ESCHENBURG

Then IIJ^(t)ll S IIJ2(t)H for t ^ О up to the first conjugate

BfiLiBJt___Lfiââê___ { ^ ) ) РГ tPC^X PQJ-nt (ç^gg (»;)) Qf îîj^ . If equality

hûiUS___£fi£___SÛJUê . tQ > 0 , it holds on [O^t^] àGU Jj/HJ^« is.

p^r^UQl ^iQnq Çjico.tQ] for: j = 1,2 .

Proof . Case (a): Let t. be the first conjugate point of ç. and put p. = y.(0) , V. =5.40) . Then in T M. there is an open neighborhood U. of 0 containing t«v. for 0 S t < t. where

exp has a smooth inverse. Let f. be as in 1.4(b). We put E.

^ 1 = (v.) and B.(t) € S(E.) as in 1.3. Now J. (considered as a

curve in E. ) solves (2). since it satisfies (4). with J.(0) = 0

and t.Bj(t) -> Id as t -> 0 (cf. 2.1). By de I'Hopital's rule,

IIJ^ ( t ) ll / IIJ2 ( t ) ll -> 1 as t -> 0 . Now the result follows from 2.4

and 2.5.

Case (b): This is immediate from the following general

comparison theorem for Jacobi fields:

3 . 4 Theorem. Let iy. be a unit speed geodesic in M. and J. i Ç.' a Jacobi field along y. with

IIJ - ( 0 ) ll = IIJ^(0)ll 9t 0 , J.40) = A..J.(0) for some A^ € З(Е^) , E^ = (с^ЧО))"*" with \^{h^) < ^J^2^ ' Then IIJ^(t)ll S llJgCt)« for. 0 S t < t^ where t^ is the

i ? ^al ] Leg1 ; __positive zero of all Jacobi fields J аЛода JSi with

J» ( 0 ) = A.«J(0) Ф 0 . Equalitv at t^ € (0,t.) implies equality along CO^tp] and J./IIJ.II is parallel along ?f.lC0,tQ] .

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