A Textbook of Matrices by Hari Kishan

By Hari Kishan

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Lfapplication exp d@finie sur un voisinage de la section montre que l'applioation exp de la section nulle de M × M . (On v@rifie que ainsi d@finie TM nulle de est un diff@omor- sur un voisinage de la diago- D exp(~,0) = Identit@). D'autre part dt--~v(% , % ) = (~ , r ( % , %)) La deuxi~me composante du vecteur tangent ~ la courbe int@grale est situ@e dans 57 l'espace de dimension finie qui ne d@pend que de l'ouvert E U contenant ~ l n, l Nous en d4duisons que : ~(~c, (~)) = ~ + ~ + ~(~,9,t) d 0%1 yi(~,~,t ) C En, l est un diff@omorphisme L'application exp 4ta14 de ~ sur tun voisinage de la diagonale de N × N o Nous noterons de M x M III.

Soit E(z) la boule de centre morphisme de classe et tel que C°° , 0 et de rayon ~ : E - 101 ~ E r de E • I1 existe un diff@o- qui est l'identit@ en dehors de (~ - id) soit localement contenu dans un e s ~ c e de dimension finie. D@monstration. Un point de E(r) E, x ~est repr@sent@ par une suite de nombre r@els 42 x = {x 1 . . . xn .... Z ) ; n£1g Soit g tun n o m b r e > 0 x 2 <~}. n qui sera choisi ult6rieurement ~Xn 2 soit E = ix, ~=(x 1 ..... x n .... ) ; o EO est un espace de Hilbert pour la norme E n6~ n2 l lo <-} d6fini par : Z o canonlque (-7) ~x ( n)2 n£~ n2 n xl 2 = I1 existe un plongement ~ i de E E ° } i(E) dans n'est pas ferm6 dans o Soit an une application an(t) = 0 de classe C~ de [0,1] dans [0,1] monotone croissante.

Et K o i(E(r)) = i ( E ) ~ Posons E (r) o ~ = K -I o f o K On v4rifie que 9 : est l'application cherch4e. Addendum au th4or~me I. Les hypoth&ses 4rant les m@mes que celle du th4or&me I. Ii existe une is~topie @ de E de classe : E X [0,1] ~ E × [0,1] C~ 4tal4e. @(x,t) = (@t(x),t) telle que : (i) ® (ii) ~t o = id . 0 $ t < I l'identit4 en dehors de (iii) @I en dehors de est un diff6omorphisme 6tal@ de dans E E qui est E(r) . est un diff6omorDhisme @tal@ de E - {01 sur E qui est l'identit4 E(r) . D4monstration.

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