It is an easy problem, but I am not sure that my argument is correct.

Suppose that f: X -> Y is a diffeomorphism, and prove that at each x

its derivative dfx is an isomorphism of tangent spaces.

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- Apr 21st 2011, 02:07 AMVevetangent spaces
It is an easy problem, but I am not sure that my argument is correct.

Suppose that f: X -> Y is a diffeomorphism, and prove that at each x

its derivative dfx is an isomorphism of tangent spaces. - Apr 21st 2011, 01:21 PMBruno J.
Let g:Y -> X be the inverse. Then d(f o g)(y) = d(1_Y)(y)=1_{Ty} and similarily d(g o f)(x) = d(1_X)(x)=1_{Tx}. Using the chain rule, you have your result!

Next time, you can post your solution and we can look at what you did.