Some Results on Radical Screen Transversal Lightlike Submanifold of an Indefinite Kaehler Norden Manifold

Abstract

In this paper, we introduce the notion of radical screen transversal and screen transversal anti-invariant lightlike submanifolds of an indefinite Kaehler Norden manifold. We investigate the geometry of distributions involved and obtain necessary and sufficient conditions for the induced connection on radical screen transversal and screen transversal anti-invariant lightlike submanifolds to be metric connection. Further, we provide the necessary and sufficient conditions for foliations determined by above distributions to be totally geodesic on radical screen transversal lightlike submanifold of an indefinite Kaehler Norden manifold. AMS 2020 Mathematics Subject Classification: 53C15, 53C40, 53C50.

Introduction

The theory of almost complex Norden manifolds, introduced by A. P. Norden in 1960 and further developed by Ganchev and others, studies manifolds with an almost complex structure J?\bar{J}J? that behaves as an anti-isometry with respect to a semi-Riemannian metric g?\bar{g}g??, unlike indefinite almost Hermitian manifolds where J?\bar{J}J? is an isometry. This difference leads to distinct geometric properties.

Lightlike submanifolds, studied extensively by Duggal and Bejancu, differ from classical non-degenerate submanifolds because their tangent and normal bundles intersect non-trivially, creating a radical (null) distribution. This introduces unique challenges in their study.

The paper introduces and investigates new types of lightlike submanifolds—radical screen transversal and screen transversal anti-invariant—in the context of indefinite Kaehler Norden manifolds. It explores the geometry of the distributions involved and derives necessary and sufficient conditions for induced connections on these submanifolds to be metric. It also examines conditions for foliations defined by these distributions to be totally geodesic.

Preliminary definitions clarify that a lightlike submanifold has a degenerate induced metric, with a nontrivial radical distribution formed by the intersection of its tangent and normal spaces. The screen distribution is a semi-Riemannian complementary distribution to the radical distribution within the tangent bundle.

References

[1] Bejancu, K. L. Duggal, Degenerate hypersurface of Semi-Riemannian manifolds, Bull. Inst. Politehnie Iasi, 37 (1991) 13-22. [2] J. K. Beem, P. E. Ehrlich, and K. L. Easley, Global Lorentzian Geometry, Marcel Dekker, Inc. New York, Second Edition, (1996). [3] A. Borisov, G. Ganchev, Curvature properties of Kaehlerian manifolds with B-metric, Proc. of 14 Spring Conference of UBM, Sunny Beach (1985), 220-226. [4] K. L. Duggal, A. Bejancu, Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications, 364 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, (1996). [5] K. L. Duggal, B. Sahin, Differential Geometry of Lightlike Submanifolds, Birkhauser, Verlag, (2010). [6] G. Ganchev, A. Borisov, Note on the almost complex manifolds with a Norden metric, Compt. Rend. Acad. Bulg. Sci., 39(5) (1986), 31-34. [7] G. Ganchev, K. Gribachev, V. Mihova, Holomorphic hypersurfaces of Kaehler manifolds with Norden metric, Plovdiv Univ. Sci. Works Math., 23(2) (1985), 221-236. [8] K. I. Gribachev, D. G. Mekerov, G. D. Dzhelepov, Generalized B-manifolds, Compt. Rend. Acad. Bulg. Sci., 38(3) (1985), 299-302. [9] A. P. Norden, On a class of four-dimensional A-spaces, Russian Math., 17(4) (1960), 145-157. [10] B. Sahin, Transversal lightlike submanifolds of indefinite Kaehler manifolds, Analele Universitatii de Vest Timisoara Seria Matematica Informatica XLIV(1) (2006), 119-145.

Copyright

Copyright © 2025 Anu Devgan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.