Tangent Bundles, T(MxN) is Diffeomorphic to TM x TN

[BrainHurts]

Homework Statement

If M and N are smooth manifolds, then T(MxN) is diffeomorphic to TM x TN

Homework Equations

The Attempt at a Solution

So I'm here

let ((p,q),v) $\in$ T(MxN)

then p $\in$ M and q $\in$ N and v $\in$ T_(p,q)(MxN).

so T_(p,q)(MxN) v = $\sum_{i=1}^{m+n}$ $v_{i}\frac{\partial}{∂x_{i}}|_{(p,q)}$

= $\sum_{i=1}^{m}$ $v_{i}\frac{\partial}{∂x_{i}}|_{p}$ + $\sum_{i=}^{m+1}$ $v_{i}\frac{\partial}{∂x_{i}}|_{q}$

not sure if this is it?

 

[micromass]

You kind of assume that $\frac{\partial}{\partial x^i}$ are vector fields on both $M\times N$ and $N$. I think you should be a little more careful than this...

 

[BrainHurts]

Hi Micro, I'm not sure I understand what you mean, am am assuming however that

$\sum_{i=1}^{m}$ $v_{i}\frac{\partial}{∂x_{i}}|_{p}$ and $\sum_{i=}^{m+1}$ $v_{i}\frac{\partial}{∂x_{i}}|_{q}$

are bases for TM and TN respectively and I'm not sure if that's good enough.

 

[micromass]

Well, you assume $\frac{\partial}{\partial x_i}\vert_p$ is both in $T_pM$ and $T_pN$. You can't do that.

 

[BrainHurts]

v = $\sum_{i=1}^{m}$ $v_{i}\frac{\partial}{∂x_{i}}|_{(p,q)}$ + $\sum_{i=}^{m+1}$ $v_{i}\frac{\partial}{∂x_{i}}|_{(p,q)}$

so i need to find a map that takes v $\rightarrow$ (w,y)

where w$\in$T_pM and y$\in$T_pN

and (w,y) $\in$ T_pM x T_qN

does this sound better? I was thinking about this problem a lot. I'm not sure if that's a diffeo right off the get.