The geometry of the Bismut Connection

This thesis concerns the study of special metrics in Hermitian and almost-Hermitian geometry, characterized by classical constraints on the curvature of their Bismut, Chern, or Gauduchon connection. More precisely, we intend to study the analogs in Hermitian and almost-Hermitian geometry of constant scalar curvature metrics, Einstein metrics, and metrics whose curvature tensor satisfies some positivity notion. We study the existence of metrics with constant scalar curvature with respect to the Gauduchon connection, which can be interpreted as a Yamabe-type problem. We then analyze the geometry of 4-dimensional compact almost-complex manifolds that carry a second-Chern–Einstein metric and we produce new examples of such spaces. With the aim of investigating the geometry of the Bismut connection, we describe the Calabi–Yau with torsion metrics of submersion type on toric bundles over Hermitian manifolds. Moreover, we analyze the cohomological properties of compact complex manifolds equipped with a Bismut flat metric. This leads to a better understanding of the evolution of the pluriclosed flow on Bismut flat manifolds. Finally, we consider a new notion of positivity for Hermitian manifolds which involves the Bismut curvature tensor, and we investigate its behavior under the action of the Hermitian curvature flows.