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Hopf-type theorems for convex surfaces

I. M. Shirokov

TL;DR

This work extends Hopf-type theorems for maps $f: M \to \mathbb{R}^n$ to the setting where $M$ is a closed convex $n$-dimensional surface, replacing geodesics with quasigeodesics in the Alexandrov–Petrunin framework. It proves a continuous Hopf-type result and its quantitative version for convex surfaces, using Gromov–Hausdorff convergence, smooth approximants, radial projections, and stability of quasigeodesics under limits. A discrete analogue is developed for affine maps on the boundary triangulation of a convex polyhedron, introducing the complex of $f$-neighbors, a resolution to handle singularities, and a Z2-equivariant argument that yields a path from antipodal to identical neighbors; it also shows the distance-set $\mathfrak{D}(\delta)$ carries nontrivial first Steenrod homology with integer coefficients. Collectively, the results broaden Borsuk–Ulam–Hopf-type phenomena to convex geometric settings and reveal rich topological structure in both continuous and discrete Hopf solutions, with potential implications for Alexandrov spaces and combinatorial geometry.

Abstract

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 $δ> 0$ (in a specific interval), has non-trivial first Steenrod homology with coefficients in $\mathbb{Z}$.

Hopf-type theorems for convex surfaces

TL;DR

This work extends Hopf-type theorems for maps to the setting where is a closed convex -dimensional surface, replacing geodesics with quasigeodesics in the Alexandrov–Petrunin framework. It proves a continuous Hopf-type result and its quantitative version for convex surfaces, using Gromov–Hausdorff convergence, smooth approximants, radial projections, and stability of quasigeodesics under limits. A discrete analogue is developed for affine maps on the boundary triangulation of a convex polyhedron, introducing the complex of -neighbors, a resolution to handle singularities, and a Z2-equivariant argument that yields a path from antipodal to identical neighbors; it also shows the distance-set carries nontrivial first Steenrod homology with integer coefficients. Collectively, the results broaden Borsuk–Ulam–Hopf-type phenomena to convex geometric settings and reveal rich topological structure in both continuous and discrete Hopf solutions, with potential implications for Alexandrov spaces and combinatorial geometry.

Abstract

In this paper we study variations of the Hopf theorem concerning continuous maps of a compact Riemannian manifold of dimension to . We investigate the case when is a closed convex -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 and are -neighbors if . We prove that if is a triangulation of a convex polyhedron in , with a metric , compatable with topology of , and is a simplicial map of general position, then there exists a polygonal path in the space of -neighbors that connects a pair of `antipodal' points with a pair of identical points. Finally, we prove that the set of -neighbors realizing a given distance (in a specific interval), has non-trivial first Steenrod homology with coefficients in .

Paper Structure

This paper contains 5 sections, 10 theorems, 6 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Generalization of the Hopf theorem for convex surfaces
  3. Quantitative version of the Hopf theorem for convex surfaces
  4. Discrete version of the Hopf theorem
  5. Acknowledgments

Key Result

Theorem 1

Let $n$ be a positive integer, let $M$ be a compact Riemannian manifold of dimension $n$, and let $f\colon M \to \mathbb{R}^n$ be a continuous map. Then for any prescribed $\delta > 0$, there exists a pair $\{x,y\} \in M \times M$ such that $f(x) = f(y)$ and $x$ and $y$ are joined by a geodesic of l

Figures (2)

  • Figure 1: Initial and 'induced by $f$' triangulations of $P$. Triangles, whose images coincide, are colored identically.
  • Figure 2: Schematic picture of a map $f\colon S^2 \to \mathbb{R}^2$. Top view on the left, front view on the right (before projection on a plane).

Theorems & Definitions (25)

  • Theorem 1: H. Hopf Hopf
  • Definition 1
  • Definition 2: Gromov-Hausdorff convergence Journey
  • Definition 3
  • Lemma 1: Journey
  • Remark 1
  • Theorem 2: Petrunin, Petrunin1
  • Proposition 1
  • proof
  • Definition 4
  • ...and 15 more