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. First, we generalize the Hopf theorem in a quantitative sense. Then we investigate the case of maps f : 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 : 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.