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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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.
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