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

Abstract

Let $(M,d,\mathfrak{m})$ be a noncompact $\text{RCD}(0,N)$ 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.