# Smooth section in principal bundle

1. Sep 8, 2009

### sperber

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