Borsuk-Ulam Type Theorems and Mountain Climbing Problem

In this paper, we present a new qualitative extension of the Hopf theorem (and a generalization of Borsuk-Ulam theorem), concerning continuous maps $f$ from a compact Riemannian manifold $M$ of dimension $n$ to $\mathbb{R}^n$. We remove the assumption of a Riemannian structure and instead consider closed triangulable manifolds $M$ equipped with a topological notion of 'distant' points. We show that for any continuous map $f \colon M \to \mathbb{R}^n$, there exists a connected component in the space of $f$-neighbors (where a pair of points $a, b$ are $f$-neighbors if $f(a) = f(b)$) that contains both a pair of 'distant' points and a pair of identical points. This result yields further consequences for Lusternik-Schnirelmann and Tucker-type theorems, as well as a multidimensional extension of the mountain-climbing lemma, which in the special case of the standard Euclidean $2$-sphere, may be stated informally as follows. For any continuous distribution of temperature and pressure on Earth (assumed time-independent), there exists a pair of antipodal points with identical values such that travelers starting from these points can move and meet while, at each moment of their journey, experiencing matching 'climatic conditions' up to an arbitrarily small constant.