Static SKT metrics on Lie groups

TL;DR

This paper studies static SKT metrics on Lie groups, classifies them in dimension 4, and constructs new examples in various dimensions, advancing understanding of Hermitian geometry and Ricci flows.

Contribution

It provides a classification of invariant static SKT metrics on simply connected Lie groups in dimension 4 and constructs new examples in higher dimensions.

Findings

Classified static SKT metrics in dimension 4.

Constructed new compact and non-compact static SKT examples.

Analyzed the Ricci form of the Bismut connection on Lie groups.

Abstract

An SKT metric is a Hermitian metric on a complex manifold whose fundamental 2-form $\omega$ satisfies $\de\debar\omega=0$. Streets and Tian introduced in \cite{sttiPlur} a Ricci-type flow that preserves the SKT condition. This flow uses the Ricci form associated to the Bismut connection, the unique Hermitian connection with totally skew-symmetric torsion, instead of the Levi-Civita connection. A SKT metric is static if the (1,1)-part of the Ricci form of the Bismut connection satisfies $\riccib=\lambda\omega$ for some real constant $\lambda$. We study invariant static metrics on simply connected Lie groups, providing in particular a classification in dimension 4 and constructing new examples, both compact and non-compact, of static metrics in any dimension.