On the Geometry of Null Real Hypersurfaces of an Indefinite Complex Contact Manifold
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On the Geometry of Null Real Hypersurfaces of an Indefinite Complex Contact Manifold Samuel Ssekajja1 Received: 10 December 2019 / Revised: 26 August 2020 / Accepted: 6 October 2020 © Iranian Mathematical Society 2020
Abstract We study the geometry of null real hypersurfaces in indefinite complex contact manifolds. We prove several classification results for a variety of well-known null hypersurfaces, including the totally umbilic, totally screen umbilic, and the screen conformal ones. Furthermore, a characterisation of the ambient space is given in case the underlying null hypersurface is totally contact umbilic, totally contact screen umbilic, or contact screen conformal, i.e., we have proved that the ambient complex contact manifold must be a space of constant G H -sectional curvature of −3. Keywords Null hypersurfaces · Totally umbilic null hypersurfaces · Complex contact manifolds Mathematics Subject Classification 53C25 · 53C40 · 53C50
1 Introduction On any semi-Riemannian manifold, there is a natural existence of null (lightlike) subspaces. In 1996, Duggal–Bejancu published a book [3] on the null geometry of submanifolds which filled an important missing part in the general theory of submanifolds. This book was later updated by Duggal–Sahin in [4], by collecting most of the new discoveries in the area since the first publication. Away from these two books, many researchers have investigated the geometry of null subspaces of semiRiemannian manifolds. On the other hand, in about the same time as in the book [3], Kupeli [8] introduces the theory of null geometry in a relatively different way. The main tool in his approach was the consideration of a factor bundle which is isomorphic
Communicated by Mohammad Koushesh.
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Samuel Ssekajja [email protected] School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, South Africa
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Bulletin of the Iranian Mathematical Society
to the screen distribution used by the authors in [3]. In Chapter 8 of [4], the authors introduces the geometry of null submanifolds of indefinite quaternion Kaehler manifolds. Therein, the authors study the geometry of real null hypersurfaces, the structure of null submanifolds, both, of indefinite quaternion Kaehler manifolds, and show that a quaternion null submanifold is always totally geodesic. This result implies that the study of null submanifolds, other than quaternion null submanifolds, is interesting. Then, they deal with the geometry of screen real submanifolds in detail. As a generalisation of real null hypersurfaces of quaternion Kaehler manifolds, they introduced Q R-null submanifolds. Furthermore, they show that the class of Q R-null submanifolds does not include quaternion null submanifolds and screen real submanifolds. They also introduced and studied the geometry of screen Q R-null and screen C R-null submanifolds as generalisations of quaternion null submanifolds and screen real submanifolds, and provided examples for each class of null submanifolds of indefinite quaternion Kaehler ma
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