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) = T n(l) , 2 f.(vv(t) - vl(0) + ( W / - W/).vi(/). Hence, with I ß - ij II, - 0, 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.æ\\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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