Almost Hypercomplex Manifolds with Hermitian-Norden Metrics and 4-dimensional Indecomposable Real Lie Algebras Depending on One Parameter

We study almost hypercomplex structure with Hermitian-Norden metrics on 4-dimensional Lie groups considered as smooth manifolds. All the basic classes of a classification of 4-dimensional indecomposable real Lie algebras depending on one parameter are investigated. There are studied some geometrical characteristics of the respective almost hypercomplex manifolds with Hermitian-Norden metrics.

A note on para-holomorphic Riemannian–Einstein manifolds

The aim of this note is the study of Einstein condition for para-holomorphic Riemannian metrics in the para-complex geometry framework. First, we make some general considerations about para-complex Riemannian manifolds (not necessarily para-holomorphic). Next, using a one-to-one correspondence between para-holomorphic Riemannian metrics and para-Kähler–Norden metrics, we study the Einstein condition for a para-holomorphic Riemannian metric and the associated real para-Kähler–Norden metric on a para-complex manifold. Finally, it is shown that every semi-simple para-complex Lie group inherits a natural para-Kählerian–Norden Einstein metric.

ON NORDEN STRUCTURES ON NEUTRAL 4-MANIFOLDS WITH ALMOST PARACOMPLEX STRUCTURES

We construct some Norden metrics on a pseudo-Riemannian 4-manifold of neutral signature (+ + - -) with respect to some compatible paracomplex structures. Also, we present some examples.