Anti-Kaehlerian Manifolds

TL;DR

This paper explores the properties of anti-Kaehlerian manifolds, showing their metric structure, topological invariants, and their relation to complex Lie groups, along with a method to generate Einstein solutions.

Contribution

It establishes that metrics on anti-Kaehlerian manifolds are real parts of holomorphic metrics, proves the vanishing of odd Chern numbers, and links complex parallelisable manifolds to anti-Kaehlerian structures, introducing a new solution generation method for Einstein equations.

Findings

Metrics are real parts of holomorphic metrics.

All odd Chern numbers vanish.

Complex parallelisable manifolds are anti-Kaehlerian.

Abstract

An anti-Kaehlerian manifold is a complex manifold with an anti-Hermitian metric and a parallel almost complex structure. It is shown that a metric on such a manifold must be the real part of a holomorphic metric. It is proved that all odd Chern numbers of an anti-Kaehlerian manifold vanish and that complex parallelisable manifolds (in particular the factor space G/D of a complex Lie group G over the discrete subgroup D) are anti-Kaehlerian manifolds. A method of generating new solutions of Einstein equations by using the theory of anti-Kaehlerian manifolds is presented.