Commutators of flows and fields

The well known formula [X,Y]=\tfrac12\tfrac{\partial^2}{\partial t^2}|_0 (\Fl^Y_{-t}\o\Fl^X_{-t}\o\Fl^Y_t\o\Fl^X_t) for vector fields $X$, $Y$ is generalized to arbitrary bracket expressions and arbitrary curves of local diffeomorphisms

Commutators Of Flows And Fields

. The well known formula [X; Y ] = 1 2 @ 2 @t 2 j 0 (Fl Y \Gammat ffi Fl X \Gammat ffi Fl Y t ffi Fl X t ) for vector fields X , Y is generalized to arbitrary bracket expressions and arbitrary curves of local diffeomorphisms. Let M be a smooth manifold. It is well known that for vector fields X;Y 2 X(M ) we have 0 = @ @t fi fi 0 (Fl Y \Gammat ffi Fl X \Gammat ffi Fl Y t ffi Fl X t ); [X; Y ] = 1 2 @ 2 @t 2 j 0 (Fl Y \Gammat ffi Fl X \Gammat ffi Fl Y t ffi Fl X t ): We give the following generalization: 1. Theorem. Let M be a manifold, let ' i : R \Theta M oe U ' i ! M be smooth mappings for i = 1; : : : ; k where each U ' i is an open neighborhood of f0g \Theta M in R \Theta M , such that each ' i t is a diffeomorphism on its domain, ' i 0 = IdM , and @ @t fi fi 0 ' i t = X i 2 X(M ). We put [' i t ; ' j t ] := (' j t ) \Gamma1 ffi (' i t ) \Gamma1 ffi ' j t ffi ' i t : Then for each formal bracket expression B of length k we hav..