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Lorentzian geometry of CR submanifolds. (English) Zbl 0692.53025

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.

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.

Reviewer: D.Allison
MSC:
53C50Lorentz manifolds, manifolds with indefinite metrics
53C55Hermitian and Kählerian manifolds (global differential geometry)
53C80Applications of global differential geometry to physics
83C50Electromagnetic fields in general relativity
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