# Recognizing the flat torus among $RCD^*(0,N)$ spaces via the study of the first cohomology group

created by gigli on 12 May 2017

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Submitted Paper

Inserted: 12 may 2017
Last Updated: 12 may 2017

Year: 2017

Abstract:

We prove that if the dimension of the first cohomology group of a $RCD^*(0,N)$ space is $N$, then the space is a flat torus. This generalizes a classical result due to Bochner to the non-smooth setting and also provides a first example where the study of the cohomology groups in such synthetic framework leads to geometric consequences.