A topological splitting theorem for weighted Alexandrov spaces

TL;DR

This paper establishes a topological splitting theorem for weighted Alexandrov spaces under volume comparison conditions, leading to an isometric splitting result for certain singular Riemannian manifolds with nonnegative Ricci curvature.

Contribution

It introduces a new topological splitting theorem for weighted Alexandrov spaces and extends it to singular Riemannian manifolds with nonnegative Bakry-Emery Ricci curvature.

Findings

Proves a topological splitting theorem under volume comparison conditions.

Derives an isometric splitting theorem for singular Riemannian manifolds.

Extends classical splitting results to weighted and singular settings.

Abstract

Under an infinitesimal version of the Bishop-Gromov relative volume comparison condition for a measure on an Alexandrov space, we prove a topological splitting theorem of Cheeger-Gromoll type. As a corollary, we prove an isometric splitting theorem for Riemannian manifolds with singularities of nonnegative (Bakry-Emery) Ricci curvature.