A topological decomposition of Busemann space

TL;DR

This paper demonstrates that Busemann spaces covered by parallel bi-infinite geodesics can be decomposed into a product space, and explores how isometries preserving geodesic foliations induce corresponding isometries on the decomposed space.

Contribution

It establishes a topological decomposition for a class of Busemann spaces and analyzes the induced isometries on the decomposed factors.

Findings

Busemann spaces with parallel bi-infinite geodesics are homeomorphic to a product with the real line.

Semi-simple isometries preserving geodesic foliations induce semi-simple isometries on the decomposed space.

The decomposition provides insights into the structure and symmetries of Busemann spaces.

Abstract

We show that a Busemann space $X$ which is covered by parallel bi-infinite geodesics is homeomorphic to a product of another Busemann space $Y$ and the real line. We also show that a semi-simple isometry on $X$ preserving the foliation by parallel geodesics canonically induces a semi-simple isometry on $Y$.