On the curvature of the Bismut connection: Bismut Yamabe problem and Calabi-Yau with torsion metrics
Giuseppe Barbaro
备注：
The uniqueness of the CYT metrics among the invariant ones on class \mathcal{C} manifolds has been acquired. Moreover, the section devoted to the CYT structure has been largely restructured in order to clarify the exposition
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.
Rigidity results for non-Kähler Calabi-Yau geometries on threefolds
Département de Mathématiques UQAM and Institute of Mathematics and Informatics Bulgarian Academy of Sciences（魁北克大学蒙特利尔分校数学系和保加利亚科学院数学与信息学研究所）；Department of Mathematics, Aarhus University（奥胡斯大学数学系）；Rowland Hall University of California, Irvine（加州大学欧文分校罗兰大厅）
