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Zbl 0692.53025
Duggal, K.L.
Lorentzian geometry of CR submanifolds. (English)
[J] Acta Appl. Math. 17, No.2, 171-193 (1989). ISSN 0167-8019; ISSN 1572-9036/e

A. Bejancu defined CR submanifolds of differentiable manifolds with a (positive definite) Riemannian metric and almost Hermitian structure as a generalization of holomorphic submanifolds and totally real submanifolds. In this article, the notion of CR submanifold is extended to orientable Lorentz submanifolds of semi-Riemannian manifolds with an almost Hermitian structure. By a Lorentz submanifold we mean an n-dimensional submanifold embedded in an $(n+p)$-dimensional semi-Riemannian manifold equipped with an indefinite metric, where the induced metric on the submanifold has Lorentzian signature (1,n-1). Definitions and theorems needed to extend the theory of CR submanifolds, contact CR submanifolds, and framed f-structures to the Lorentzian case are given. \par Many proofs are omitted as they are straight-forward generalizations but details of new results are presented. The primary new contribution is the study of CR submanifolds with a distribution D which is everywhere light- like, i.e. the metric restricted to D is degenerate. Several examples are presented and a discussion of how these notions relate to general relativity are included. For example, a four dimensional Lorentz CR manifold with a light-like distribution is related to the class of spacetimes representing null electromagnetic fields with the energy momentum tensor of a pure radiation field. Finally, a research problem to find the relation between Lorentzian geometry and pseudo conformal geometry is proposed.
[D.Allison]

MSC 2000:: *53C50 Lorentz manifolds, manifolds with indefinite metrics
53C55 Complex differential geometry (global)
53C80 Appl. of global differential geometry to physics
83C50 Electromagnetic fields

Keywords: CR submanifolds; contact CR submanifolds; framed f-structures; null electromagnetic fields; pure radiation field; Lorentzian geometry; pseudo conformal geometry

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