Maximal first Betti number rigidity of noncompact RCD(0,𝑁) spaces

Abstract

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