CLOSED GEODESICS ON RIEMANNIAN MANIFOLDS

11

PROOF.

Using the local representation of 7%(c) (see 1.13), we obtain for exp

the representation

(£(0 . i?c(0) *-

((exP

°

T

*^r

1

°

e x

P ° Tr*cHt)2exp.r*cjic(t)).

This is a differentiable fibre map from 6C X c*TM into c*rM. Now apply 1.9.

F maps the zero section of rH\(i ^ bijectively onto the diagonal of Hl(I, Ì) ×

H\I, M). It only remains to show that TF at Oc Å TCH](I, M) is a bijection.

But this follows by looking at the local representation: It carries (£(/), 0) *

n t o

(€(/), | ( 0 ) and (0, ç(0 ) into (0, ç(/)) . D

1.15

COROLLARY.

For eüery e Å / / ' ( / , Ë/ ) i/i^re ernte a natural chart

(exp;1,

9t (e)) w/7A

exp, = exp | Ñ'Ð ^ / / ' ( / , Ë/) : 6' Ð Ã,AE/'ß Ë Ë/ ) - 9i,(e). D

1.16

PROPOSITION.

The mapping

3: //'(Ë Ë/ ) - //°(//•(/ , M)*TM)\ (e(t)) \-+(de(t) =e(t))

is á differentiable section in the bündle a°. For e Å 9l( c), £ = expc_)e, //*e represen-

tation of'de in the bündle chart over %(c), is given by

a,.f(/)= v€(/) + 0,.|(o

with

È,Ì) = T*c-l{Tr.vtnttxpyl ° (r

T M ( o J

exp ) ï

T*C9/.

PROOI

. We have

Mt) = r

T M ( / )

exp.ir*^(0/ , + T*c|i/)J

= (7;-,€(i,jexp) ï T*C3/ + (r

T V | i o 2

exp) ï

T

*c.v|f/).

This gives the expressions for dc£(t) and 0c£(t). In particular, #c: e, — c*7*A/ is a

fibre mapping. Hence, the mapping which associates to £ = exp~!e Å H](£c.) the

principal part, v£ + i

(

.{ 6 H()(c*TM), of the representation of 3e is differentia-

ble. Ð

1.17

THEOREM.

The bündle a} of H{)-vector fields along H]-curves on Ì has á

Riemannian metric which is characterized by the property that on (a°)"1(c) =

H()(c*TM), c Å C y ( / , Ì) , // is given by ( ,

0

. We therefore denote this metric

by ( , )

0

in general.

PROOF.

With £ as above define G: c C TM - L(TM. ÃÌ ) by

(GUI T U , = (F^

2

exp. r

c2

exp)

exp |

From the properties of c it follows that G(f) is a positive selfadjoint Operator of

dass C*\

Gc:

(T*C) _ 1 Ï G Ï (

T

*C )

: c\. - L(c*TM. c*TM)