B-CONNECTIONS AND THEIR CONFORMAL INVARIANTS ON CONFORMALLY KAEHLER MANIFOLDS WITH B-METRIC

Georgi Ganchev, Kostadin Gribachev and Vesselka Mihova

Abstract: On a Kaehler manifold there is not a complete analogue of the conformal geometry on a Riemannian manifold. In this paper, we consider a class of complex manifolds with B-metric (including Kaehler manifolds with B-metric). The general conformal group and its special subgroups are determined. The Bochner curvature tensor of the manifold is shown to be a conformal invariant. The zero Bochner curvature tensor is proved to be an integrability condition of a geometrical system of partial differential equations and a characterization condition of a conformally flat manifold. Holomorphically umbilic submanifolds (Holomorphic spheres) are conformal invariants. The manifolds satisfying the axiom of holomorphic spheres are also characterized by zero Bochner curvature tensor. Thus, on the considered manifolds, there is a complete analogue of the conformal geometry on a Riemannian manifold.

Classification (MSC2000): 55B05

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