Gray-Hervella classes on product twistor spaces

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.

• The paper classifies Gray-Hervella types for almost Hermitian structures on product twistor spaces by deriving explicit curvature and integrability conditions.
• It details the construction of natural Riemannian metrics and four canonical almost complex structures, mirroring classical Atiyah-Hitchin-Singer and Eells-Salamon models.
• The results reveal strong rigidity phenomena, showing that only specific classes occur under strict conditions such as anti-self-duality, scalar flatness, or Einstein metrics.

Gray-Hervella Classes on Product Twistor Spaces

Introduction

This paper investigates the Gray-Hervella classification of almost Hermitian structures arising on the product bundle $\mathcal{Z} \times_M \mathcal{Z}$, where $\mathcal{Z}$ denotes the twistor space of a Riemannian $4$-manifold $(M,g)$ (2604.12849). Motivations stem from generalized complex geometry in the sense of Hitchin, notably the observation that for a generalized Riemannian metric, the corresponding generalized twistor space is diffeomorphic to this product bundle. The work provides a comprehensive analysis of compatible almost complex structures—analogs of the classic Atiyah-Hitchin-Singer and Eells-Salamon structures—on the product twistor space, classifying their Gray-Hervella types in dimension four, and deriving explicit curvature and integrability conditions underpinning these classifications.

Product Twistor Spaces and Natural Almost Complex Structures

Let $(M,g)$ be a Riemannian $4$-manifold. The twistor space $\mathcal{Z}$ is the fibre bundle over $M$ whose fibre at $p \in M$ consists of $g$-orthogonal complex structures on $\mathcal{Z}$0, with $\mathcal{Z}$1 diffeomorphic to $\mathcal{Z}$2. The product twistor space $\mathcal{Z}$3 is then a fibre bundle over $\mathcal{Z}$4 whose fibre is the product of two $\mathcal{Z}$5-spheres, corresponding to pairs of compatible complex structures at each point.

The paper constructs a natural family of Riemannian metrics $\mathcal{Z}$6 on the product space, parametrized by positive weights $\mathcal{Z}$7 on each factor, and four natural almost complex structures $\mathcal{Z}$8 ($\mathcal{Z}$9), mirroring the classic Atiyah-Hitchin-Singer and Eells-Salamon structures. Each such $4$0 acts in a specific way on both the horizontal distribution (using one of the complex structures from $4$1) and the vertical distribution (using both factors from the fibre).

Technical Framework for Gray-Hervella Classification

The Gray-Hervella framework partitions almost Hermitian structures into sixteen classes $4$2. The structure $4$3 lies in a class according to the properties of the fundamental $4$4-form $4$5, its exterior derivative, and the Nijenhuis tensor. The paper provides explicit formulas for the Levi-Civita connection on $4$6, the decomposition of the curvature operator, and computations of the covariant derivatives and codifferentials of the fundamental forms for each almost complex structure. This technical development expresses all relevant torsion and integrability properties in terms of curvature components of the underlying $4$7-manifold.

Main Classification Results and Rigidity Phenomena

The main results classify which Gray-Hervella classes can occur for the almost Hermitian structures $4$8 on $4$9, focusing on the two components $(M,g)$0 and $(M,g)$1, where $(M,g)$2 is the positive/negative twistor bundle. The principal findings are:

• For $(M,g)$3:
  • On $(M,g)$4: Only the classes $(M,g)$5 (Kähler) and $(M,g)$6 (Hermitian semi-Kähler) can appear. However, $(M,g)$7 is never Kähler on $(M,g)$8. On $(M,g)$9, the structure is Kähler precisely when $(M,g)$0 is Einstein with positive scalar curvature, anti-self-dual, and $(M,g)$1 (where $(M,g)$2 is the scalar curvature) with a further rigidity condition on the Weyl tensor.
  • The structure is of class $(M,g)$3 on $(M,g)$4 if and only if $(M,g)$5 is anti-self-dual and scalar flat, and on $(M,g)$6 if and only if $(M,g)$7 is anti-self-dual.
• For $(M,g)$8 (quasi-Eells-Salamon analogs):
  • On $(M,g)$9, only the semi-Kähler class ($4$0) can occur when $4$1 is anti-self-dual and scalar flat.
  • On $4$2, several mixed classes can occur, with precise curvature conditions: for example, nearly-Kähler ($4$3), almost-Kähler ($4$4), or quasi-Kähler ($4$5) are realized only under stringent conditions (Einstein, anti-self-dual, and determined relations among the curvature components and the parameters $4$6).
  • Notably, some classes, such as nearly-Kähler and almost-Kähler, are shown never to occur on $4$7 for any of the four canonical almost complex structures.

These theorems establish strong rigidity: most Gray-Hervella classes are excluded, and the allowable ones have explicit, sharp characterizations in terms of the curvature decomposition of $4$8.

Implications and Directions

The results have salient implications for the geometric analysis of generalized and classical twistor fibrations. They indicate that very few almost Hermitian types are realized on product twistor spaces, and when they are, they signal strong geometric properties (e.g., anti-self-duality, Einstein condition) of the underlying four-manifold. The identification of stringent curvature conditions for the Kähler, (quasi-)Kähler, and nearly-Kähler classes signifies that the geometry of the base manifold essentially controls the structure of compatible almost Hermitian geometries on the associated twistor bundles.

From a theoretical standpoint, this study reinforces and refines the correspondence between the geometry of twistor spaces, complex and almost complex structures, and four-dimensional Riemannian geometry. The explicit description of which classes may (or may not) occur can inform future developments in the study of special metrics, Hamiltonian structures, and their moduli on associated bundles.

Potential future work includes an extension to higher-dimensional generalized twistor spaces, analysis in the presence of torsion or weaker holonomy conditions, and a systematic study of the (generalized) Hermitian structures in the context of generalized complex geometry.

Conclusion

This work provides a complete and rigorous classification of Gray-Hervella classes for almost Hermitian structures on the product twistor space $4$9 in dimension four. The study establishes that only specific classes—chiefly those associated with strong curvature and holonomy constraints on $\mathcal{Z}$0—are realized. The results clarify the interaction between Riemannian, complex, and almost complex geometry in the context of twistor theory, and contribute concrete obstructions and existence results for special geometric structures on bundles over four-manifolds (2604.12849).