Vector field on $S^2$ with exactly one zero

Question

My homework problem is to construct a vector field on sphere (and torus, but I guess I will be able to extend the idea if I figure out it for sphere) with exactly one zero. I don't know how to approach this so I'm asking for any explanation. (Btw, this problem is from problemsheet on index of a vector fields, but I doubt that index would help somehow)

Answer 1

Hint Puncturing the sphere $\Bbb S^2$ at a point, say, $N$, leaves the space $\Bbb S^2 - \{N\} \cong \Bbb R^2$; we can make this identification explicit, e.g., with stereographic projection from the point $N$. Now, $\Bbb R^2$ admits plenty of vector fields that vanish nowhere, and some of them can be used (via that identification) to construct a vector field on $\Bbb S^2$ that vanishes only at $N$.

Answer 2

I bet the reason this is included in a problem sheet on vector field index is that if you restrict to vector fields with nondegenerate zeros, by Poincare-Hopf your vector field had better have at least two zeros. To have a vector field with one zero on the sphere, it needs to have index equal to two.

Here's an example of a vector field with a zero of index two:

To formalize this, imagine you're looking down on the north pole and use stereographic projection.