Left-invariant almost para-complex structures on six-dimensional nilpotent Lie groups

TL;DR

This paper explores new geometric structures on five specific six-dimensional nilpotent Lie groups that lack traditional symplectic or complex structures, introducing multiparametric metrics and para-complex structures to enrich their geometric understanding.

Contribution

The paper introduces novel left-invariant almost para-complex structures and multiparametric metrics on these Lie groups, filling a gap where classical structures are absent.

Findings

Existence of multiparametric families of (3,3) signature metrics.

Construction of almost para-complex pseudo-Riemannian half-flat structures.

Metrics with diagonal Ricci operator having two eigenvalues differing in sign.

Abstract

There are five six-dimensional nilpotent Lie groups G, which do not admit neither symplectic, nor complex structures and, therefore, can be neither almost pseudo-Kahler, nor almost Hermitian. In this work, these Lie groups are being studied. The aim of the paper is to define new left-invariant geometric structures on the Lie groups under consideration that compensate, in some sense, the absence of symplectic and complex structures. New examples of multiparametric families of metrics of signature (3,3) and almost para-complex pseudo-Riemannian half-flat structures on six-dimensional nilmanifolds are obtained. These metrics have a diagonal Ricci operator with two eigenvalues, which differ only in sign.