nLab para-Hermitian manifold

Idea

The analogue of Hermitian manifold for para-complex structure.

Definition

Definition

An almost para-Hermitian manifold $(M,J,\eta)$ is an almost para-complex manifold $(M,J)$ endowed with a pseudo-Riemannian metric $\eta$ of signature $(d,d)$, for $\text{dim}(M)=2d$, such that

$$\eta(J(X),J(Y))=-\eta(X,Y)$$

Related concepts

• Hermitian manifold
• para-complex structure

References

General:

• Stefan Ivanov, Simeon Zamkovy, ParaHermitian and paraquaternionic manifolds, Differential Geometry and its Applications 23 2 (2005) 205-234 [doi:10.1016/j.difgeo.2005.06.002]

As target spacetimes in a context of type II geometry:

• Vincenzo Emilio Marotta, Richard J. Szabo, Born Sigma-Models for Para-Hermitian Manifolds and Generalized T-Duality, Reviews in Mathematical Physics 33 09 (2021) 2150031 [arXiv:1910.09997, doi:10.1142/S0129055X21500318]