Lorentzian Geometry and CR-Submanifolds

This paper contains an up-to-date information on the Lorentzian geometry of CR-submanifolds, contact CR-submanifolds and globally framed CR-submanifolds (M, g) of an indefinite semi-Riemannian manifold. In view of the large number of excellent paper appearing in this field, we focus on those key results whose Lorentzian geometry is different than their corresponding Riemannian geometry. Contrary to the case of Riemannian CR-submanifolds and above-stated two other classes, the induced distribution D of M, has three subcases, namely, (a) $$g_{|D}$$ g | D is spacelike or (b) $$g_{|D}$$ g | D is Lorentzian or (c) $$g_{|D}$$ g | D is lightlike. For the first two subcases D is an invariant submanifold of M, but, the third subcase need not be invariant. We notice that the geometry of the subcase (a) is mostly similar with the Riemannian case, but, the subcase (b) still remains an open problem since $$g_{|D}$$ g | D Lorentzian is not compatible with the required Hermitian structure of D. Moreover, the geometry of subcase (c) is quite different for which we provide several new results. We also give interesting physical examples used in relativity.