CLOSED GEODESICS ON RIEMANNIAN MANIFOLDS 7
denotes the decomposition into the restrictions upon the horizontal and vertical
subspaces, respectively, we have
v ( / ° V)(t) - V ( / ° £)(') = W / . V U W - Tn(l)2fM(t)
= Tn(l),2f.(vv(t) - vl(0) + ( W / - W/).vi(/).
Hence, with I I ß - ij II, - 0, I I v/(ij) - V/(£)ll
0
- 0 also. D
1.8
PROPOSITION.
Lei ç: Å -* É; ö : F- I be vector bundles as in 1.7. Lei
L(rr\ ö) : L(E; F) -* I be the associated bündle of linear mappings with the induced
Riemannian metric and Riemannian connection. Then the canonical inclusions
H\L(E; F)) «* L(H°(E); H°(F)) and
Hl(L(E;
F)) =* L(H\E); H\F)),
given by
A = (A(t))»{Ä:=(t(t))»(A(t).ttt))},
are continuous. More precisely,
(*) \\Á(î)\\ 2 0^\\Á\\2÷Ç\\1^2\\Á\\ú\\î\\1
and
(**) M"(i)llf8IMIIf||{||f.
PROOF.
We use 1.5. Then (*) is obvious. To prove (**) we note that
VA(i)(t)=VA(tU(t)+A(t).vt(t).
Hence,
IIV/i(€)Ho = HíË. î + A.vt\\l 2\\íÁ.ae\\1 + 2IU.V|II^
4|íËâÉÉßÉÉ+ ? 4ÌÉÉ?|ÉßÉÉ?8ÌÉÉ?ÉÉ*||?D.
REMARK.
Á similar result holds for the canonical inclusions
Hl(L(E],E2,...,Ek;F))^L(H((E]),H(E2),...,H (Ek)-!H0(F))
and
/ /
,
( L ( £
l
, £
2
, . . . , £ , ; F ) ) - L ( / /
!
( £
1
) , / /
,
( £
2
) , . . . , / /
,
( £ , ) ; / /
l
( F ) ) .
1.9
LEMMA.
Let /:â - F be á fibre map\ cf. 1.6. Then f: Ç
](â)
-
HX(F)
is
once continuously differentiable with the tangential given by T2 /"" .
Note. The differentiability of arbitrarily high order is proved along the same
lines starting with the differentiable fibre map T2f: £ -* L(E\ F). See Klingen-
berg [3] and Flaschel und Klingenberg [1] for details. 1.9 is a slight generalization
of a result contained in Palais [1] and Eliasson [1].
PROOF.
From 1.7 we know that/is continuous. The Taylor formula gives
/(u(0) -/(€(')) - W / · MO - €(0) = Mt), ç(0). MO - «(0).
Here,
Ã(Ý(/),ç(/) ) = [Ti(t)+5(1lin-t0))2fds- Ti(t)2f
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