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Thread: Embedding of manifolds

  1. #1
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    Embedding of manifolds

    Hello,

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

    If we have such a embedding, then dj:TM->$\displaystyle T\mathbb{R}^n$ 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:
    $\displaystyle dj(p)[f]=\sum_{i=1}^n X^j \frac{d}{dx_i}_{|p}$, with $\displaystyle <x,X^j>=0$

    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 $\displaystyle (U,j)$ for $\displaystyle M$, and focus on $\displaystyle U$ as the manifold in question.
    At $\displaystyle p\in U$, if $\displaystyle j=(x^1,\ldots,x^n)$ and for $\displaystyle (v^i)=V\in T_pM$, we will have

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