Maximal first Betti number rigidity of noncompact $\texttt{RCD}(0,N)$ spaces

TL;DR

This paper characterizes the structure of noncompact $ exttt{RCD}(0,N)$ spaces with maximal first Betti number, showing they are either flat manifolds with a torus or a product with a ray, extending rigidity results in metric geometry.

Contribution

It establishes a rigidity theorem for noncompact $ exttt{RCD}(0,N)$ spaces with maximal first Betti number, identifying their precise geometric and measure-theoretic structure.

Findings

Spaces with maximal first Betti number are either flat manifolds or products with a ray.

The measure is a multiple of the Riemannian volume in these cases.

The result extends classical rigidity theorems to the non-smooth setting.

Abstract

Let $(M,d,\mathfrak{m})$ be a noncompact $\texttt{RCD}(0,N)$ space with $N\in\mathbb{N}_+$ and $\text{supp}\mathfrak{m}=M$. We prove that if the first Betti number of $M$ equals $N-1$, then $(M,d,\mathfrak{m})$ is either a flat Riemannian $N$-manifold with a soul $T^{N-1}$ or the metric product $[0,\infty)\times T^{N-1}$, both with the measure a multiple of the Riemannian volume, where $T^{N-1}$ is a flat torus.