Almost Complex and Almost Product Einstein Manifolds from a Variational Principle

Details Almost Complex and Almost Product Einstein Manifolds from a Variational Principle Almost Complex and Almost Product Einstein Manifolds from a Variational Principle

1996-12-05

arxiv.org/abs/dg-ga/9612009v1

J.Math.Phys.40:3446-3464,1999

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

31 pages, Latex

Scientific paper

10.1063/1.532899

It is shown that the first order (Palatini) variational principle for a generic nonlinear metric-affine Lagrangian depending on the (symmetrized) Ricci square invariant leads to an almost-product Einstein structure or to an almost-complex anti-Hermitian Einstein structure on a manifold. It is proved that a real anti-Hermitian metric on a complex manifold satisfies the K\"ahler condition on the same manifold treated as a real manifold if and only if the metric is the real part of a holomorphic metric. A characterisation of anti-K\"ahler Einstein manifolds and almost-product Einstein manifolds is obtained. Examples of such manifolds are considered.

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