Quasi-Kähler Chern-flat manifolds and complex 2-step nilpotent Lie algebras (Q2883319)

The authors study quasi-Kähler Chern-flat almost Hermitian manifolds. An almost complex manifold is called Chern flat if its Chern connection has trivial holonomy. The authors show that quasi-Kähler Chern-flat almost Hermitian manifolds are strictly related with anti-bi-invariant almost complex Lie algebras.NEWLINENEWLINEThe main result of the paper is the following theorem.NEWLINENEWLINELet \(G\) be a complex Lie group with complex structure \(J\) and a left invariant Hermitian metric \(g\). Let \(\Gamma\subset G\) be a lattice and \(M=G\slash \Gamma\). The following facts are equivalent.NEWLINENEWLINE1. The curvature of the metric \(g\) satisfies the second Gray identity.NEWLINENEWLINE2. The curvature of any left-invariant Hermitian metric on \(M\) satisfies the second Gray identity.NEWLINENEWLINE3. \(g\) is quasi Kähler with respect to \(J^{-}\).NEWLINENEWLINE4. The Lie group \(G\) is a 2-step nilpotent.NEWLINENEWLINEMoreover, if one of the above conditions is satisfied, then \(g\) is Chern flat with respect to \(J\) and \(J^-\), where \(J^-(X) = J(X)\) if \(X\in\mathfrak z\) and \(J^-(X)=-J(X)\) if \(X\in \mathfrak z ^{\perp}\) and \(\mathfrak z\) is the center of the Lie algebra \(\mathfrak g\) of \(G\).NEWLINENEWLINE The authors also prove that on compact manifolds there is a natural one-to-one correspondence between quasi-Kähler Chern-flat almost Hermitian structures and Chern-flat Hermitian structures whose curvature tensor satisfies the second Gray condition. Compact quasi-Kähler Chern-flat manifolds are all 2-step nilmanifolds whose Lie algebras are endowed with an anti-bi-invariant structures. The authors also study some algebraic problems concerning the existence of such Lie algebras and give many examples of them.