Smoothing 3-dimensional polyhedral spaces

TL;DR

This paper demonstrates that 3D polyhedral spaces with nonnegative curvature can be closely approximated by smooth Riemannian manifolds maintaining the same curvature condition.

Contribution

It introduces a method to approximate nonnegatively curved polyhedral manifolds with smooth Riemannian manifolds, bridging discrete and continuous geometric models.

Findings

Polyhedral manifolds can be approximated by smooth manifolds.

The approximation preserves nonnegative curvature.

This advances understanding of curvature in geometric topology.

Abstract

We show that 3-dimensional polyhedral manifolds with nonnegative curvature in the sense of Alexandrov can be approximated by nonnegatively curved 3-dimensional Riemannian manifolds.