Abstract
There is one-to-one correspondence between contact semi-Riemannian structures ( η , ξ , φ , g ) and non-degenerate almost CR structures ( H , ϑ , J ) . In general, a non-degenerate almost CR structure is not a CR structure, that is, in general the integrability condition for H 1 , 0 : = X - i J X , X ∈ H is not satisfied. In this paper we give a survey on some known results, with the addition of some new results, on the geometry of contact semi-Riemannian manifolds, also in the context of the geometry of Levi non-degenerate almost CR manifolds of hypersurface type, emphasizing similarities and differences with respect to the Riemannian case.
Highlights
ContactRiemannian geometry and CR geometry are two fields of research that have been developed independently of each other, and with different motivations
We note that there is not a monograph dedicated to contact semi-Riemannian structures which emphasizes its connection with the non-degenerate almost CR structures
We can say that the contact geometry begins with Sophus Lie (1872) when he introduced the notion of a contact transformation as a geometric tool to study systems of differential equations
Summary
Contact (semi-)Riemannian geometry and (almost) CR geometry are two fields of research that have been developed independently of each other, and with different motivations. We note that there is not a monograph dedicated to contact semi-Riemannian structures which emphasizes its connection with the non-degenerate almost CR structures. Contact manifolds equipped with semi-Riemannian metrics were first introduced and studied by T. There is one-to-one correspondence between contact semi-Riemannian structures and non-degenerate almost CR structures. In this paper (which reflects the interests and knowledge of the author) we give a survey on some known results, with additions of some new result, on the geometry of contact semi-Riemannian manifolds, in the context of the geometry of Levi non-degenerate almost CR manifolds of hypersurface type, emphasizing similarities and differences with respect to the Riemannian case. We explain the relationship between contact semi-Riemannian structures and non-degenerate pseudohermitian structures, describing in some detail several important examples, like hypersurfaces of indefinite Kähler manifolds, and tangent hyperquadric bundles over semi-Riemannian manifolds. Blair’s book [2], who want to have a comprehensive look at the main differences between the strictly pseudo-convex setting and the semi-Riemannian setting
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