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Thread: Tangent bundle of submanifold

  1. #1
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    Tangent bundle of submanifold

    Hello,

    I'm trying to do this problem

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

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

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