On the curvature of the Bismut connection: Bismut Yamabe problem and Calabi-Yau with torsion metrics

DOI10.1007/S12220-023-01203-2arXiv2109.06159MaRDI QIDQ6377561

Publication date: 13 September 2021

Abstract: We study two natural problems concerning the scalar and the Ricci curvatures of the Bismut connection. Firstly, we study an analog of the Yamabe problem for Hermitian manifolds related to the Bismut scalar curvature, proving that, fixed a conformal Hermitian structure on a compact complex manifold, there exists a metric with constant Bismut scalar curvature in that class when the expected constant scalar curvature is non-negative. A similar result is given in the general case of Gauduchon connections. We then study an Einstein-type condition for the Bismut Ricci curvature tensor on principal bundles over Hermitian manifolds with complex tori as fibers. Thanks to this analysis we construct explicit examples of Calabi--Yau with torsion Hermitian structures and prove a uniqueness result for them.

Mathematics Subject Classification ID

Differential geometry of homogeneous manifolds (53C30) Global differential geometry of Hermitian and Kählerian manifolds (53C55) Calabi-Yau theory (complex-analytic aspects) (32Q25) Local differential geometry of Hermitian and Kählerian structures (53B35) General geometric structures on manifolds (almost complex, almost product structures, etc.) (53C15) Homogeneous complex manifolds (32M10) Other connections (53B15)

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