By Cho M. H., Cai W.

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Example text

_ X~dT and 0 t exp~ XTdT the right and left chronological exponentials, 0 respectively, generated by the (nonstationary) vector field Xt, t 9 R, on M. The arrow in the notation for exp indicates the direction in which the field X t is "taken out" when the exponential is differentiated with respect to t: f t t d -4 -4 g-F exp S X , d , = exp j' X , d z o X t , 0 0 d . . ~ exp ; X , d * = 0 t Xtoexp S X~d~'. 0 We now consider the analogous notions for operator equations on a finitely generated projective C~(M)-module {L A flow on ~ is an arbitrary absolutely continuous family of semi-isomorphisms Pt, t 9 R, of the module such that P0 = id.

The left translations described above yield a canonical representation of an arbitrary Lie group as a Lie subgroup of a suitable automorphism group. Let D e r M c ~ be an arbitrary set of vector fields on M. X~d~ I X~E~, Exp ~ = exp Define t, ~ 1 t c I s o M. I t i s e a s y t o s e e t h t Exp ~ i s a l w a y s an a r c w i s e c o n n e c t e d s u b g r o u p o f I s o M ( b u t n o t n e c e s sarily a Lie subgroup). On t h e o t h e r h a n d , l e t I s o M c ~ be an a r b i t r a r y subset of lsoM.

Existence Theorem for a Finitely Generated Projective Module. Let ~ be a finitely generated projective C~(M)-module, embedded as a direct sun,hand in the free module C~(M; Rm): there exists a submodule @ c C~(M; R m) such that C~(M; R m) = ~ @ @ Let Dr, t 9 R, be a locally integrable family of derivations of ~, 8(D t) = X t. This family is extended to a family Dt, t 9 ~, of derivations of C~(M; R m) by the formula D , = D t e p r ~ V x t, w h e r e p r @: nection C~(M; on C~(M; R m) --+@ i s t h e projection o f C~(M; k s) o n t o O, V i s t h e c a n o n i c a l con- ~m).

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