Contact Semi-Riemannian Structures in CR Geometry: Some Aspects

There is one-to-one correspondence between contact semi-Riemannian structures (eta,xi,phi,g) and non-degenerate almost CR structures (H,& thetasym;,J). In general, a non-degenerate almost CR structure is not a CR structure, that is, in general the integrability condition for H1,0:=X-iJX,X is an element of 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.