
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!

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