I have this question: Prove that there exist a point $p\in S^2$ such that $f(p)=\pm p$ for $f:S^2\to S^2$ a smooth function.
$S^2$ is the unit sphere. The question can be found in the book Differential Geometry of Curves and Surfaces of Kristopher Tapp. It's in the section 6.3 called Compact Surfaces. I assumed that I should use the global Gauss-Bonnet theorem, but I don't if I'm on the right path or how to use it. Can someone help me?
Suppose there is a function such that $f(p)\ne\pm p$. Then there is always a shortest nontrivial geodesic from $p$ to $f(p)$. Let $g(p)$ be its unit tangent vector at the point $p$. Then $g$ violates the Hairy Ball Theorem.