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Half lightlike submanifolds of codimension 2. (English) Zbl 0995.53051

The theory of lightlike (degenerate) submanifolds in semi-Riemannian manifolds is relatively new (see, for instance, T. Ikawa [Tsukuba J. Math. 9, 353-371 (1965; Zbl 0588.53017)]). The paper under review deals with the geometry of totally umbilical half-lightlike submanifolds of codimension 2 in a semi-Riemannian manifold. Necessary and sufficient conditions that the Ricci tensor of such a submanifold is symmetric are obtained.
Interesting examples of totally umbilical half-lightlike surfaces in \({\mathbb{R}}^4_2\) and \({\mathbb R}^4_1\), respectively, are given. In particular, the authors investigate totally umbilic half-lightlike surfaces in a 4-dimensional space-time manifold. The null Gauss curvature of such a surface vanishes. Sufficient conditions for the vanishing of the null sectional curvature functions are proved.
Finally, certain possible research problems are stated.

MSC:

53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
53B25 Local submanifolds

Citations:

Zbl 0588.53017
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