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Scalar curvature and symmetry properties of lightlike submanifolds

In this paper, the induced Ricci tensor and the extrinsic scalar curvature on lightlike submanifolds are obtained. This scalar quantity extend the result given by C. Atindogbe in. An example of extrinsic scalar curvature on one class of 2-degenerate manifolds is provided. | Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ Research Article Turk J Math (2013) 37: 95 – 113 ¨ ITAK ˙ c TUB doi:10.3906/mat-1106-8 Scalar curvature and symmetry properties of lightlike submanifolds Cyriaque ATINDOGBE1 , Oscar LUNGIAMBUDILA2,∗, Jo¨ el TOSSA3 Institut de Math´ematiques et de Sciences Physiques (IMSP) Universit´e d’Abomey-Calavi, 01 B.P. 613, Porto-Novo, B´enin 2 D´epartement de Math´ematiques et Informatique, Facult´e des Sciences Universit´e de Kinshasa, B.P. 190 KINSHASA XI R.D. Congo 3 Institut de Math´ematiques et de Sciences Physiques (IMSP) Universit´e d’Abomey-Calavi, 01 B.P. 613, Porto-Novo, B´enin 1 Received: 08.06.2011 • Accepted: 28.09.2011 • Published Online: 17.12.2012 • Printed: 14.01.2013 Abstract: In this paper, the induced Ricci tensor and the extrinsic scalar curvature on lightlike submanifolds are obtained. This scalar quantity extend the result given by C. Atindogbe in [1]. An example of extrinsic scalar curvature on one class of 2 -degenerate manifolds is provided. We investigate lightlike submanifolds which are locally symmetric, semi-symmetric, Ricci semi-symmetric in indefinite spaces form. In the coisotropic case, we show that, under some conditions, these lightlike submanifolds are totally geodesic. Key words: Extrinsic scalar curvature, locally symmetric lightlike submanifold, semi-symmetric lightlike submanifold, Ricci semi-symmetric lightlike submanifold 1. Introduction The scalar curvature is one of the most important concepts in semi-Riemannian geometry and its connected areas such as General Relativity. This scalar quantity, under the geometric point of view, is just the contraction of the symmetric Ricci tensor Ric with a non-degenerate metric g , that is S = gαβ Ricαβ . (1.1) In geometry of the lightlike submanifolds, two difficulties arise: since the induced connection is not a Levi-Civita connection (unless M be totally geodesic) the induced Ricci tensor is not symmetric in .