Hopf-type theorems for \(f\)-neighbors (Q6587374)

The authors 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.\N\NIn this paper, the authors operate in the Borsuk-Ulam subclass of theorems and concentrate on extensions related to the Hopf theorem:\N\N\textbf{Theorem 1.} [\textit{H. Hopf}, Port. Math. 4, 129--139 (1944; Zbl 0060.40612)]: Let \(n\) be a positive integer, let \(M\) be a compact Riemannian manifold of dimension \(n\), and let \(f:M\rightarrow \mathbb{R}^n\) 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 length \(\delta\).\N\NOne of the main points of the present paper is the further study of properties of sets \(\Omega_f^{sph},\Omega_f^{vis}, \Omega_f^{top}\) for the case of continuous maps \(f:M\rightarrow \mathbb{R}^m\), where \(M\) is a compact Riemannian manifold of dimension \(\dim M<m\). The main results in this direction are as follows.\N\N\textbf{Theorem 3.} Let \(M\) be a compact Riemannian manifold, let \(d:M\times M\rightarrow [0,\infty)\) be any intrinsic metric on \(M\) compatible with its topology, and let \(f:M\rightarrow \mathbb{R}^m\) be a continuous map. Then \(\Omega_f^{vis}\) for \((M,d)\) is infinite.\N\N\textbf{Theorem 4.} Let \(S^1\) be a Euclidean circle in \(\mathbb{R}^2\), let \(d\) be the angular distance on \(S^1\), and let \(f:S^1\rightarrow \mathbb{R}^2\) be a continuous map. Then \(\Omega_f^{sph}\) for \((S^1,d)\) is infinite.\N\NAnother driving idea of the program is to `count' `solutions' of the Hopf theorem, that is to investigate the cardinality of the set of pairs of points satisfying the Hopf theorem when \(\delta\) is fixed. The main result in the quantitative direction is as follows:\N\N\textbf{Theorem 5.} Let \(n\) be a positive integer such that \(n>1\), and let \(\delta\) be a positive real number. Let \(M\) be a compact Riemannian manifold of dimension \(n\), and let \(f:M\rightarrow \mathbb{R}^n\) be a continuous map. If \(\mathcal{C}(\delta)\) is empty, then \(\mathcal{F}(\delta)\) is uncountable.