Hopf-type theorems for convex surfaces

In this paper we study variations of the Hopf theorem concerning continuous maps $f$ of a compact Riemannian manifold $M$ of dimension $n$ to $\mathbb{R}^n$. We investigate the case when $M$ is a closed convex $n$-dimensional surface and prove that the Hopf theorem (as well its quantitative generalization) is still valid but with the replacement of geodesic to quasigeodesic in the sense of Alexandrov (and Petrunin). Besides, we study a discrete version of the Hopf theorem. We say that a pair of points $a$ and $b$ are $f$-neighbors if $f(a) = f(b)$. We prove that if $(P,d)$ is a triangulation of a convex polyhedron in $\mathbb{R}^3$, with a metric $d$, compatable with topology of $P$, and $f \colon P \to \mathbb{R}^2$ is a simplicial map of general position, then there exists a polygonal path in the space of $f$-neighbors that connects a pair of `antipodal' points with a pair of identical points. Finally, we prove that the set of $f$-neighbors realizing a given distance $\delta > 0$ (in a specific interval), has non-trivial first Steenrod homology with coefficients in $\mathbb{Z}$.