We show that if a torsion tensor of anti-Hermitian metric connection is pure, then the given anti-Hermitian manifold is anti-Kähler. We prove that if an anti-Hermitian manifold is a conformally flat anti-Kähler–Codazzi manifold, then the scalar curvature vanishes, if and only if the given manifold is isotropic anti-Kähler. We also consider anti-Hermitian metrics of Hessian type defined by holomorphic Hamiltonian functions. Finally, we consider an example of anti-Kähler metrics on Walker 4-manifold.
Anti-Kähler–Codazzi manifold isotropic anti-Kähler manifold holomorphic function metric connection with torsion Walker metric
Mathematics Subject Classification
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The authors would like to express their gratitude to the referee for valuable suggestions and comments.
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