Yoshinobu Kamishima

Uniformization of Kähler manifolds with vanishing Bochner tensor

In 1948, S. Bochner introduced a curvature tensor on Hermitian manifolds [1]. He defined it as an analogue to the Weyl conformal curvature tensor. When, on a Riemannian manifold M n, the Weyl conformal curvature tensor (n>3) or the Schouten-Weyl tensor (n--3) vanishes, then M n is said to be a conformally flat manifold. In this case, M n can be uniformized over the n-sphere S n with respect to the group of conformal t ransformations Conf(Sn). It is natural in Geometry to determine the class of compact K~ihler manifolds for which the Bochner curvature tensor vanishes. The Bochner curvature tensor B on a complex manifold with a K/ihler metric is defined as follows:

Standard pseudo-Hermitian structure on manifolds and seifert fibration

A strictly pseudoconvex pseudo-Hermitian manifoldM admits a canonical Lorentz metric as well as a canonical Riemannian metric. Using these metrics, we can define a curvaturelike function Λ onM. AsM supports a contact form, there exists a characteristic vector field ξ dual to the contact structure. If ξ induces a local one-parameter group ofCR transformations, then a strictly pseudoconvex pseudo-Hermitian manifoldM is said to be a standard pseudo-Hermitian manifold. We study topological and geometric properties of standard pseudo-Hermitian manifolds of positive curvature Λ or of nonpositive curvature Λ. By the definition, standard pseudo-Hermitian manifolds are calledK-contact manifolds by Sasaki. In particular, standard pseudo-Hermitian manifolds of constant curvature Λ turn out to be Sasakian space forms. It is well known that a conformally flat manifold contains a class of Riemannian manifolds of constant curvature. A sphericalCR manifold is aCR manifold whose Chern-Moser curvature form vanishes (equivalently, Weyl pseudo-conformal curvature tensor vanishes). In contrast, it is emphasized that a sphericalCR manifold contains a class of standard pseudo-Hermitian manifolds of constant curvature Λ (i.e., Sasakian space forms). We shall classify those compact Sasakian space forms. When Λ≤0, standard pseudo-Hermitian closed aspherical manifolds are shown to be Seifert fiber spaces. We consider a deformation of standard pseudo-Hermitian structure preserving a sphericalCR structure.