Some Remarks Concerning Anti-Hermitian Metrics

Abstract

We show that if a torsion tensor of anti-Hermitian metric connection is pure, then the given anti-Hermitian manifold is anti-Kahler. We prove that if an anti-Hermitian manifold is a conformally flat anti-Kahler–Codazzi manifold, then the scalar curvature vanishes, if and only if the given manifold is isotropic anti-Kahler. We also consider anti-Hermitian metrics of Hessian type defined by holomorphic Hamiltonian functions. Finally, we consider an example of anti-Kahler metrics on Walker 4-manifold.