SELF-DUAL GEOMETRY OF GENERALIZED HERMITIAN SURFACES

Several results on the geometry of conformally semiflat Hermitian surf aces of both classical and hyperbolic types (generalized Hermitian sur faces) are obtained. Some of these results are generalizations and cla rifications of already known results in this direction due to Koda, It oh, and other authors. They reveal some unexpected beautiful connectio ns between such classical characteristics of conformally semiflat (gen eralized) Hermitian surfaces as the Einstein property, the constancy o f the holomorphic sectional curvature, and so on. A complete classific ation of compact self-dual Hermitian RK-surfaces that are at the same time generalized Hopf manifolds is obtained. This provides a complete solution of the Chen problem in this class of Hermitian surfaces.