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Screen Cauchy Riemann lightlike submanifolds. (English) Zbl 1083.53063

The notion of light-like submanifold has been introduced and studied by K. L. Duggal and A. Bejancu in [Light-like Submanifolds of Semi-Riemannian Manifolds and Applications, Kluwer Academic, 364 (1996; Zbl 0848.53001)]. A result in this article shows that, for an indefinite Kähler manifold, the Cauchy Riemann (CR) light-like submanifolds do not include invariant (complex) and real light-like submanifolds.

The paper under review gives an affirmative answer to the following question: “Are there any light-like submanifolds of an indefinite Kähler manifold which contain invariant (complex) and real light-like submanifolds ?” The main tool is the notion of Screen Cauchy Riemann (SCR)-light-like submanifolds of an indefinite Kähler manifold. For such submanifolds the authors prove two existence theorems, show that the class of SCR-light-like submanifolds contains complex and screen real subcases, and find the integrability condition of all the distributions. Totally umbilical proper SCR-light-like submanifolds are also studied. Some new results on irrotational screen real light-like submanifolds are proved and examples are provided.


MSC:
53C50Lorentz manifolds, manifolds with indefinite metrics
53C15Differential geometric structures on manifolds
53C40Global submanifolds (differential geometry)