I would like to construct an explitic smooth vector field on the sphere such that the north and south pole are hyperbolic equilibrium points. Mi idea is this
To construct a vector field in $R^2$ with the origin as the hyperbolic point. Say $V: R^2 \to R^2$ is the vector field with the property that $\lim \|V(x)\| =0$ as $\|x\|$ goes to $\infty$. Then, consider the inverse function of the stereographic projection, say $f:R^2 \to S \subset R^3$, where $S$ is a sphere and $f(0)=$south pole and the north pole does not have a preimage (as you know, by considering the one point compactification of $R^2$, $\infty \to$ north pole). Now, define the vector field $W: S \to R^2$ by $W(p)=df(f^{-1}(p))(V(f^{-1}(p))$ if $p $ is not the north pole and set North pole $\to 0$.
Is this construction correct? Thank you