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

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.