308 7. THE REDUCED DISTANCE
where
0 = 2 e4Coc:+6C 0 T (e2C 0 T 2 d~(T2) (p, q) + 2n0o T + c'#,T)
(^3). 4T2 3 2 1205
(using la - a*I :S 2VT). Combining this with (7.50), we obtain^7
where
L (q, Ti) - L (q, T2)
< - --2n0o 3 ( T 2 3/2 - (^2 Tl-1'2 )3/2)
- --2n0o 3 ( T 1 3/2 - (^2 Tl -T2 )3/2)
- 203 ( Ti/^2 - (2T1 - T2)l/^2 )
:S 01 ( T2 - Tl) ,
D
When combined with the uniqueness of the boundary-value problem for
the £-geodesic equation on small time intervals, Lemma 7.13 can also be
used to prove the following; we shall give the proof elsewhere.
LEMMA '7.29 (Short £-geodesics are minimizing). Given VE TpM, there
exists T > 0 such that 'YVl[o,T] is a minimal £-geodesic.
The next lemma and Rademacher's Theorem (Lemma 7.110) imply thatLis differentiable almost everywhere on M x (0, T).
LEMMA '7.30 (Lis locally Lipschitz). The function L: M x (0, T) ~IR
is Lipschitz with respect to the metric g (T) + dT^2 defined on space-time.
PROOF. For any 0 < To < T and qo E M' let E ~ min { rg' TlOTQ ' 1~} > 0.
Then for any Tl < T2 in (To - E, To+ c) and qi, q2 E B 9 (o) (qo, c),
IL (q1, T1) - L (q2, T2)1 :SIL (qi, Ti) - L (q2, Ti)I +IL (q2, Ti) - L (q2, T2)l -
To prove that L is Lipschitz near (qo, To), it suffices to prove (1) and (2)
below.
(1) L (·,Ti) is locally Lipschitz in the space variables uniformly in Tl E
(To - E, To+ c). Let dT denote the distance function with respect to the
metric g ( T) and let ')' : [O, Ti] ~ M be a minimal £-geodesic from p to q 1.