Geometric Realizations of para-Hermitian curvature models

TL;DR

This paper establishes conditions under which para-Hermitian algebraic curvature models can be realized by actual para-Hermitian manifolds, extending curvature decomposition techniques and ensuring constant scalar curvatures.

Contribution

It extends the Tricerri-Vanhecke curvature decomposition to para-Hermitian geometry and characterizes geometric realizability of curvature models with constant scalar curvatures.

Findings

Para-Hermitian algebraic curvature models satisfy the para-Gray identity if and only if realizable.

Realizations can have constant scalar and *-scalar curvature.

Extension of curvature decomposition to para-Hermitian setting.

Abstract

We show that a para-Hermitian algebraic curvature model satisfies the para-Gray identity if and only if it is geometrically realizable by a para-Hermitian manifold. This requires extending the Tricerri-Vanhecke curvature decomposition to the para-Hermitian setting. Additionally, the geometric realization can be chosen to have constant scalar curvature and constant *-scalar curvature.