γ : [a, b] → M. The bijection φ is called an isometry iﬀ it satisﬁes these equivalent conditions. In the case M = M the isometries φ : M → M form a group denoted by I(M ) and called the isometry group of M. The proof is based on the following lemma. Lemma 5.1.2. For every p ∈ M there exists a constant ε > 0 such that, for all v, w ∈ TpM with 0 < \|w\| < \|v\| < ε, we have d(expp(w), expp(v)) = \|v\| − \|w\| =⇒ w = \|w\| \|v\| v. (5.1.1) Remark 5.1.3. It follows from the triangle inequality and Theorem 4.4.4 that d(expp(v), expp(w
)) ≥ d(expp(v), p) − d(expp(w), p) = \|v\| − \|w\| whenever 0 < \|w\| < \|v\| < inj(p). Lemma 5.1.2 asserts that equality can only hold when w is a positive multiple of v or, to put it diﬀerently, that the distance between expp(v) and expp(w) must be strictly bigger that \|v\| − \|w\| whenever w is not a positive multiple of v. 5.1. ISOMETRIES 225 Proof of Lemma 5.1.2. As in Corollary 4.3.8 we denote Bε(p) := {v ∈ TpM \| \|v\| < ε}, Uε(p) := {q ∈ M \| d(p, q) < ε}. By Theorem 4.4.4 and the deﬁnition of the injectivity radius, the exponential map at p is a diﬀeomorphism expp : Bε(p) → Uε(p) for ε < inj(p). Choose 0 < r < inj(p). Then the closure of Ur(p) is a compact subset of M. Hence there is a constant ε > 0 such that ε < r and ε < inj(p) for every p ∈ Ur(p). Since ε < r we have ε < inj(p) ∀ p ∈ Uε(p). (5.1.2) : Bε(p) → Uε(p) is a diﬀeomorphism for every p ∈ Uε(p). Thus expp Deﬁne p1 := expp(w) and p2 := expp(v). Then, by assumption, we have d(p1, p2) = \|v\| − \|w\| < ε. Since p1 ∈ Uε(p) it follows from our choice of ε that ε < inj(p1). Hence there is a unique tangent vector v1 ∈ Tp1M such that \|v1\| = d(p1, p2) = \|v\| − \|w\|, expp1(v1) = p2. Following ﬁrst the shortest geodesic from p to

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