Exist vector field having only finitely many zeros, all lying in open set of compact connected manifold?

Question

Let $U$ be any open set on the compact connected manifold $X$. Does there exist a vector field having only finitely many zeros, all of which lie in $U$?

Answer 1

This depends of the topology of $U$, if a vector field have finitely many zeros, their number are related to the Euler characteristic see

https://en.wikipedia.org/wiki/Poincar%C3%A9%E2%80%93Hopf_theorem

So you have to check that the Euler characteristic of the complement of $U$ (which is a manifold with boundary) is zero.

Answer 2

This is always possible. Start with a vector field on $X$ that has only finitely many zeros, and then apply a diffeomorphism of $X$ that takes all of the zeros into $U$. (For any one of the zeros, you can find a diffeomorphism of $X$ that takes it to a point in $U$; and if you've already put some of the zeros in $U$, you can arrange that the diffeomorphism doesn't move those points.)