1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

8. .C-JACOBI FIELDS AND THE £-EXPONENTIAL MAP 355

COROLLARY 7.93. Fix f E (0, T) and consider the £-exponential map

Dr exp : TpM ---+ M. Then V is a critical point of the map Dr exp if and

only if there is a nontrivial £-Jacobi field J(T) along £exp(V,T), TE [O,f],

such that J(O) = 0 and J(f) = 0.

PROOF. If V is a critical point of £ 7 exp, then there exists W such that

D [£exp(V, f)] (W) = 0. By the lemma, ls \s=O £exp(V + sW, T) is the

required £-Jacobi field.

On the other hand, if we have a nontrivial £-Jacobi field J 1 ( T) with

Ji(O) = 0 and J1(f) = 0, then we define W ~ d~\ 7 = 0 Ji(T). Since J1(T) is

nontrivial, we have W # 0. By the uniqueness of solutions of the initial-value

problem for £-Jacobi fields, we know that

D [£ 7 exp(V)] (W) = J(f) = J1(f) = 0.

We see that V is a critical point of £ 7 exp. D

The Hopf-Rinow theorem in Riemannian geometry can be generalized

for £-geodesics and the £-exponential map. The proof of this result will

appear elsewhere.

LEMMA 7.94 (£-Hopf-Rinow). Suppose (Nn, h (T)), TE [O, T], is a solu-

tion to the backward Ricci flow satisfying the curvature bound !Rm (x, T)I .:::;

Co < oo for ( x, T) E N x [O, T]. The following are equivalent:

(1) for every T E [O, T), the metric h ( T) is complete;

(2) £,exp is defined on all of TpN x [O, T) for some p E N;

(3) £exp is defined on all of TpN x [O, T) for all p EN.

Moreover,

(4) any of the above statements implies that given any two points p, q

and 0 ~ T 1 < T2 < T, there is a minimal £-geodesic / : [T1, T2] ---+ N joining

p and q.

8.4. £-cut locus. We have the following simple lemma which is anal-
ogous to the corresponding theorem in Riemannian geometry.

LEMMA 7.95 (When £-geodesics stop minimizing). Given V E TpM,

there exists TV E (0, T] such that

£ ( /Vl[o,r]) = L(tv (T) ,T) for all TE [0,Tv)

and

£ ( /Vl[o,r]) > L (tv (T), T) for any TE (Tv, T),

where /V : [O, T) ---+ M is the £-geodesic with limr->O .JT!fi!:. ( T) = V.

PROOF. The existence of TV follows from the additivity property for
concatenated paths (7.20). The fact that TV> 0 follows from Lemma 7.29.
D