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\inT_pM and y\inT_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.