Static SKT metrics on Lie groups

Abstract

An SKT metric is a Hermitian metric on a complex manifold whose fundamental 2-form ω satisfies $${\partial \overline{\partial} \omega=0}$$ . Streets and Tian introduced in Streets and Tian (Int Math Res Not IMRN 16:3101–3133, 2010) 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 called static if the (1, 1)-part of the Ricci form of the Bismut connection satisfies (ρ B )(1, 1) = λω for some real constant λ. 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.