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Totally umbilical lightlike submanifolds. (English) Zbl 1049.53011

In the paper under review the authors study the geometry of totally umbilical light-like submanifolds. First they recall some results for light-like submanifolds and their structure equations. Next they prove several new theorems on such submanifolds in semi-Riemannian manifolds of constant curvature. The authors show that the induced Ricci tensor of any totally umbilical light-like submanifold in a manifold of constant curvature is symmetric if and only if its screen distribution is integrable.

It is worth noticing that many well-known spacetimes have light-like hypersurfaces with an integrable 2-dimensional screen distribution.

MSC:
53B25Local submanifolds
53C40Global submanifolds (differential geometry)
53C50Lorentz manifolds, manifolds with indefinite metrics