CR- Submanifoldsof a Nearly Hyperbolic Cosymplectic Manifold

IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 6, Issue 3 (May. - Jun. 2013), PP 74-77 www.iosrjournals.org CR- Submanifoldsof a Nearly Hyperbolic Cosymplectic Manifold Mobin Ahmad and Kashif Ali Department of Mathematics, Integral University, Kursi Road, Lucknow-226026, India. Abstract:In the present paper, we study some properties of CR-submanifolds of a nearly hyperbolic cosymplectic manifold. We also obtain some results on 𝜉 −horizontal and 𝜉 −vertical CR- submanifolds of a nearly hyperboliccosymplectic manifold. Keywords: CR-submanifolds, nearlyhyperbolic cosymplectic manifold, totally geodesic,parallel distribution. I. Introduction The notion of CR-submanifolds of Kaehler manifold was introduced and studied by A. Bejancu in ([1], [2]). Since then, several papers on Kaehler manifolds were published. CR-submanifolds of Sasakian manifold was studied by C.J. Hsu in [3] and M. Kobayashi in [4].Later, several geometers (see, [5], [6] [7], [8] [9], [10]) enriched the study of CR-submanifolds of almost contact manifolds. On the other hand,almost hyperbolic (𝑓, 𝑔, 𝜂, 𝜉)-structure was defined and studied by Upadhyay and Dube in [11]. Dube and Bhatt studied CRsubmanifolds of trans-hyperbolic Sasakian manifold in [12]. In this paper, we study some properties of CRsubmanifolds of a nearly hyperbolic cosymplectic manifold. The paper is organized as follows. In section 2, we give a brief description of nearly hyperbolic cosymplectic manifold.In section 3, some properties of CR-submanifolds of nearly hyperbolic cosymplectic manifold are investigated. In section 4, some results on parallel distribution on ξ −horizontal and ξ −vertical CR- submanifolds of a nearly cosymplectic manifold are obtained. II. Nearly Hyperbolic Cosymplectic manifold Let𝑀 be an 𝑛-dimensional almost hyperbolic contact metric manifold with the almost hyperbolic contact metric 𝜙, 𝜉, 𝜂, 𝑔 - structure, where a tensor 𝜙 of type 1,1 a vector field 𝜉, called structure vector field and 𝜂, the dual 1-form of 𝜉 satisfying the followings: 𝜙 2 𝑋 = 𝑋 + 𝜂 𝑋 𝝃, 𝑔 𝑋, 𝜉 = 𝜂 𝑋 ,(2.1) 𝜂 𝜉 = −1, 𝜙 𝜉 = 0, 𝜂𝑜𝜙 = 0, (2.2) 𝑔 𝜙𝑋, 𝜙𝑌 = −𝑔 𝑋, 𝑌 − 𝜂 𝑋 𝜂 𝑌 (2.3) for any 𝑋, 𝑌 tangent to𝑀 [11]. In this case 𝑔 𝜙𝑋, 𝑌 = −𝑔 𝑋, 𝜙𝑌 .(2.4) An almost hyperbolic contact metric 𝜙, 𝜉, 𝜂, 𝑔 -structure on𝑀 is called nearly hyperbolic cosymplecticstructure if and only if ∇𝑋 𝜙 𝑌 + ∇𝑌 𝜙 𝑋 = 0,(2.5) ∇𝑋 𝜉 = 0(2.6) for all 𝑋, 𝑌 tangent to 𝑀 and Riemannian Connection ∇. III. CR-Submanifolds of Nearly Hyperbolic Cosymplectic Manifold Let 𝑀 be a submanifold immersed in 𝑀. We assume that the vector field 𝜉 is tangent to 𝑀. Then 𝑀 is called a CR-submanifold [13] of 𝑀 if there exist two orthogonal differentiable distributions 𝐷 and 𝐷⊥ on 𝑀 satisfying (i) 𝑇𝑀 = 𝐷⨁𝐷⊥ , (ii) the distribution 𝐷 is invariant by 𝜙, that is, 𝜙𝐷𝑋 = 𝐷𝑋 for each 𝑋𝜖𝑀, (iii) the distribution 𝐷⊥ is anti-invariant by 𝜙,that is, 𝜙𝐷𝑋⊥ ⊂ 𝑇𝑋 𝑀⊥ for each𝑋𝜖𝑀, where𝑇𝑀𝑎𝑛𝑑𝑇 ⊥ 𝑀 be the Lie algebra of vector fields tangential to 𝑀 and normal to 𝑀 respectively. If dim 𝐷𝑥⊥ = 0 (𝑟𝑒𝑠𝑝. , dim 𝐷𝑥 = 0), then the CR-submanifold is called an invariant (resp., anti-invariant) submanifold. The distribution 𝐷 𝑟𝑒𝑠𝑝. , 𝐷⊥ is called the horizontal (resp., vertical)distribution. Also, the pair 𝐷, 𝐷⊥ is called 𝜉 − 𝑕𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑟𝑒𝑠𝑝. , 𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑖𝑓𝜉𝑋 ∈ 𝐷𝑋 𝑟𝑒𝑠𝑝. , 𝜉𝑋 ∈ 𝐷𝑋⊥ . Let the Riemannian metric induced on 𝑀 is denoted by the same symbol 𝑔 and ∇ be the induced LeviCivita connection on 𝑁, then the Gauss and Weingarten formulas are given respectively by [14] ∇𝑋 𝑌 = ∇𝑋 𝑌 + 𝑕 𝑋, 𝑌 , (3.1) ∇𝑋 𝑁 = −𝐴𝑁 𝑋 + ∇𝑋⊥ 𝑁(3.2) www.iosrjournals.org 74 | Page Cr- Submanifolds of a Nearly Hyperbolic Cosymplectic Manifold for any 𝑋, 𝑌 ∈ 𝑇𝑀𝑎𝑛𝑑𝑁 ∈ 𝑇 ⊥ 𝑀, where ∇⊥ is a connection on the normal bundle 𝑇 ⊥ 𝑀, 𝑕 is the second fundamental form and 𝐴𝑁 is the Weingarten map associated with N as 𝑔 𝐴𝑁 𝑋, 𝑌 = 𝑔 𝑕 𝑋, 𝑌 , 𝑁 (3.3) for any 𝑥 ∈ 𝑀𝑎𝑛𝑑𝑋 ∈ 𝑇𝑥 𝑀. We write 𝑋 = 𝑃𝑋 + 𝑄𝑋,(3.4) where𝑃𝑋 ∈ 𝐷𝑎𝑛𝑑𝑄𝑋 ∈ 𝐷⊥ . Similarly, for 𝑁 normal to 𝑀, we have 𝜙𝑁 = 𝐵𝑁 + 𝐶𝑁,(3.5) where𝐵𝑁 𝑟𝑒𝑠𝑝. 𝐶𝑁 is the tangential component 𝑟𝑒𝑠𝑝. 𝑛𝑜𝑟𝑚𝑎𝑙𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡 of 𝜙𝑁. Lemma 3.1.Let 𝑀 be a CR- submanifold of a nearly hyperbolic cosymplectic manifold 𝑀.Then 𝜙𝑃 ∇𝑋 𝑌 + 𝜙𝑃 ∇𝑌 𝑋 = 𝑃∇𝑋 𝜙𝑃𝑌 + 𝑃∇𝑌 𝜙𝑃𝑋 − 𝑃𝐴𝜙𝑄𝑌 𝑋 − 𝑃𝐴𝜙𝑄𝑋 𝑌,(3.6) 2𝐵𝑕 𝑋, 𝑌 = 𝑄∇𝑋 𝜙𝑃𝑌 + 𝑄∇𝑌 𝜙𝑃𝑋 − 𝑄𝐴𝜙𝑄𝑋 𝑌 − 𝑄𝐴𝜙𝑄𝑌 𝑋,(3.7) 𝜙𝑄∇𝑋 𝑌 + 𝜙𝑄∇𝑌 𝑋 + 2𝐶𝑕 𝑋, 𝑌 = 𝑕 𝑋, 𝜙𝑃𝑌 + 𝑕 𝑌, 𝜙𝑃𝑋 + ∇𝑋⊥ 𝜙𝑄𝑌 + ∇⊥𝑌 𝜙𝑄𝑋(3.8) 𝑓𝑜𝑟𝑎𝑛𝑦𝑋, 𝑌 ∈ 𝑇𝑀. Proof.Using (2.4), (2.5) and (2.6), we get ∇𝑋 𝜙 𝑌 + 𝜙 ∇𝑋 𝑌 + 𝜙𝑕 𝑋, 𝑌 = ∇𝑋 𝜙𝑃𝑌 + 𝑕 𝑋, 𝜙𝑃𝑌 − 𝐴𝜙𝑄𝑌 𝑋 + ∇𝑋⊥ 𝜙𝑄𝑌. Interchanging 𝑋&𝑌and adding, we have ∇𝑋 𝜙 𝑌 + ∇𝑌 𝜙 𝑋 + 𝜙 ∇𝑋 𝑌 + 𝜙 ∇𝑌 𝑋 + 2𝜙𝑕 𝑋, 𝑌 = ∇𝑋 𝜙𝑃𝑌 + ∇𝑌 𝜙𝑃𝑋 + 𝑕 𝑋, 𝜙𝑃𝑌 + 𝑕 𝑌, 𝜙𝑃𝑋 −𝐴𝜙𝑄𝑌 𝑋 − 𝐴𝜙𝑄𝑋 𝑌 + ∇𝑋⊥ 𝜙𝑄𝑌 + ∇⊥𝑌 𝜙𝑄𝑋. Using (2.5) in above equation, we have 𝜙𝑃 ∇𝑋 𝑌 + 𝜙𝑄 ∇𝑋 𝑌 + 𝜙𝑃 ∇𝑌 𝑋 + 𝜙𝑄 ∇𝑌 𝑋 + 2𝐵𝑕 𝑋, 𝑌 +2𝐶𝑕 𝑋, 𝑌 = 𝑃∇𝑋 𝜙𝑃𝑌 + 𝑄∇𝑌 𝜙𝑃𝑋 + 𝑕 𝑋, 𝜙𝑃𝑌 +𝑕 𝑌, 𝜙𝑃𝑋 − 𝑃𝐴𝜙𝑄𝑌 𝑋 − 𝑄𝐴𝜙𝑄𝑌 𝑋 − 𝑃𝐴𝜙𝑄𝑋 𝑌 −𝑄𝐴𝜙𝑄𝑋 𝑌 + ∇𝑋⊥ 𝜙𝑄𝑌 + ∇⊥𝑌 𝜙𝑄𝑋.(3.9) Comparing the horizontal, vertical and normal components, we get (3.6) — (3.8). Hence the Lemma is proved. ⧠ Lemma 3.2.Let 𝑀 be a CR- submanifold of a nearly hyperbolic cosymplectic manifold 𝑀. Then 2 ∇𝑋 𝜙 𝑌 = ∇𝑋 𝜙𝑌 − ∇𝑌 𝜙𝑋 + 𝑕 𝑋, 𝜙𝑌 − ∇𝑌 𝜙𝑋 − 𝜙 𝑋, 𝑌 (3.10) 𝑓𝑜𝑟𝑎𝑛𝑦𝑋, 𝑌 ∈ 𝐷. Proof.From Gauss formula (3.1), we have ∇𝑋 𝜙𝑌 − ∇𝑌 𝜙𝑋 = ∇𝑋 𝜙𝑌 + 𝑕 𝑋, 𝜙𝑌 − ∇𝑌 𝜙𝑋 − 𝑕 𝑌, 𝜙𝑋 .(3.11) Also, we have ∇𝑋 𝜙𝑌 − ∇𝑌 𝜙𝑋 = ∇𝑋 𝜙 𝑌 − ∇𝑌 𝜙 𝑋 + 𝜙 𝑋, 𝑌 . (3.12) From (3.11) and (3.12), we get ∇𝑋 𝜙 𝑌 − ∇𝑌 𝜙 𝑋 = ∇𝑋 𝜙𝑌 + 𝑕 𝑋, 𝜙𝑌 − ∇𝑌 𝜙𝑋 − 𝑕 𝑌, 𝜙𝑋 − 𝜙 𝑋, 𝑌 . Adding (3.15) and (2.5), we obtain 2 ∇𝑋 𝜙 𝑌 = ∇𝑋 𝜙𝑌 + 𝑕 𝑋, 𝜙𝑌 − ∇𝑌 𝜙𝑋 − 𝑕 𝑌, 𝜙𝑋 − 𝜙 𝑋, 𝑌 . Hence the Lemma is proved. ⧠ Lemma 3.3.Let 𝑀 be a CR- submanifold of a nearly hyperbolic cosymplectic manifold 𝑀. Then 2 ∇𝑋 𝜙 𝑌 = 𝐴𝜙𝑋 𝑌 − 𝐴𝜙𝑌 𝑋 + ∇𝑋⊥ 𝜙𝑌 − ∇⊥𝑌 𝜙𝑋 − 𝜙 𝑋, 𝑌 (3.14) 𝑓𝑜𝑟𝑎𝑛𝑦𝑋, 𝑌 ∈ 𝐷⊥ . Proof.From Weingarten formula (3.2), we have ∇𝑋 𝜙𝑌 − ∇𝑌 𝜙𝑋 = 𝐴𝜙𝑋 𝑌 − 𝐴𝜙𝑌 𝑋 + ∇𝑋⊥ 𝜙𝑌 − ∇⊥𝑌 𝜙𝑋.(3.15) Also, ∇𝑋 𝜙𝑌 − ∇𝑌 𝜙𝑋 = ∇𝑋 𝜙 𝑌 − ∇𝑌 𝜙 𝑋 + 𝜙 𝑋, 𝑌 .(3.16) From (3.15) and (3.16), we get ∇𝑋 𝜙 𝑌 − ∇𝑌 𝜙 𝑋 = 𝐴𝜙𝑋 𝑌 − 𝐴𝜙𝑌 𝑋 + ∇𝑋⊥ 𝜙𝑌 − ∇⊥𝑌 𝜙𝑋 − 𝜙 𝑋, 𝑌 . (3.17) Adding (3.17) and (2.5), we obtain 2 ∇𝑋 𝜙 𝑌 = 𝐴𝜙𝑋 𝑌 − 𝐴𝜙𝑌 𝑋 + ∇𝑋⊥ 𝜙𝑌 − ∇⊥𝑌 𝜙𝑋 − 𝜙 𝑋, 𝑌 . Hence the Lemma is proved. ⧠ Lemma 3.4.Let 𝑀 be a CR- submanifold of a nearly hyperbolic cosymplectic manifold 𝑀. Then 2 ∇𝑋 𝜙 𝑌 = −𝐴𝜙𝑌 𝑋 + ∇𝑋⊥ 𝜙𝑌 − ∇𝑌 𝜙𝑋 − 𝑕 𝑌, 𝜙𝑋 − 𝜙 𝑋, 𝑌 (3.18) 𝑓𝑜𝑟𝑎𝑛𝑦𝑋 ∈ 𝐷𝑎𝑛𝑑𝑌 ∈ 𝐷⊥ . www.iosrjournals.org (3.13) 75 | Page Cr- Submanifolds of a Nearly Hyperbolic Cosymplectic Manifold Proof.Using Gauss and Weingarten formula for ∈ 𝐷𝑎𝑛𝑑𝑌 ∈ 𝐷⊥ , we have ∇𝑋 𝜙𝑌 − ∇𝑌 𝜙𝑋 = −𝐴𝜙𝑌 𝑋 + ∇𝑋⊥ 𝜙𝑌 − ∇𝑌 𝜙𝑋 + 𝑕 𝑌, 𝜙𝑋 . (3.19) Also, we have ∇𝑋 𝜙𝑌 − ∇𝑌 𝜙𝑋 = ∇𝑋 𝜙 𝑌 − ∇𝑌 𝜙 𝑋 + 𝜙 𝑋, 𝑌 .(3.20) By virtue of (3.19) and (3.20), we get ∇𝑋 𝜙 𝑌 − ∇𝑌 𝜙 𝑋 = −𝐴𝜙𝑌 𝑋 + ∇𝑋⊥ 𝜙𝑌 − ∇𝑌 𝜙𝑋 + 𝑕 𝑌, 𝜙𝑋 − 𝜙 𝑋, 𝑌 .(3.21) Adding (3.21) and (2.5), we obtain 2 ∇𝑋 𝜙 𝑌 = −𝐴𝜙𝑌 𝑋 + ∇𝑋⊥ 𝜙𝑌 − ∇𝑌 𝜙𝑋 + 𝑕 𝑌, 𝜙𝑋 − 𝜙 𝑋, 𝑌 . Hence the Lemma is proved. ⧠ IV. Parallel Distribution Definition4.1.The horizontal (resp., vertical) distribution 𝐷(𝑟𝑒𝑠𝑝. , 𝐷⊥ ) is said to be parallel [13] with respect to the connectionon 𝑀𝑖𝑓∇𝑋 𝑌 ∈ 𝐷 (𝑟𝑒𝑠𝑝. , ∇𝑍 𝑊 ∈ 𝐷⊥ ) for any vector field 𝑋, 𝑌 ∈ 𝐷 𝑟𝑒𝑠𝑝. , 𝑊, 𝑍 ∈ 𝐷⊥ . Theorem 4.2.Let 𝑀 be a𝜉 − 𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙 CR-submanifold of a nearly hyperbolic cosymplecticmanifold𝑀. If the horizontal distribution 𝐷 is parallel,then 𝑕 𝑋, 𝜙𝑌 = 𝑕 𝑌, 𝜙𝑋 . (4.1) 𝑓𝑜𝑟𝑎𝑛𝑦𝑋, 𝑌 ∈ 𝐷. Proof.Using parallelism of horizontal distribution D, we have ∇𝑋 𝜙𝑌 ∈ 𝐷𝑎𝑛𝑑∇𝑌 𝜙𝑋 ∈ 𝐷𝑓𝑜𝑟𝑎𝑛𝑦𝑋, 𝑌 ∈ 𝐷.(4.2) Now, by virtue of (3.7), we have 𝐵𝑕 𝑋, 𝑌 = 0.(4.3) From (3.5) and (4.3), we get 𝜙𝑕 𝑋, 𝑌 = 𝐶𝑕 𝑋, 𝑌 (4.4) 𝑓𝑜𝑟𝑎𝑛𝑦𝑋, 𝑌 ∈ 𝐷. From (3.8), we have 𝑕 𝑋, 𝜙𝑌 + 𝑕 𝑌, 𝜙𝑋 = 2𝐶𝑕 𝑋, 𝑌 (4.5) 𝑓𝑜𝑟𝑎𝑛𝑦𝑋, 𝑌 ∈ 𝐷. Replacing 𝑋𝑏𝑦𝜙𝑋𝑖𝑛 (4.5) and using (4.4), we have 𝑕 𝜙𝑋, 𝜙𝑌 + 𝑕 𝑌, 𝑋 = 𝜙𝑕 𝜙𝑋, 𝑌 . (4.6) Now, replacing 𝑌𝑏𝑦𝜙𝑌 in (4.6), we get 𝑕 𝑋, 𝑌 + 𝑕 𝜙𝑌, 𝜙𝑋 = 𝜙𝑕 𝑋, 𝜙𝑌 .(4.7) Thus from (4.6) and (4.7), we find 𝑕 𝑋, 𝜙𝑌 = 𝑕 𝑌, 𝜙𝑋 . Hence the Theorem is proved. ⧠ Theorem 4.3.Let𝑀 be aCR-submanifold of a nearly hyperbolic cosymplecticmanifold𝑀. If the distribution 𝐷⊥ is parallel with respect to the connection on M, then 𝐴𝜙𝑌 𝑍 + 𝐴𝜙𝑍 𝑌 ∈ 𝐷⊥ 𝑓𝑜𝑟𝑎𝑛𝑦𝑌, 𝑍 ∈ 𝐷⊥ . ⊥ Proof.Let 𝑌, 𝑍 ∈ 𝐷 , then using (3.1) and (3.2), we have −𝐴𝜙𝑍 𝑌 − 𝐴𝜙𝑌 𝑍 + ∇⊥𝑌 𝜙𝑍 + ∇⊥𝑍 𝜙𝑌 = 𝜙 ∇𝑌 𝑍 + 𝜙∇𝑍 𝑌 + 2𝜙𝑕 𝑌, 𝑍 .(4.8) Taking inner product with 𝑋 ∈ 𝐷𝑖𝑛 4.8 , we get 𝑔 𝐴𝜙𝑌 𝑍 + 𝐴𝜙𝑍 𝑌 = 0 which is equivalent to 𝐴𝜙𝑌 𝑍 + 𝐴𝜙𝑍 𝑌 ∈ 𝐷⊥ 𝑓𝑜𝑟𝑎𝑛𝑦𝑌, 𝑍 ∈ 𝐷⊥ . Definition 4.4.A CR-submanifold is said to be mixed-totally geodesic if𝑕 𝑋, 𝑍 = 0 𝑓𝑜𝑟𝑎𝑙𝑙𝑋 ∈ 𝐷𝑎𝑛𝑑𝑍 ∈ 𝐷⊥ . Lemma 4.5.Let 𝑀 be a CR-submanifold of a nearly hyperbolic cosymplecticmanifold 𝑀. Then 𝑀is mixed totally geodesic if and only if 𝐴𝑁 𝑋 ∈ 𝐷 for all 𝑋 ∈ 𝐷. Definition 4.6.A Normal vector field 𝑁 ≠ 0 is called𝐷 − 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 normal section if ∇𝑋⊥ 𝑁 = 0for all 𝑋 ∈ 𝐷. Theorem 4.7.Let 𝑀 be a mixed totally geodesic CR-submanifold of a nearly hyperbolic cosymplecticmanifold𝑀. Then the normal section𝑁 ∈ 𝜙𝐷⊥ is 𝐷 − 𝑝𝑎𝑟𝑎𝑙𝑙𝑒l if and only if ∇𝑋 𝜙𝑁 ∈ 𝐷𝑓𝑜𝑟𝑎𝑙𝑙𝑋 ∈ 𝐷. Proof.Let 𝑁 ∈ 𝜙𝐷⊥ , then from (3.7), we have 𝑄∇𝑌 𝜙𝑋 = 0. In particular, we have 𝑄∇𝑌 𝜙𝑋 = 0. Using it in (3.8), we have 𝜙𝑄∇𝑋 𝜙𝑁 = ∇𝑋⊥ 𝑁. (4.9) www.iosrjournals.org 76 | Page Cr- Submanifolds of a Nearly Hyperbolic Cosymplectic Manifold Thus, if the normal section 𝑁 ≠ 0 is D-parallel, then using ‘definition 4.6’and (4.9), we get 𝜙∇𝑋 𝜙𝑁 = 0 which is equivalent to ∇𝑋 𝜙𝑁 ∈ 0𝑓𝑜𝑟𝑎𝑙𝑙𝑋 ∈ 𝐷. The converse part easily follows from (4.9). This completes the proof of the theorem. References [1]. [2]. [3]. [4]. [5]. [6]. [7]. [8]. [9]. [10]. [11]. [12]. [13]. [14]. A.Bejancu, CR- submanifolds of a Kaehler manifold I, Proc. Amer. Math. Soc. 69 (1978), 135-142. CR- submanifolds of a Kaehler manifold II, Trans. Amer. Math. Soc. 250 (1979), 333-345. C.J. Hsu, On CR-submanifolds of Sasakian manifolds I, Math. Research Center Reports, Symposium Summer 1983, 117-140. M. Kobayash, CR-submanifolds of a Sasakian manifold, Tensor N.S. 35 (1981), 297-307. A. Bejancuand N.Papaghuic, CR-submanifolds of Kenmotsu manifold, Rend. Mat.7 (1984), 607-622. Lovejoy S.K. 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