Tangent bundle of submanifold

• Dec 4th 2010, 09:50 AM
void
Tangent bundle of submanifold
Hello,

I'm trying to do this problem

"Prove that if $M$ is a submanifold of a manifold $N$, then $TM$ is a submanifold of $TN$"

what I've thought so far is following:
Let $(U,k)$ be charts of $N$. Since $M$ is a submanifold of $N$, $M$ can be covered by a collection of $(U,k)$ with the property $M\cap U=k^{-1}(R^m\times {0})$ ( $m$ is the dimension of $M$). Let $p\in M$, I think I have to prove that $T_p M$ can be covered by the collection of $T_p U$ with the property $T_pM\cap T_pU=Dk^{-1}(R^m\times {0})$

Could anyone give me suggestions or comments?
• Dec 4th 2010, 06:44 PM
xxp9
since M is a submanifold of N, near any point p of M, there is a chart $(x^1, x^2, ..., x^n)$ of N so that M is defined by $x^{m+1}=...=x^n=0$. TN can be parametrized by $(x^1, ...,x^n, y^1, ..., y^n)$ where any tangent vector $v=\sum y_i \frac{\partial}{\partial{x_i}}$. And TM can be parametrized by $(x^1, ..., x^m, y^1, ..., y^m)$, that is, $x^{m+1}=...=x^n=y^{m+1}=...=y^n=0$