Riemannian Geometry – Lecture 22 Exponential Map Dr. Emma Carberry October 26, 2015 Lemma 22.1 (Rescaling lemma). For X 2 TpM and c 2 R n f0g, if the geodesic γX is defined on the interval (−, ) then the geodesic γcX is defined on interval − c ; c and γcX (t) = γX (ct): Proof: Define γ~ : − c ; c ! M by γ~(t) = γX (ct): _ _ Then writing re γ~_ for covariant differentiation along γ~, since γ~(t) = cγ_ (ct) and γ~(0) = cX _ ¨k k _ i _ j re γ~_ γ~(t) = γ~ (t) + Γij(~γ(t))γ~ (t)γ~ (t) @k 2 = c rγ_ γ_ X (ct) = 0: Define E ⊂ TM by E = f(p; X) 2 TM j γX is defined on an interval containing [0; 1]g; and denote by Ep the intersection of E with the set TpM. Then the exponential map is exp : E! M X 7! γX (1): 1 Proposition 22.2. 1. For each p 2 M and X 2 TpM, γX (t) = exp(tX) whenever these are defined 2. E is an open subset of TM and for each p 2 M, we have 0 2 Ep. 3. For each p 2 M, Ep is a star-shaped region about 0. That is, whenever X 2 Ep then also cX 2 Ep for all c 2 [0; 1]. 4. In fact for each p 2 M there exists a neighbourhood V of p in M and ~ > 0 such that for every (q; X) 2 TV with jXj < ~, we have (q; X) 2 E. 5. exp is a smooth map. 6. For each p 2 M there exists > 0 such that expp : fX 2 TpM j jXj < g ! M is a diffeomorphism onto its image. Proof: (1) is simply the rescaling lemma with t = 1: exp(cX) = γcX (1) = γX (c): (2) Clearly E contains the 0 section of TM. To prove that E is open, we will use the stronger version of the existence and uniqueness theo- rem. From Theorem 21.10, we have: For each p 2 M there exists an open neighbourhood U of p in M and an open neighbourhood U of (p; 0) in TU together with > 0 and a smooth map γ :(−, ) × U ! TU such that t 7! γ(t; q; X) is the unique trajectory of the geodesic vector field G which for t = 0 passes through the point q with velocity vector X 2 TqM. (3) This follows immediately from the rescaling lemma; since γcX (t) = γX (ct) 2 is defined whenever ct 2 [0; 1] and c 2 [0; 1], it is certainly defined whenever t 2 [0; 1]. (4) Taking a compact subset of the open neighbourhood U of p and then an open subset V of this compact set, then there exists δ > 0 such that f(q; X) 2 TV j X 2 TqM satisfies jXj < δg ⊂ U: Then for q 2 V , the geodesic γX (t) is defined whenever jtj < and jXj < δ. δ By the rescaling lemma then the geodesic γY is defined whenever jtj < 2 and jY j < 2 , where X Y = 2 . (5): The exponential map (p; X) 7! exp(p; X) is the evaluation of the smooth map γ(t; p; X) at t = 1, and hence is smooth. (6): The differential of expp at 0 2 TpM is just the identity map, since d d(expp)0(X) = (expp(tX)) dt t=0 d = (γ(1; p; tX)) dt t=0 d = (γ(t; p; X)) dt t=0 = X: Then by the inverse function theorem, the exponential map is locally a diffeomorphism. Definition 22.3. If V is a neighbourhood of 0 in TpM such that expp restricted to V is a diffeo- morphism onto its image, then we call expp(V ) =: U a normal neighbourhood of p. A choice of orthonormal basis E1;:::;En for TpM is equivalent to an isomorphism E : T M ! n p R ; i 1 n X Ei 7! (X ;:::;X ) and so on a normal neighbourhood V of p 2 M we have local coordinates −1 n E ◦ expp : V ! R : Such coordinates are called normal coordinates centered at p. Normal coordinates are extremely useful, in particular they yield a very simple representation for geodesics. 3 Lemma 22.4. Let (M; g) be a Riemannian manifold and (V; fφig) be a normal coordinate chart centred at p 2 M. i For any X = X @i 2 TpM, the geodesic γX is given in normal coordinates by −1 1 n γX (t) = φ (tX ; : : : ; tX ): Question 22.5. Prove this. 4.