sperber
Sep8-09, 02:23 PM
Hi!
I have the following problem: Let P be a principal bundle over a manifold M, p: P -> M. Let G be the lie group acting on P from the right. Now let U be an open set on M and
s : U -> P
a smooth section. Now it has been said that we can define a local trivialization of P by
t : p^{-1}(U) -> U x G, t( s(x)*g ) = (x,g)
It is clear to me that t is well-defined, bijective and that the inverse t^{-1} given by
t^{-1}(x,g) = s(x)*g
is smooth. What I do not understand is why t itself is smooth. Can somebody help?
Thanks, Sperber
I have the following problem: Let P be a principal bundle over a manifold M, p: P -> M. Let G be the lie group acting on P from the right. Now let U be an open set on M and
s : U -> P
a smooth section. Now it has been said that we can define a local trivialization of P by
t : p^{-1}(U) -> U x G, t( s(x)*g ) = (x,g)
It is clear to me that t is well-defined, bijective and that the inverse t^{-1} given by
t^{-1}(x,g) = s(x)*g
is smooth. What I do not understand is why t itself is smooth. Can somebody help?
Thanks, Sperber