Abstract
We prove that if the dimension of the first cohomology group of a \(\mathsf{RCD}^*(0,N)\) 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.
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Acknowledgements
This work has been supported by the MIUR SIR-Grant ‘Nonsmooth Differential Geometry’ (RBSI147UG4). We wish to thank the referee for the very careful reading and the detailed report.
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Communicated by L. Ambrosio.
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Gigli, N., Rigoni, C. Recognizing the flat torus among \(\mathsf{RCD}^*(0,N)\) spaces via the study of the first cohomology group. Calc. Var. 57, 104 (2018). https://doi.org/10.1007/s00526-018-1377-z
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DOI: https://doi.org/10.1007/s00526-018-1377-z