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

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.