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Differential Geometry Lecture 17: Geodesics and the Exponential

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Differential Geometry Lecture 17: Geodesics and the Exponential Differential geometry Lecture 17: Geodesics and the exponential map David Lindemann University of Hamburg Department of Mathematics Analysis and Differential Geometry & RTG 1670 3. July 2020 David Lindemann DG lecture 17 3. July 2020 1 / 44 1 Geodesics 2 The exponential map 3 Geodesics as critical points of the energy functional 4 Riemannian normal coordinates 5 Some global Riemannian geometry David Lindemann DG lecture 17 3. July 2020 2 / 44 Recap of lecture 16: constructed covariant derivatives along curves defined parallel transport studied the relation between a given connection in the tangent bundle and its parallel transport maps introduced torsion tensor and metric connections, studied geometric interpretation defined the Levi-Civita connection of a pseudo-Riemannian manifold David Lindemann DG lecture 17 3. July 2020 3 / 44 Geodesics Recall the definition of the acceleration of a smooth curve n 00 n γ : I ! R , that is γ 2 Γγ (T R ). Question: Is there a coordinate-free analogue of this construc- tion involving connections? Answer: Yes, uses covariant derivative along curves. Definition Let M be a smooth manifold, r a connection in TM ! M, 0 and γ : I ! M a smooth curve. Then rγ0 γ 2 Γγ (TM) is called the acceleration of γ (with respect to r). Of particular interest is the case if the acceleration of a curve vanishes, that is if its velocity vector field is parallel: Definition A smooth curve γ : I ! M is called geodesic with respect to a 0 given connection r in TM ! M if rγ0 γ = 0. David Lindemann DG lecture 17 3. July 2020 4 / 44 Geodesics In local coordinates (x 1;:::; x n) on M we obtain for any smooth curve γ : I ! M the local formula n 2 k n i j ! 0 X @ γ X @γ @γ k @ r 0 γ = + Γ : γ @t2 @t @t ij @x k k=1 i;j=1 As a consequence we obtain a local form of the geodesic equa- tion: Corollary γ is a geodesic if and only if in all local coordinates covering a nontrivial subset of the image of γ it holds that 2 k n i j @ γ X @γ @γ k + Γ = 0 81 ≤ k ≤ n; @t2 @t @t ij i;j=1 k k where one usually writes Γij instead of Γij ◦ γ. k i j k Alternative notation: x¨ +x _ x_ Γij = 0 81 ≤ k ≤ n. David Lindemann DG lecture 17 3. July 2020 5 / 44 Geodesics Example Geodesics w.r.t. the Levi-Civita connection r of n n P i 2 R ; (du ) are affine lines of constant speed. For any i=1 n arbitrary curve γ : I ! R we have 2 1 2 n 0 00 @ γ @ γ r 0 γ = γ = γ; ;:::; : γ @t2 @t2 If a curve is a geodesic with respect to a metric connection, e.g. the Levi-Civita connection, it automatically has the following property: Lemma Let γ : I ! M be a geodesic on a pseudo-Riemannian manifold (M; g) with respect to a metric connection r. Then 0 0 g(γ ; γ ): I ! R is constant. @(g(γ0,γ0)) 0 0 0 0 0 0 Proof: @t = rγ (g(γ ; γ )) = (rγ g)(γ ; γ ) + 0 0 2g(rγ0 γ ; γ ) = 0 David Lindemann DG lecture 17 3. July 2020 6 / 44 Geodesics Corollary A geodesic γ in w.r.t. the Levi-Civita connection of a Riemannian manifold (M; g) has constant speed pg(γ0; γ0). Next, we need to study if geodesics always exist and determine their uniqueness. Proposition A Let M be a smooth manifold and r a connection in TM ! M. Let further p 2 M and v 2 TpM. Then there exists " > 0 and a smooth curve γ :(−"; ") ! M, γ(0) = p, γ0(0) = v, such that γ is a geodesic. If γ1 : I1 ! M and γ2 : I2 ! M are geodesics on M such that I1 \ I2 6= ; and for some point t0 2 I1 \ I2, 0 0 γ1(t0) = γ2(t0) and γ1(t0) = γ2(t0), then γ1 and γ2 coincide on I1 \ I2, i.e. γ1jI1\I2 = γ2jI1\I2 . Proof: (next page) David Lindemann DG lecture 17 3. July 2020 7 / 44 Geodesics (continuation of proof) suffices to prove this proposition in local coordinates the differential equation for a geodesic in local k i j k coordinatesx ¨ +x _ x_ Γij = 0, 1 ≤ k ≤ n, is a nonlinear system of second order ODEs turn this system of n second order ODEs into a system of 2n first order system of ODEs system of equations k k i i j k x_ = v ; v_ = −x_ x_ Γij 81 ≤ k ≤ n with fitting initial values first thing in need of clarification: the symbols v k the v k are precisely the induced coordinates on k k TU ⊂ TM, so that v (V ) = V (x ) for all V 2 TqM with q 2 U in the above eqn., the x k and v k are, however, to be read as components of a curve (x = x(t); v = v(t)) : I ! TM (continued on next page) David Lindemann DG lecture 17 3. July 2020 8 / 44 Geodesics (continuation of proof) in local coordinates x k , v k , the loc. geod. eqn. can be viewed as integral curve of a vector field on TU, G 2 X(TU), that is a smooth section in TTU ! TU k k to see this first observe that since the x and v are coordinate functions on TU, they induce coordinates on TTU the corresponding local frame in TTU ! TU is given by @ @ @ @ @x1 ;:::; @xn ; @v 1 ;:::; @v n @ one can imagine each @xk as being \horizontal" and each @ @v k as being \vertical" using Einstein summation convention, G is given by k @ i j k @ G = v @xk − v v Γij @v k (continued on next page) David Lindemann DG lecture 17 3. July 2020 9 / 44 Geodesics (continuation of proof) by covering M with charts and thus TM with induced charts, G extends to a vector field on TM, G 2 X(TM) since any integral curve of G,(x; v): I ! TM, t 7! (x(t); v(t)), fulfilsx _ = v, it is precisely the velocity vector field of the curve x : I ! M, t 7! x(t) hence, the projection of any integral curve of G to M via the bundle projection π : TM ! M is a geodesic by existence and uniqueness properties of integral curves of vector fields, the statement of this proposition follows Definition The (local) flow of the vector field G 2 X(TM), locally given k @ i j k @ by G = v @xk − v v Γij @v k as in Proposition A, is called geodesic flow with respect to r. David Lindemann DG lecture 17 3. July 2020 10 / 44 Geodesics Local uniqueness of geodesics allows us to define a maximality property for geodesics: Definition A geodesic γ : I ! M is called maximal if there exists no strictly larger interval Ie⊃ I and a geodesic γe : Ie! M, such that γejI = γ. This means that γ cannot be extended to a larger domain while still keeping its geodesic property. A smooth manifold with connection r in TM ! M is called geodesically complete if every maximal geodesic is defined on I = R. A pseudo-Riemannian manifold (M; g), respectively the metric g, is called geodesically complete if its Levi-Civita connection is complete. David Lindemann DG lecture 17 3. July 2020 11 / 44 Geodesics Question: Which reparametrisation of geodesics are allowed so that the result is still a geodesic? Answer: Lemma Let γ : I ! M be a geodesic with non-vanishing speed with respect to r and f : I ! I 0 a diffeomorphism. Then γ ◦ f is a geodesic with non-vanishing speed if and only if f is affine-linear, that is of the form f (t) = at + b for a 2 R n f0g, b 2 R. Proof: local formula & chain rule 0 00 0 0 2 0 00 0 r(γ◦f )0 (γ ◦ f ) = f · γ ◦ f + (f ) · (rγ0 γ ) ◦ f = f · γ ◦ f ; where the last equality comes from assumption that γ is a geodesic hence, γ ◦ f is a geodesic if and only if f 00 = 0, that is if f = at + b with a 2 R n f0g, b 2 R David Lindemann DG lecture 17 3. July 2020 12 / 44 Geodesics An immediate consequence is: Corollary A Maximal geodesics are unique up to affine reparametrisation. If γ : I ! M is a geodesic with initial value γ(0) = p, 0 γ (0) = v 2 TpM, t 7! γa(t) := γ(at) is a geodesic with initial 0 value γa(0) = p, γa(t) = av for all a 2 R. If a = 0, the domain −1 of γa is R. If a 6= 0, the domain of γa is a · I . In the case of pseudo-Riemannian manifolds, we have the fol- lowing additional result: Corollary A geodesic in a pseudo-Riemannian manifold with respect to the Levi-Civita connection with nonvanishing velocity can always be parametrised to be of unit speed, that is either g(γ0; γ0) ≡ 1 or g(γ0; γ0) ≡ −1. David Lindemann DG lecture 17 3. July 2020 13 / 44 Geodesics Example n Each maximal geodesics of R equipped with the canonical connection with initial condition n 0 n ∼ n γ(0) = p 2 R , γ (0) = v 2 TpR = R , is of the form n γ : R ! R ; t 7! p + tv: This in particular means that the canonical connection on n R is geodesically complete.
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