Anti-Kahlerian manifolds

An anti-Kahlerian 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-Kahlerian 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-Kahlerian manifolds. A method of generating new solutions of Einstein equations by using the theory of anti-Kahlerian manifolds is presented.