ISSN 1995-0802, Lobachevskii Journal of Mathematics, 2013, Vol. 34, No. 1, pp. 76–84. c  Pleiades Publishing, Ltd., 2013. Integrability of Distributions in GCR GCR GCR-Lightlike Submanifolds of Indeﬁnite Cosymplectic Manifolds Varun Jain 1* , Rakesh Kumar 2** , and Rakesh Kumar Nagaich 3*** (Submitted by P. N. Ivanshin) 1 Multani Mal Modi College, Patiala, Punjab, India 2 University College of Engineering, Punjabi University, Patiala, India 3 Department of Mathematics, Punjabi University, Patiala, India Abstract—We give necessary and suﬃcient conditions for the integrability of various distributions of GCR-lightlike submanifold of an indeﬁnite Cosymplectic manifold. We also ﬁnd the conditions for each leaf of holomorphic distribution and radical distribution to be totally geodesic in submanifold. DOI: 10.1134/S1995080213010071 Keywords and phrases: Indeﬁnite Cosymplectic manifolds, lightlike submanifolds, GCR- lightlike submanifolds. 1. INTRODUCTION The geometry of CR-lightlike submanifolds of indeﬁnite Kaehler manifolds was introduced and studied by Duggal and Bejancu [4] as a generalization of CR-submanifolds of Kaehler manifolds [1, 10]. But this class of submanifolds exclude the complex and totally real submanifolds as subcases. Later on, Duggal and Sahin [5] introduced Screen Cauchy-Riemann (SCR)-lightlike submanifolds of indeﬁnite Kaehler manifolds. Since there was no inclusion relation between CR and Screen CR cases therefore Duggal and Sahin [6] introduced a new class called generalized Cauchy-Riemann (GCR)-lightlike submanifolds of indeﬁnite Kaehler manifolds. Duggal and Sahin [7] also introduced the theory of contact CR and contact SCR-lightlike submanifolds of indeﬁnite Sasakian manifolds. To ﬁnd an umbrella of invariant, screen real, contact CR-lightlike submanifolds and real hypersurfaces, Duggal and Sahin [8] introduced a new class of submanifolds called generalized Cauchy-Riemann (GCR)-lightlike submanifolds of indeﬁnite Sasakian manifolds. Since contact geometry has vital role in the theory of diﬀerential equations, optics and phase spaces of a dynamical system, therefore contact geometry with deﬁnite and indeﬁnite metric becomes the topic of main discussion. In [9], Sangeet et.al. established the conditions for the integrability of various distributions of GCR- lightlike submanifolds of indeﬁnite Kaehler manifolds and obtained conditions for the distributions to deﬁne totally geodesic foliations in submanifolds. In this paper, initially we deﬁne GCR-lightlike submanifolds of indeﬁnite Cosymplectic manifolds and then give necessary and suﬃcient conditions for the integrability of various distributions of GCR-lightlike submanifold of an indeﬁnite Cosymplectic manifold. We also ﬁnd the conditions for each leaf of holomorphic distribution and radical distribution to be totally geodesic in submanifold. * E-mail: varun82jain@gmail.com ** E-mail: dr_rk37c@yahoo.co.in *** E-mail: nagaichrakesh@yahoo.com 76