Bismut connection on Vaisman manifolds

TL;DR

This paper investigates the holonomy of the Bismut connection on Vaisman manifolds, showing it is contained in U(n-1), and computes it explicitly for certain solvmanifolds and Hopf manifolds.

Contribution

It provides a new understanding of the Bismut connection's holonomy on Vaisman manifolds and explicit calculations for specific classes of manifolds.

Findings

Holonomy group of Bismut connection is contained in U(n-1) for Vaisman manifolds.

Explicit computation of the holonomy group for solvmanifolds.

Explicit computation of the holonomy group for some classical Hopf manifolds.

Abstract

The holonomy of the Bismut connection on Vaisman manifolds is studied. We prove that if $M^{2n}$ is endowed with a Vaisman structure, then the holonomy group of the Bismut connection is contained in U$(n-1)$. We compute explicitly this group for particular types of manifolds, namely, solvmanifolds and some classical Hopf manifolds.