Nearly para-Kähler geometry on Lie groups

Abstract

In this paper, we introduce left-invariant (strict) nearly para-Kaähler structures on Lie groups (nearly para-Kähler Lie algebras) and present some properties of them. We prove the existence of a unique connection in terms of the characteristic connection on these Lie groups with totally skew-symmetric torsion tensor and we show that the Nijenhuis tensor is parallel with respect to the characteristic connection. We determine conditions that allow the curvature tensor of the characteristic connection satisfies the first Bianchi identity. Finally, we focus on anti-abelian almost para-complex structures on Lie groups and we study nearly para-Kähler conditions for these structures. In this study we encounter bi-invariant metrics as pseudo-Riemannian metrics that accept nearly para-Kähler structures. Moreover, we present some examples of nearly para-Kähler Lie algebras.