I have an injective immersion $i:N\rightarrow M$ and a vector field $X$ on $N$. How to find a field $Y$ on $M$ which is $i$-linked to $X$?
Reminder: $X,Y$ vector fields on $N,M$ respectively and $\psi:N\rightarrow M$ a differential map. Then $X$ and $Y$ are $\psi$ linked if $\psi_{*_p}(X_p) = Y_{\psi(p)}$
To do this, I define on $i(N), Y_{i(p)} := i_{*_p}(X_p)$ and this works. But I have to extend it to the whole of $M$ and I don't see how I can do that.
I don't quite understand the argument extending a vector field defined on a closed submanifold, so if someone could help I would be gratefull.
Given a smooth map $i : N \rightarrow M$ and a vector field $X : N \rightarrow TN$, we can't always have a vector field $Y$ on $M$ which is related (or linked in your terms) to $X$. This is simply because there may be points in the codomain $M$ which is not in the image of $i$ or points of intersection because $i$ may be not injective. The existence of the vector field related to $X$ guaranteed if the map $i : N \rightarrow M$ is a diffeomorphism (look Lee's smooth manifold p.183).