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Standard pseudo-Hermitian structure on manifolds and Seifert fibration. (English) Zbl 0841.53028

Summary: A strictly pseudoconvex pseudo-Hermitian manifold \(M\) admits a canonical Lorentz metric as well as a canonical Riemannian metric. Using these metrics, we can define a curvaturelike function \(\Lambda\) on \(M\). As \(M\) supports a contact form, there exists a characteristic vector field \(\xi\) dual to the contact structure. If \(\xi\) induces a local one-parameter group of CR transformations, then a strictly pseudoconvex pseudo-Hermitian manifold \(M\) is said to be a standard pseudo-Hermitian manifold.
We study topological and geometric properties of standard pseudo-Hermitian manifolds of positive curvature \(\Lambda\) or of nonpositive curvature \(\Lambda\). By the definition, standard pseudo-Hermitian manifolds are called \(K\)-contact manifolds by Sasaki. In particular, standard pseudo-Hermitian manifolds of constant curvature \(\Lambda\) 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 spherical CR manifold whose Chern-Moser curvature form vanishes (equivalently, the Weyl pseudo-conformal curvature tensor vanishes). In contrast, it is emphasized that a spherical CR manifold contains a class of standard pseudo-Hermitian manifolds of constant curvature \(\Lambda\) (i.e., Sasakian space forms). We shall classify those compact Sasakian space forms. When \(\Lambda \leq 0\), standard pseudo-Hermitian closed aspherical manifolds are shown to be Seifert fiber spaces. We consider a deformation of the standard pseudo-Hermitian structure preserving a spherical \(CR\) structure.

MSC:

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
57S25 Groups acting on specific manifolds
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