Gray-Hervella classes on product twistor spaces

TL;DR

This paper investigates the Gray-Hervella classes of almost Hermitian structures on the product of twistor spaces over four-dimensional Riemannian manifolds, extending generalized geometry concepts.

Contribution

It determines the Gray-Hervella classes of these structures in the specific case when the base manifold has dimension four.

Findings

Gray-Hervella classes are explicitly characterized for the product twistor space.

The study extends the understanding of almost complex structures in generalized geometry.

Results are specific to four-dimensional base manifolds.

Abstract

Motivated by generalized geometry (in the sense of Hitchin), the product bundle ${\mathcal Z}\times_{M} {\mathcal Z}$ of the twistor space ${\mathcal Z}$ of a Riemannian manifold $(M,g)$ is considered. The product twistor space admits a natural family of Riemannian metrics and four compatible almost complex structures, analogs of the Atiyah-Hitchin-Singer and Eells-Salamon almost complex structures on the twistor space. The Gray-Hervellal classes of these almost Hermitian structures are determined in the case when the dimension of the base manifold $M$ is four.