I was doing revision for my differential geometry class, and came upon the following questions:
If a vector field on a sphere only vanishes at points where the index is $1$, how many such points are there? For the standard torus $T^{2} \hookrightarrow \mathbb{R}^{3}$, is it possible to find a vector field which vanishes nowhere? Prove or disprove.
I was trying to attempt to solve these questions, but I'm not sure on how to begin. What would be a good way to get started? Any help would be greatly appreciated.
You have to use the Poincare theorem. Since the Euler characteristic of the 2-sphere is 2, you have two points.
For the torus, its Euler characteristic is zeroo, so ther exists a vector field which does not has zero on the torus.
https://en.wikipedia.org/wiki/Poincar%C3%A9%E2%80%93Hopf_theorem