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

1. Feb 26, 2013

BrainHurts

1. The problem statement, all variables and given/known data

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

2. Relevant equations

3. 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?

2. Feb 26, 2013

micromass

Staff Emeritus
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...

3. Feb 26, 2013

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.

4. Feb 26, 2013

micromass

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

5. Feb 26, 2013

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$\in$TpM and y$\in$TpN

and (w,y) $\in$ TpM x TqN

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.