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, + 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)
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