Static SKT metrics 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.