Hopf-type Theorems For $f$-neighbors

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 $\mathbb{R}^n$. We investigate the case of maps $f\colon M \to \mathbb{R}^m$ 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 $f\colon M \to \mathbb{R}^{m}$ be a continuous map. We say that two distinct points $a$ and $b$ in $M$ are visual $f$-neighbors if the segment in $\mathbb{R}^{m}$ 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.