Totally umbilical lightlike submanifolds. (English) Zbl 1049.53011

Duggal, K. L.; Jin, D. H.

Kodai Math. J. 26, No. 1, 49-68 (2003).

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.

Reviewer: Marian Hotloś (Wrocław)

MSC:

53B25	Local submanifolds
53C40	Global submanifolds
53C50	Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics

Keywords:

degenerate metric; totally umbilical submanifold; light-like submanifold