Locally $G$-homogeneous Busemann $G$-spaces

TL;DR

This paper proves that small metric spheres in locally G-homogeneous Busemann G-spaces are topologically equivalent and homogeneous, advancing understanding of the Busemann conjecture on characterizing topological manifolds.

Contribution

It provides new proofs of known properties and establishes that locally G-homogeneous Busemann G-spaces have topologically homogeneous small spheres, supporting the Busemann conjecture.

Findings

Small metric spheres are homeomorphic and strongly topologically homogeneous.

Locally G-homogeneous Busemann G-spaces are topologically well-behaved.

Spaces that are uniformly locally G-homogeneous on an orbal subset are finite-dimensional.

Abstract

We present short proofs of all known topological properties of general Busemann $G$-spaces (at present no other property is known for dimensions more than four). We prove that all small metric spheres in locally $G$-homogeneous Busemann $G$-spaces are homeomorphic and strongly topologically homogeneous. This is a key result in the context of the classical Busemann conjecture concerning the characterization of topological manifolds, which asserts that every $n$-dimensional Busemann $G$-space is a topological $n$-manifold. We also prove that every Busemann $G$-space which is uniformly locally $G$-homogeneous on an orbal subset must be finite-dimensional.