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,Æ/'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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