Kiliç, Erol; Şahin, Bayram Radical anti-invariant lightlike submanifolds of semi-Riemannian product manifolds. (English) Zbl 1165.53033 Turk. J. Math. 32, No. 4, 429-449 (2008). A submanifold \(M\) of a pseudo-Riemannian (simple) product manifold \((\overline M = M_1 \times M_2, \overline g)\) of \((M_1,g_1)\) and \((M_2,g_2)\) is said to be invariant if \(F(TM) = TM\) and it is called anti-invariant if \(F(TM) \subset TM^\perp\) (see [K. Yano and M. Kon, Structures on manifolds. Series in Pure Mathematics, Vol. 3. Singapore: World Scientific. Distr. by John Wiley & Sons Ltd., Chichester. (1984; Zbl 0557.53001)]) where \(\pi:\overline M \to M_1\), \(\sigma:\overline M \to M_2\) are usual projection maps and \(F= \pi_{\ast} - \sigma_{\ast}\).In [K. L. Duggal and A. Bejancu, Light-like submanifolds of semi-Riemannian manifolds and applications. Dordrecht: Kluwer Academic Publishers (1996; Zbl 0848.53001)] and [K. L. Duggal and D. H. Jin, Null curves and hypersurfaces of semi-Riemannian manifolds. Hackensack, NJ: World Scientific (2007; Zbl 1144.53002)], the many aspects of the geometry of light-like submanifolds of pseudo-Riemannian manifolds were studied. In the paper under review, the authors consider some geometrical concepts of light-like submanifolds of pseudo-Riemannian manifolds mostly developed in the books mentioned above to the setting of (simple) product of pseudo-Riemannian manifolds. More explicitly, they introduce the concept of anti-invariant light-like submanifolds of a pseudo-Riemannian (simple) product manifold. They study the geometry of leaves of distributions and establish conditions for the integrability of distributions of such structures. They obtain that the induced connection is indeed a metric connection and find conditions implying that a radical anti-invariant light-like submanifold is a product manifold. Moreover, they provide a necessary condition for a totally umbilical radical anti-invariant light-like submanifold to be totally geodesic. Reviewer: Bülent Ünal (Ankara) Cited in 6 Documents MSC: 53C40 Global submanifolds 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) 53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics Keywords:degenerate metric; semi-Riemannian product manifolds; \(r\)-light-like submanifolds; locally Riemannian products Citations:Zbl 0557.53001; Zbl 0848.53001; Zbl 1144.53002 PDFBibTeX XMLCite \textit{E. Kiliç} and \textit{B. Şahin}, Turk. J. Math. 32, No. 4, 429--449 (2008; Zbl 1165.53033)