2022/08/29 by A. V. Malyutin, Malyutin, A. V., I. M. Shirokov +1
Mathematics · #47H10 #54H25 #55M20 #55Mxx #Advanced Topology and Set Theory #Algebraic Topology (math.AT) #FOS: Mathematics #General Topology (math.GN) #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.2208.13554
openalex publication_date 2022/08/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We work within the framework of a program aimed at exploring various extended versions for theorems from a class containing Borsuk-Ulam type theorems, some fixed point theorems, the KKM lemma, Radon, Tverberg, and Helly theorems. In this paper we study variations of the Hopf theorem concerning continuous maps of a compact Riemannian manifold M of dimension n to ℝn. We investigate the case of maps f\colon M → ℝm with n < m and introduce several notions of varied types of f-neighbors, which is a pair of distinct points in M such that f takes it to a 'small' set of some type. Next for each type, we ask what distances on M are realized as distances between f-neighbors of this type and study various characteristics of this set of distances. One of our main results is as follows. Let f\colon M → ℝm be a continuous map. We say that two distinct points a and b in M are visual f-neighbors if the segment in ℝm with endpoints f(a) and f(b) intersects f(M) only at f(a) and f(b). Then the set of distances that are realized as distances between visual f-neighbors is infinite. Besides we generalize the Hopf theorem in a quantitative sense.