Riemannian Geometry – Lecture 22 Exponential Map

Dr. Emma Carberry

October 26, 2015

Lemma 22.1 (Rescaling lemma). For X ∈ TpM and c ∈ R \{0}, 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 ∇e γ˜˙ for covariant differentiation along γ˜, since γ˜(t) = cγ˙ (ct) and γ˜(0) = cX

˙ ¨k k ˙ i ˙ j ∇e γ˜˙ γ˜(t) = γ˜ (t) + Γij(˜γ(t))γ˜ (t)γ˜ (t) ∂k 2 = c ∇γ˙ γ˙ X (ct) = 0.

Deﬁne E ⊂ TM by

E = {(p, X) ∈ TM | γX is deﬁned on an interval containing [0, 1]}, 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 ∈ M and X ∈ TpM, γX (t) = exp(tX) whenever these are deﬁned

2. E is an open subset of TM and for each p ∈ M, we have 0 ∈ Ep.

3. For each p ∈ M, Ep is a star-shaped region about 0. That is, whenever X ∈ Ep then also

cX ∈ Ep for all c ∈ [0, 1].

4. In fact for each p ∈ M there exists a neighbourhood V of p in M and ˜ > 0 such that for every (q, X) ∈ TV with |X| < ˜, we have (q, X) ∈ E.

5. exp is a smooth map.

6. For each p ∈ M there exists > 0 such that

expp : {X ∈ TpM | |X| < } → 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 theorem. From Theorem 21.10, we have:

For each p ∈ 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 ∈ TqM.

(3) This follows immediately from the rescaling lemma; since

γcX (t) = γX (ct)

2 is deﬁned whenever ct ∈ [0, 1] and c ∈ [0, 1], it is certainly deﬁned whenever t ∈ [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

{(q, X) ∈ TV | X ∈ TqM satisﬁes |X| < δ} ⊂ U.

Then for q ∈ V , the geodesic γX (t) is deﬁned whenever |t| < and |X| < δ.

δ By the rescaling lemma then the geodesic γY is deﬁned whenever |t| < 2 and |Y | < 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 ∈ 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 diffeomorphism 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 ∈ 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, {φi}) be a normal coordinate chart centred at p ∈ M.

i For any X = X ∂i ∈ TpM, the geodesic γX is given in normal coordinates by

−1 1 n γX (t) = φ (tX , . . . , tX ).

Question 22.5. Prove this.