3

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.

Kelvin Lois
  • 7,430
user405156
  • 1,619

1 Answers1

3

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).

Kelvin Lois
  • 7,430
  • Surely if I could extend the vector field $Y$ as I defined it on $i(N)$ to the whole of $M$ it would work no? – user405156 Nov 01 '17 at 15:32
  • @tomak : we can extend a vector field along asubmanifold to the whole $M$ if the submanifold is properly embedded (https://math.stackexchange.com/questions/247240/proving-the-extension-lemma-for-vector-fields-on-submanifolds?rq=1). For your case the image of an injective smooth immersion $i(N)$ is just a immersed submanifold. So generally we can't extend it to the whole $M$. If you wanna know just ask a new question about it. Surely many expert will help. – Kelvin Lois Nov 01 '17 at 15:41