Recognizing the flat torus among $RC{D}^{\ast }(0,N)$ spaces via the study of the first cohomology group

Abstract

We prove that if the dimension of the first cohomology group of a $RC{D}^{\ast }(0,N)$ 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.

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Cited by in corpus (4)

• Maximal volume entropy rigidity for ${\mathsf{RCD}}^{\ast }(-(N-1),N)$ spaces
• An upper bound on the revised first Betti number and a torus stability result for RCD spaces
• Representation theorems for normed modules
• Convergence to the Product of the Standard Spheres and Eigenvalues of the Laplacian