Self-dual geometry of generalized Hermitian surfaces.

*(English. Russian original)*Zbl 0907.53023
Sb. Math. 189, No. 1, 21-44 (1998); translation from Mat. Sb. 189, No. 1, 19-41 (1998).

Let \((M,g)\) be a pseudo-Riemannian manifold endowed with a generalized almost Hermitian structure, i.e., a tensor field \(J\) of type \((1,1)\) such that \(g(JX, JY)= -\alpha g(X,Y)\) and \(J^2= \alpha I\) where \(\alpha= -1\) for the classical type and \(\alpha= 1\) for the hyperbolic type. \((M,g,J)\) is called a generalized Hermitian manifold if the structure is integrable, i.e., its Nijenhuis tensor vanishes.

The main content of this paper is a study of self-dual and anti-self-dual generalized Hermitian surfaces aiming at a classification of these four-dimensional spaces. It contains several known and new results about this problem. In particular, the authors consider the subclasses of generalized Hermitian surfaces of constant holomorphic sectional curvature, generalized Hermitian surfaces belonging to the class \({\mathcal {AH}}_1\) (i.e., \(R(JX,JY)= R(X,Y))\) or to the class \({\mathcal {AH}}_3\) (i.e., \(R(JX, JY, JZ, JW)= R(X,Y,Z,W))\), where \(R\) denotes the Riemannian curvature operator, and generalized Hermitian surfaces which are locally conformally Kählerian, generalized Hopf manifolds or Einstein manifolds. The study yields several characterizations and some classification results are derived.

The main content of this paper is a study of self-dual and anti-self-dual generalized Hermitian surfaces aiming at a classification of these four-dimensional spaces. It contains several known and new results about this problem. In particular, the authors consider the subclasses of generalized Hermitian surfaces of constant holomorphic sectional curvature, generalized Hermitian surfaces belonging to the class \({\mathcal {AH}}_1\) (i.e., \(R(JX,JY)= R(X,Y))\) or to the class \({\mathcal {AH}}_3\) (i.e., \(R(JX, JY, JZ, JW)= R(X,Y,Z,W))\), where \(R\) denotes the Riemannian curvature operator, and generalized Hermitian surfaces which are locally conformally Kählerian, generalized Hopf manifolds or Einstein manifolds. The study yields several characterizations and some classification results are derived.

Reviewer: L.Vanhecke (Leuven)

##### MSC:

53C15 | General geometric structures on manifolds (almost complex, almost product structures, etc.) |

53C55 | Global differential geometry of Hermitian and Kählerian manifolds |

53B35 | Local differential geometry of Hermitian and Kählerian structures |