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Osserman lightlike hypersurfaces of indefinite S -manifolds

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We mainly deal with the problem of admissibility for screen distributions on a lightlike hypersurface of both a semi-Riemannian manifold and an indefinite S -manifold. In the latter case, we first show that a characteristic screen distribution is never admissible, and then we provide a characterization for admissible screen distributions on proper totally umbilical lightlike hypersurfaces. | Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ Research Article Turk J Math (2014) 38: 340 – 352 ¨ ITAK ˙ c TUB ⃝ doi:10.3906/mat-1206-1 Osserman lightlike hypersurfaces of indefinite S -manifolds Letizia BRUNETTI∗ Department of Mathematics, University of Bari “Aldo Moro” Bari, Italy Received: 01.06.2012 • Accepted: 25.09.2013 • Published Online: 27.01.2014 • Printed: 24.02.2014 Abstract: We mainly deal with the problem of admissibility for screen distributions on a lightlike hypersurface of both a semi-Riemannian manifold and an indefinite S -manifold. In the latter case, we first show that a characteristic screen distribution is never admissible, and then we provide a characterization for admissible screen distributions on proper totally umbilical lightlike hypersurfaces. Finally, in studying Osserman conditions, we characterize Osserman totally umbilical hypersurfaces of a semi-Riemannian manifold, obtaining explicit results on the eigenvalues of the pseudoJacobi operators in the case of lightlike hypersurfaces with Lorentzian screen leaves. Key words: Lightlike hypersurface, semi-Riemannian manifold, admissible screen distribution, pseudo-Jacobi operator, Osserman condition, indefinite S -manifold 1. Introduction The study of sectional curvature has always been one of the most interesting topics, since it can provide information about the properties of manifolds. The study of Osserman conditions perfectly falls within this area of interest. In Riemannian geometry, the original problem involving the Osserman conditions is known as the Osserman Conjecture. Namely, let (M, g) be a Riemannian manifold, with curvature tensor R , and X a unit vector of Tp M , p ∈ M . It is then possible to define the symmetric endomorphism RX : X ⊥ → X ⊥ such that RX = Rp (·, X)X , called the Jacobi operator with respect to X at p. A Riemannian manifold (M, g) whose Jacobi operators have eigenvalues independent of X ∈ Tp M and p ∈ M is said to be an .