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Math Help - Embedding of manifolds

  1. #1
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    Embedding of manifolds

    Hello,

    Let M be a manifold and j:M-> \mathbb{R}^n a smooth embedding.
    What is the local structure of dj?

    If we have such a embedding, then dj:TM-> T\mathbb{R}^n is a map between the Tangent bundles. But what about the local structure?

    I have found this equation, but i'm not sure, whether it is true in general or not:
    dj(p)[f]=\sum_{i=1}^n X^j \frac{d}{dx_i}_{|p}, with <x,X^j>=0

    Thanks in advance!
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  2. #2
    Super Member Rebesques's Avatar
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    You can just consider the example of a smooth coordinate chart (U,j) for M, and focus on U as the manifold in question.
    At p\in U, if j=(x^1,\ldots,x^n) and for (v^i)=V\in T_pM, we will have

    dj_p(V)=\langle \nabla j,V\rangle_p=v^i\partial_i|p.
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