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Math Help - 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 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?
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  2. #2
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    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
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