Regarding diffeomorphism on manifolds

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Sarosh

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I am trying to show that:
If M, N are smooth manifolds without boundary.
Then T(M × N) is diffeomorphic to TM × TN.

an element of TM is of the form (x,v) where x is in M, and same for TN: (y,u) where y is in N.
An elemnt of TM × TN is: ((x,v),(y,u)).
And of T(M × N) is: ((x,y),(v,u)).
Now we define a map
f: TM × TN —>T(M × N) by
f((x,v),(y,u))=((x,y),(v,u)).

We want to show that f is a diffeomorphism, i.e. it is smooth, a bijection and its inverse is smooth.
It is obvious that f is injective and surjective. I am stuck of how to show formally smoothness on a chart.
 
If you consider manifold charts containing [imath]x[/imath] and [imath]y[/imath] then [imath]f[/imath] can be represented by [imath]\tilde f : \mathbb R^{2n+2m} \rightarrow \mathbb R^{2n+2m}[/imath], where [imath]m[/imath] and [imath]n[/imath] are the dimensions of [imath]M[/imath] and [imath]N[/imath].
 
What is f^~ here?
Can you please provide additional information?
I also saw that the diffeomorphism
T_(p,q)(M×N) = T_p M×T_q N implies the diffeo I need to show, but I had no idea how it is related!
 
Sarosh said:
What is f^~ here?
Can you please provide additional information?
I also saw that the diffeomorphism
T_(p,q)(M×N) = T_p M×T_q N implies the diffeo I need to show, but I had no idea how it is related!
Click to expand...
[imath]\tilde f[/imath] is the "affine version", or induced map, of [imath]f[/imath]. A map from one [imath]k[/imath]-dimensional manifold to another is considered smooth iff the corresponding charts induce smooth maps in [imath]\mathbb R^k[/imath]. In your case one chart maps [imath]T_{p,q}(M\times N)[/imath] to [imath]\mathbb R^{2(m+n)}[/imath] and another maps [imath]T_p(M) \times T_q(N)[/imath] to [imath]\mathbb R^{2n} \times \mathbb R^{2m}[/imath], so you have to show that the induced map (called [imath]\tilde f[/imath] in my previous post) from (neighborhood of) [imath]\mathbb R^{2(m+n)}[/imath] to [imath]\mathbb R^{2n} \times \mathbb R^{2m}[/imath] is smooth.
 
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