Maximal first Betti number rigidity for open manifolds of nonnegative Ricci curvature

TL;DR

This paper proves that open Riemannian manifolds with nonnegative Ricci curvature and a maximal first Betti number are necessarily flat, establishing a rigidity result in geometric analysis.

Contribution

It establishes a new rigidity theorem linking maximal first Betti number to flatness in open manifolds with nonnegative Ricci curvature.

Findings

Manifolds with first Betti number n-1 are flat.

Rigidity result for open manifolds with maximal first Betti number.

Extension of classical results in Ricci curvature geometry.

Abstract

Let $M$ be an open Riemannian $n$-manifold with nonnegative Ricci curvature. We prove that if the first Betti number of $M$ equals $n-1$, then $M$ is flat.