Embedding of manifolds

**Sogan** · December 13th 2010, 11:22 AM

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!

**Rebesques** · December 22nd 2010, 10:57 PM

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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