To browse Academia. Skip to main content. Log In Sign Up. Generalised Cauchy-Riemann lightlike submanifolds of indefinite cosymplectic manifolds Annals of the Alexandru Ioan Cuza University - Mathematics, Abhitosh Upadhyay. Generalised Cauchy-Riemann lightlike submanifolds of indefinite cosymplectic manifolds. In this paper, we study generalised Cauchy-Riemann GCR lightlike submanifold of an indefinite cosymplectic manifold and give a few examples also.

Key words: cosymplectic manifold, CR-submanifold, lightlike submanifold, screen and lightlike transversal vector bundles.

Gigabyte debug code d3Introduction The study of lightlike submanifolds is interesting due to the fact that the intersection of the normal vector bundle and the tangent bundle is non- trivial making it remarkably different from non-degenerate submanifolds. Duggal and Bejancu [4] have studied lightlike submanifolds of indefinite Kaehler manifolds.

### Lightlike Submanifolds of Indefinite Sasakian Manifolds

On the other hand, a general notion of lightlike subman- ifold and generalized Cauchy-Riemann GCR lightlike submanifold of in- definite Sasakian manifold was introduced by Duggal and Sahin [6], [7].

Recently, we have studied Cauchy-Riemann and screen Cauchy-Riemann lightlike submanifolds of indefinite cosymplectic manifolds [9]. We have shown that there does not exist inclusion relation between these two classes.

The objective of this paper is to define a generalised Cauchy-Riemann light- like submanifold of an indefinite cosymplectic manifold, which includes in- variant, screen real, contact CR lightlike subcases and real hypersurfaces. Authenticated An indefinite almost contact metric manifold M is called an indefinite cosymplectic manifold if [3], [8] 1.

Then according to the decomposition 1. Using 1. Then from the decomposition of the tangent bundle of a lightlike submanifold, we have 1. From 1. We have the following definition due to Bejan and Duggal: Definition 1. Following [5], we have: Definition 1. Thus, we have the following decomposition: 2. Thus, from Definition 2. Proposition 2. Let M be a contact CR-lightlike submanifold. Then from 2. We have: Proposition 2.

There exist no coisotropic, isotropic or totally light- like proper GCR-lightlike submanifold M of an indefinite cosymplectic mani- fold M. Hence, conditions A and B of Definition 2. It is easy to see that any contact CR-lightlike three-dimensional sub- manifold is 1-lightlike real hypersurface [9].

Moreover, it has also been Authenticated Thus, from Proposition 2. Example 2.We introduce and study generalized transversal lightlike submanifold of indefinite Sasakian manifolds which includes radical and transversal lightlike submanifolds of indefinite Sasakian manifolds as its trivial subcases. A characteristic theorem and a classification theorem of generalized transversal lightlike submanifolds are obtained.

The theory of submanifolds in Riemannian geometry is one of the most important topics in differential geometry for years. We see from [ 1 ] that semi-Riemannian submanifolds have many similarities with the Riemannian counterparts.

However, it is well known that the intersection of the normal bundle and the tangent bundle of a submanifold of a semi-Riemannian manifold may be not trivial, so it is more difficult and interesting to study the geometry of lightlike submanifolds than nondegenerate submanifolds.

Rajasthan ka sabse garm jilaThe two standard methods to deal with the above difficulties were developed by Kupeli [ 2 ] and Duggal-Bejancu [ 34 ], respectively.

The study of CR-lightlike submanifolds of an indefinite Kaehler manifold was initiated by Duggal-Bejancu [ 3 ]. Since the book was published, many geometers investigated the lightlike submanifolds of indefinite Kaehler manifolds by generalizing the CR-lightlike submanifold [ 3 ], SCR-lightlike submanifolds [ 5 ] to GCR-lightlike submanifolds [ 6 ], and discussing the integrability and umbilication of these lightlike submanifolds.

We also refer the reader to [ 7 ] for invariant lightlike submanifolds and to [ 8 ] for totally real lightlike submanifolds of indefinite Kaehler manifolds. On the other hand, after Duggal-Sahin introduced screen real lightlike submanifolds and contact screen CR-lightlike submanifolds [ 9 ] of indefinite Sasakian manifolds by studying the integrability of distributions and the geometry of leaves of distributions as well as other properties of this submanifolds, the generalized CR-lightlike submanifold which contains contact CR and SCR-lightlike submanifolds were introduced in [ 4 ].

However, all these submanifolds of indefinite Sasakian manifolds mentioned above have the same geometric conditionwhere is the almost contact structure on indefinite Sasakian manifolds, is the radical distribution, and is the tangent bundle. The purpose of this paper is to generalize the radical and transversal lightlike submanifolds of indefinite Sasakian manifolds by introducing generalized transversal lightlike submanifolds.

The paper is arranged as follows. In Section 2we give the preliminaries of lightlike geometry of Sasakian manifolds needed for this paper. In Section 3we introduce the generalized transversal lightlike submanifolds and obtain a characterization theorem for such lightlike submanifolds. Section 4 is devoted to discuss the integrability and geodesic foliation of distributions of generalized transversal lightlike submanifolds.

In Section 5we investigate the geometry of totally contact umbilical generalized transversal lightlike submanifolds and obtain a classification theorem for such lightlike submanifolds. A submanifold of dimension immersed in a semi-Riemannian manifold of dimension is called a lightlike submanifold if the metric induced from ambient space is degenerate and its radical distribution is of rankwhere and.

It is well known that the radical distribution is given bywhere is called normal bundle of in. Thus there exist the nondegenerate complementary distribution and of in andrespectively, which are called the screen and screen transversal distribution onrespectively. Thus we have where denotes the orthogonal direct sum. Considering the orthogonal complementary distribution of init is easy to see that is a subbundle of.

As is a nondegenerate subbundle ofthe orthogonal complementary distribution of in is also a nondegenerate distribution. Clearly, is a subbundle of. Since for any local basis ofthere exists a local null frame of sections with values in the orthogonal complement of in such that andit follows that there exists a lightlike transversal vector bundle locally spanned by see [ 3 ].

Then we have that.We first prove some results on invariant lightlike submanifolds of indefinite Sasakian manifolds. Then, we introduce a general notion of contact Cauchy-Riemann CR lightlike submanifolds and study the geometry of leaves of their distributions.

We also study a class, namely, contact screen Cauchy-Riemann SCR lightlike submanifolds which include invariant and screen real subcases. Finally, we prove characterization theorems on the existence of contact SCR, screen real, invariant, and contact CR minimal lightlike submanifolds. Duggal and B. This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We will be providing unlimited waivers of publication charges for accepted articles related to COVID Sign up here as a reviewer to help fast-track new submissions. Journal overview. Special Issues. Duggal 1 and B. Academic Editor: Ingo Witt. Received 17 Feb Revised 31 Oct Accepted 24 Jan Published 25 Mar Abstract We first prove some results on invariant lightlike submanifolds of indefinite Sasakian manifolds. References K. Duggal and A.

Nazaikinskii, V. Shatalov, and B. Kang, S. Jung, B.In this paper, we introduce and study a new class of CR-lightlike submanifold of an indefinite nearly Sasakian manifold, called quasi generalized Cauchy—Riemann QGCR lightlike submanifold. We give some characterization theorems for the existence of QGCR-lightlike submanifolds and finally derive necessary and sufficient conditions for some distributions to be integrable. Download to read the full article text.

Ahmad, M.

Convince chiara to investigate together answersBlair D. Kodai Math. Blair, D. In: Progress in Mathematics, vol. Birkhauser Boston, Inc. Bonome A.

**Cauchy Riemann Equation II Numericals #3**

Calin, C. Cappelletti-Montano, B. Duggal, K. In: Mathematics and its Applications. Kluwer Academic Publishers, Dordrecht World Scientific Publishing Co. Ltd, Singapore In: Frontiers in Mathematics. Duggal K. Acta Math.

Mesa early learning networkGoogle Scholar. Jin D. Korean Math. Massamba F. Slovaca 62 2— Cuza Iasi.

Police brutality articlesYano, K. In: Series in Pure Mathematics, vol.We study a class of submanifolds, called Generalized Cauchy-Riemann GCR lightlike submanifolds of indefinite Sasakian manifolds as an umbrella of invariant, screen real, contact CR lightlike subcases [8] and real hypersurfaces [9]. We prove existence and non-existence theorems and a characterization theorem on minimal GCR-lightlike submanifolds.

This is a preview of subscription content, log in to check access. Rent this article via DeepDyve. Bejan and K. Duggal, Global lightlike manifolds and harmonicity, Kodai Math. Google Scholar. Duggal, Spacetime manifolds and contact structures, Int. Duggal and A. Duggal and B. Sahin, Lightlike submanifolds of indefinite Sasakian manifolds, Int. ID21 pp.

Kang, S. Jung, B. Kim, H. Pak and J. Pak, Lightlike hypersurfaces of indefinite Sasakian manifolds, Indian J.

## Lightlike hypersurfaces of metallic semi-Riemannian manifolds

Pure and Appl. Download references. Correspondence to K. Reprints and Permissions. Duggal, K. Generalized Cauchy-Riemann lightlike submanifolds of indefinite Sasakian manifolds. Acta Math Hung45—58 Download citation. Received : 16 November Accepted : 18 December Published : 04 June Issue Date : January Search SpringerLink Search. Abstract We study a class of submanifolds, called Generalized Cauchy-Riemann GCR lightlike submanifolds of indefinite Sasakian manifolds as an umbrella of invariant, screen real, contact CR lightlike subcases [8] and real hypersurfaces [9].

References [1] A. Sahin Authors K. Duggal View author publications. View author publications. Rights and permissions Reprints and Permissions. About this article Cite this article Duggal, K.Search for more papers by this author. In our paper, we introduce and study lightlike hypersurfaces of a metallic semi-Riemannian manifold. We examine some geometric properties of invariant lightlike hypersurfaces.

## Generalized Transversal Lightlike Submanifolds of Indefinite Sasakian Manifolds

We show that the induced structure on an invariant lightlike hypersurface is also metallic. We also define screen semi-invariant lightlike hypersurfaces, investigate integrability conditions for the distributions and give some examples.

Downloaded 20 times. Received 18 April Accepted 7 September System Upgrade on Fri, Jun 26th, at 5pm ET During this period, our website will be offline for less than an hour but the E-commerce and registration of new users may not be available for up to 4 hours. For online purchase, please visit us again. Contact us at customercare wspc. View article.

Keywords: Metallic structure lightlike hypersurface screen semi-invariant lightlike hypersurface. References 1. Duggal and A. Duggal and B.

7-1/2l x 7-1/2w x 3h cotton waffle zip pouch w/ tassel, 4GallowayLecture notes on spacetime geometryBeijing Int. Research Cent. Google Scholar 6. Hretcanu and M. CrasmareanuApplications of the golden ratio on Riemannian manifoldsTurk.

Acet, S. Google Scholar 9. World J. Google Scholar Ovidius Const. Gunes and B. Poyraz and E. AcetLightlike hypersurfaces of para-Sasakian space formGulf. GunesLightlike surfaces with planar normal sections in Minkowski 3-SpacesInt.

Stakhov and B.

Goldberg and K. YanoPolynomial structures on manifoldsKodai Math.Publication Date : December 31, Hacettepe Journal of Mathematics and Statistics. YearVolumeIssuePages 1 - 9 Zotero Mendeley EndNote. Abstract In this article, we examine the term of screen pseudo-slant lightlike submanifolds of a golden semi-Riemannian manifold. Also, we obtain an example. We give some characterizations about the geometry of such submanifolds.

Keywords Semi-Riemannian manifold, Golden ratio, Lightlike submanifold. References [1] M. Crasmareanu and C. Gezer, N. Cengiz and A. Akyol, Golden maps between golden Riemannian manifolds andconstancy of certain maps, Math Commun. Duggal and A. Bejancu, Lightlike submanifolds of semi-Riemannian manifoldsand applications, Mathematics and Its Applications. Kluwer Publisher, Duggal and B. Article ID Acet, S. Cuza Iasi N. Chen, Geometry of slant submanifolds, Leuven, Katholike Univ.

Carriazo, New developments in slant submanifolds theory, New Delhi, India, Hacettepe Journal of Mathematics and Statistics, DOI: Vancouver Acet B. Full Text File. Authors of the Article. Acet, B. Hacettepe Journal of Mathematics and Statistics :

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