CR-SUBMANIFOLDS OF A NEARLY TRANS-SASAKIAN MANIFOLD

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1 IJMMS 31:3 2002) PII. S Hindawi Publishing Corp. CR-SUBMANIFOLDS OF A NEARLY TRANS-SASAKIAN MANIFOLD FALLEH R. AL-SOLAMY Received 26 September 2001 This paper considers the study of CR-submanifolds of a nearly trans-sasakian manifold, generalizing the results of trans-sasakian manifolds and thus those of Sasakian manifolds Mathematics Subject Classification: 53C Introduction. In 1978, Bejancu introduced the notion of CR-submanifold of a Kaehler manifold [1]. Since then several papers on CR-submanifolds of Kaehler manifold have been published. On the other hand, CR-submanifolds of a Sasakian manifold have been studied by Kobayashi [6], Shahid et al. [10], Yano and Kon [11], and others. Bejancu and Papaghuic [2] studied CR-submanifolds of a Kenmotsu manifold. In 1985, Oubina introduced a new class of almost contact metric manifold known as trans- Sasakian manifold [7]. This class contains α-sasakian and β-kenmotsu manifold [5]. Geometry of CR-submanifold of trans-sasakian manifold was studied by Shahid [8, 9]. A nearly trans-sasakian manifold [4] is a more general concept. The results of this paper are the generalization of the results obtained earlier by several authors, namely [6, 8, 9] and others. 2. Preliminaries. Let M be an n-dimensional almost contact metric manifold with an almost contact metric structure φ,ξ,η,g) satisfying [3] φ 2 X = X +ηx)ξ, ηξ) = 1, φ ξ = 0, gφx,φy) = gx,y) ηx)ηy ), gx,ξ) = ηx), 2.1) where X and Y are vector fields tangent to M. An almost contact metric structure φ,ξ,η,g) on M is called trans-sasakian if [7] X φ ) Y ) = α { gx,y)ξ ηy )X } +β { gφx,y)ξ ηy )φx }, 2.2) where α and β are nonzero constants, denotes the Riemannian connection of g on M, and we say that the trans-sasakian structure is of type α,β). Further, an almost contact metric manifold Mφ,ξ,η,g) is called nearly trans- Sasakian if [4] X φ ) Y )+ Y φ ) X) = α { 2gX,Y)ξ ηy )X ηx)y } β { ηy )φx +ηx)φy }. 2.3) It is clear that any trans-sasakian manifold, and thus any Sasakian manifold, satisfies the above relation.

2 168 FALLEH R. AL-SOLAMY Let M be an m-dimensional isometrically immersed submanifold of a nearly trans- Sasakian manifold M and denote by the same g the Riemannian metric tensor field induced on M from that of M. Definition 2.1. An m-dimensional Riemannian submanifold M of a nearly trans- Sasakian manifold M is called a CR-submanifold if ξ is tangent to M and there exists a differentiable distribution D : x M D x T x M such that 1) the distribution D x is invariant under φ, that is, φd x D x for each x M; 2) the complementary orthogonal distribution D : x M Dx T x M of D is anti-invariant under φ, thatis,φdx Tx M for all x M, where T x M and Tx M are the tangent space and the normal space of M at x, respectively. If dimd x = 0 resp., dimd x = 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-horizontal resp., vertical) if ξ x D x resp., ξ x D x )[6]. For any vector field X tangent to M, weput[6] where PX and QX belong to the distribution D and D. For any vector field N normal to M, weput[6] X = PX+QX, 2.4) φn = BN +CN, 2.5) where BN resp., CN) denotes the tangential resp., normal) component of φn. Let resp., ) be the covariant differentiation with respect to the Levi-Civita connection on M resp., M). The Gauss and Weingarten formulas for M are respectively given by X Y = X Y +hx,y ); X N = A N X + XN, 2.6) for X,Y TM and N T M, where h resp., A) is the second fundamental form resp., tensor) of M in M, and denotes the normal connection. Moreover, we have g hx,y ),N ) = g A N X,Y ). 2.7) 3. Some basic lemmas. First we prove the following lemma. Lemma 3.1. Let M be a CR-submanifold of a nearly trans-sasakian manifold M. Then P X φpy ) +P Y φpx ) P A φqx Y ) P A φqy X ) = φp X Y +φp Y X +2αgX,Y )Pξ αηy )PX αηx)py βηy )φpx +βηx)φpy, Q X φpy ) +Q Y φpx ) Q A φqx Y ) Q A φqy X ) = 2BhX,Y ) αηy )QX αηx)qy +2αgX,Y )Qξ, hx,φpy )+hy,φpx)+ XφQY + Y φqx = φq Y X +φq X Y +2ChX,Y ) βηy )φqx βηx)φqy 3.1) 3.2) 3.3) for X,Y TM.

3 CR-SUBMANIFOLDS OF A NEARLY TRANS-SASAKIAN From the definition of the nearly trans-sasakian manifold and using 2.4), 2.5), and 2.6), we get X φpy +hx,φpy ) A φqy X + X φqy φ X Y +hx,y ) ) + Y φpx +hy,φpx) A φqx Y + Y φqx φ Y X +hx,y ) ) = α { 2gX,Y)ξ ηy )X ηx)y } β { ηy )φx +ηx)φy } 3.4) for any X,Y TM. Now using 2.4) and equaling horizontal, vertical, and normal components in 3.4), we get the result. Lemma 3.2. Let M be a CR-submanifold of a nearly trans-sasakian manifold M. Then 2 X φ ) Y ) = X φy Y φx +hx,φy ) hy,φx) φ[x,y ] +α { 2gX,Y)ξ ηy )X ηx)y } β { ηy )φx +ηx)φy } 3.5) for any X,Y D. By Gauss formula 2.6), we get X φy Y φx = X φy +hx,φy ) Y φx hy,φx). 3.6) Also, we have X φy Y φx = X φ ) Y ) Y φ ) X)+φ[X,Y ]. 3.7) From 3.6) and3.7), we get X φ ) Y ) Y φ ) X) = X φy +hx,φy ) Y φx hy,φx) φ[x,y ]. 3.8) Also for nearly trans-sasakian manifolds, we have X φ ) Y )+ Y φ ) X) = α { 2gX,Y)ξ ηy )X ηx)y } β { ηy )φx +ηx)φy }. 3.9) Combining 3.8) and3.9), the lemma follows. In particular, we have the following corollary. Corollary 3.3. manifold, then Let M be a ξ-vertical CR-submanifold of a nearly trans-sasakian 2 X φ ) Y ) = X φy Y φx +hx,φy ) hy,φx) φ[x,y ]+2αgX,Y )ξ 3.10) for any X,Y D. Similarly, by Weingarten formula 2.6), we get the following lemma.

4 170 FALLEH R. AL-SOLAMY Lemma 3.4. Let M be a CR-submanifold of a nearly trans-sasakian manifold M, then 2 Y φ ) Z) = A φy Z A φz Y + Y φz Z φy φ[y,z] +α { 2gY,Z)ξ ηy )Z ηz)y } β { ηy )φz +ηz)φy } 3.11) for any Y,Z D. Corollary 3.5. manifold, then Let M be a ξ-horizontal CR-submanifold of a nearly trans-sasakian 2 Y φ ) Z) = A φy Z A φz Y + Y φz Z φy φ[y,z]+2αgy,z)ξ 3.12) for any Y,Z D. Lemma 3.6. Let M be a CR-submanifold of a nearly trans-sasakian manifold M, then 2 X φ ) Y ) = α { 2gX,Y)ξ ηy )X ηx)y } β { ηy )φx +ηx)φy } A φy X ) XφY Y φx hy,φx) φ[x,y ] for any X D and Y D. 4. Parallel distributions Definition 4.1. The horizontal resp., vertical) distribution D resp., D )issaidto be parallel [1] with respect to the connection on M if X Y D resp., Z W D ) for any vector field X,Y D resp., W,Z D ). Now we prove the following proposition. Proposition 4.2. Let M be a ξ-vertical CR-submanifold of a nearly trans-sasakian manifold M. If the horizontal distribution D is parallel, then for all X,Y D. hx,φy ) = hy,φx) 4.1) Using parallelism of horizontal distribution D, we have X φy D, Y φx D for any X,Y D. 4.2) Thus using the fact that QX = QY = 0forY D, 3.2) gives BhX,Y ) = gx,y)qξ for any X,Y D. 4.3) Also, since then φhx,y ) = BhX,Y )+ChX,Y ), 4.4) φhx,y ) = gx,y)qξ+chx,y ) for any X,Y D. 4.5) Next from 3.3), we have hx,φy )+hy,φx) = 2ChX,Y ) = 2φhX,Y ) 2gX,Y)Qξ, 4.6)

5 CR-SUBMANIFOLDS OF A NEARLY TRANS-SASAKIAN for any X,Y D. Putting X = φx D in 4.6), we get hφx,φy )+h Y,φ 2 X ) = 2φhφX,Y ) 2gφX,Y )Qξ 4.7) or hφx,φy ) hy,x) = 2φhφX,Y ) 2gφX,Y)Qξ. 4.8) Similarly, putting Y = φy D in 4.6), we get hφy,φx) hx,y ) = 2φhX,φY ) 2gX,φY)Qξ. 4.9) Hence from 4.8) and4.9), we have φhx,φy ) φhy,φx) = gx,φy )Qξ gφx,y)qξ. 4.10) Operating φ on both sides of 4.10) and using φξ = 0, we get hx,φy ) = hy,φx) 4.11) for all X,Y D. Now, for the distribution D, we prove the following proposition. Proposition 4.3. Let M be a ξ-vertical CR-submanifold of a nearly trans-sasakian manifold M. If the distribution D is parallel with respect to the connection on M, then AφY Z +A φz Y ) D for any Y,Z D. 4.12) Let Y,Z D, then using Gauss and Weingarten formula 2.6), we obtain A φz Y + Y φz A φy Z + Z φy = φ Y Z +φ Z Y +2φhY,Z) +α { 2gX,Y)ξ ηy )Z ηz)y } β { ηy )φz +ηz)φy } 4.13) for any Y,Z D. Taking inner product with X D in 4.13), we get g A φy Z,X ) +g A φz Y,X ) = g Y Z,φX ) +g Z Y,φX ). 4.14) If the distribution D is parallel, then Y Z D and Z Y D for any Y,Z D. So from 4.14) weget g A φy Z,X ) +g A φz Y,X ) = 0 or g A φy Z +A φz Y,X ) = ) which is equivalent to AφY Z +A φz Y ) D for any Y,Z D, 4.16) and this completes the proof. Definition 4.4 [6]. A CR-submanifold is said to be mixed totally geodesic if hx,z) = 0 for all X D and Z D.

6 172 FALLEH R. AL-SOLAMY The following lemma is an easy consequence of 2.7). Lemma 4.5. Let M be a CR-submanifold of a nearly trans-sasakian manifold M. Then M is mixed totally geodesic if and only if A N X D for all X D. Definition 4.6 [6]. A normal vector field N 0 is called D-parallel normal section if XN = 0 for all X D. Now we have the following proposition. Proposition 4.7. Let M be a mixed totally geodesic ξ-vertical CR-submanifold of a nearly trans-sasakian manifold M. Then the normal section N φd is D-parallel if and only if X φn D for all X D. Let N φd. Then from 3.2) we have Q Y φx ) = 0 for any X D, Y D. 4.17) In particular, we have Q Y X) = 0. By using it in 3.3), we get XφQY = φq X Y or X N = φq X φn. 4.18) Thus, if the normal section N 0isD-parallel, then using Definition 4.6 and 4.18), we get φq X φn ) = ) which is equivalent to X φn D for all X D. The converse part easily follows from 4.18). 5. Integrability conditions of distributions. First we calculate the Nijenhuis tensor N φ X,Y ) on a nearly trans-sasakian manifold M. For this, first we prove the following lemma. Lemma 5.1. Let M be a nearly trans-sasakian manifold, then φx φ ) Y ) = 2αgφX,Y )ξ ηy )φx +βηy )X βηx)ηy )ξ ηx) Y ξ +φ Y φ ) X)+η Y X ) ξ 5.1) for any X,Y T M. From the definition of nearly trans-sasakian manifold M, we have φx φ ) Y ) = 2αgφX,Y )ξ ηy )φx +βηy )X βηy )ηx)ξ Y φ ) φx). 5.2) Also, we have Y φ ) φx) = Y φ 2 X φ Y φx = Y φ 2 X φ Y φx +φ φ Y X ) φ φ Y X ) = Y X +ηx) Y ξ φ Y φx φ Y X ) φ φ Y X ) = Y X +ηx) Y ξ φ Y φ ) X) Y X η Y X ) ξ. 5.3)

7 CR-SUBMANIFOLDS OF A NEARLY TRANS-SASAKIAN Using 5.3) in5.2), we get φx φ ) Y ) = 2αgφX,Y )ξ ηy )φx +βηy )X βηx)ηy )ξ ηx) Y ξ +φ Y φ ) X)+η Y X ) ξ 5.4) for any X,Y T M, which completes the proof of the lemma. On a nearly trans-sasakian manifold M, Nijenhuis tensor is given by N φ X,Y ) = φx φ ) Y ) φy φ ) X) φ X φ ) Y )+φ Y φ ) X) 5.5) for any X,Y T M. From 5.1) and5.5), we get N φ X,Y ) = 4αgφX,Y )ξ ηy )φx +ηx)φy +βηy )X βηx)y ηx) Y ξ +ηy ) X ξ +η Y X ) ξ 5.6) η X Y ) ξ +2φ Y φ ) X) 2φ X φ ) Y ). Thus using 2.3) in the above equation and after some calculations, we obtain N φ X,Y ) = 4αgφX,Y )ξ +2α 1)ηY )φx +2α+1)ηX)φY βηy )X 3βηX)Y +4βηX)ηY )ξ ηx) Y ξ +ηy ) X ξ 5.7) +η Y X ) ξ η X Y ) ξ +4φ Y φ ) X) for any X,Y T M. Now we prove the following proposition. Proposition 5.2. Let M be a ξ-vertical CR-submanifold of a nearly trans-sasakian manifold M. Then, the distribution D is integrable if the following conditions are satisfied: SX,Z) D, hx,φz) = hφx,z) 5.8) for any X,Z D. given by The torsion tensor SX,Y) of the almost contact structure φ,ξ,η,g) is Thus, we have SX,Y) = N φ X,Y )+2dηX,Y )ξ = N φ X,Y )+2gφX,Y )ξ. 5.9) SX,Y) = [φx,φy ] φ[φx,y ] φ[x,φy ]+2gφX,Y)ξ 5.10) for any X,Y TM. Suppose that the distribution D is integrable. So for X,Y D, Q[X,Y ] = 0 and η[x,y ]) = 0asξ D. If SX,Y) D, then from 5.7) and5.9) we have { ) 22α+1)gφX,Y )ξ+η [X,Y ] ξ+4 φ Y φx +φhy,φx)+q Y X +hx,y ) )} D 5.11)

8 174 FALLEH R. AL-SOLAMY or 22α+1)gφX,Y )Qξ+η [X,Y ] ) Qξ+4 φq Y φx+φhy,φx)+q Y X+hX,Y ) ) =0 5.12) for any X,Y D. Replacing Y by φz for Z D in the above equation, we get 22α+1)gφX,φZ)Qξ +4 φq φy φx +φhφz,φx)+q φz X +hx,φz) ) = ) Interchanging X and Z for X,Z D in 5.13) and subtracting these relations, we obtain φq[φx,φz]+q[x,φz]+hx,φz) hz,φx) = ) for any X,Z D and the assertion follows. Now, we prove the following proposition. Proposition 5.3. Let M be a CR-submanifold of a nearly trans-sasakian manifold M. Then A φy Z A φz Y = α ηy )Z ηz)y ) + 1 φp[y,z] 5.15) 3 for any Y,Z D. For Y,Z D and X TM),weget 2gA φz Y,X)= 2g hx,y ),φz ) = g hx,y ),φz ) +g hx,y ),φz ) = g X Y,φZ ) +g Y X,φZ ) = g X Y + Y X,φZ ) = g φ X Y + Y X ),Z ) = g Y φx + X φy +αηx)y +αηy )X 2αgX,Y )ξ,z ) = g Y φx,z ) g X φy,z ) αηx)gy,z) αηy )gx,z)+2αηz)gx,y ) = g Y Z,φX ) +g A φy Z,X ) αηx)gy,z) αηy )gx, Z) +2αηZ)gX, Y ). 5.16) The above equation is true for all X TM), therefore, transvecting the vector field X both sides, we obtain 2A φz Y = A φy Z φ Y Z αgy,z)ξ αηy )Z +2αηZ)Y 5.17) for any Y,Z D. Interchanging the vector fields Y and Z, weget 2A φy Z = A φz Y φ Z Y αgy,z)ξ αηz)y +2αηY )Z. 5.18) Subtracting 5.17) and 5.18), we get A φy Z A φz Y = α ηy )Z ηz)y ) + 1 φp[y,z] 5.19) 3 for any Y,Z D, which completes the proof.

9 CR-SUBMANIFOLDS OF A NEARLY TRANS-SASAKIAN Theorem 5.4. Let M be a CR-submanifold of a nearly trans-sasakian manifold M. Then, the distribution D is integrable if and only if A φy Z A φz Y = α ηy )Z ηz)y ), for any Y,Z D. 5.20) First suppose that the distribution D is integrable. Then [Y,Z] D for any Y,Z D.SinceP is a projection operator on D, sop[y,z]= 0. Thus from 5.15) we get 5.20). Conversely, we suppose that 5.20) holds. Then using 5.15), we have φp[y,z] = 0 for any Y,Z D. Since rank φ = 2n. Therefore, either P[Y,Z] = 0or P[Y,Z] = kξ. ButP[Y,Z] = kξ is not possible as P is a projection operator on D. Thus, P[Y,Z]= 0, which is equivalent to [Y,Z] D for any Y,Z D and hence D is integrable. Corollary 5.5. Let M be a ξ-horizontal CR-submanifold of a nearly trans-sasakian manifold M. Then, the distribution D is integrable if and only if A φy Z A φz Y = ) for any Y,Z D. References [1] A. Bejancu, CR submanifolds of a Kaehler manifold. I, Proc. Amer. Math. Soc ), no. 1, [2] A. Bejancu and N. Papaghuic, CR-submanifolds of Kenmotsu manifold, Rend. Mat ), no. 4, [3] D. E. Blair, Contact Manifolds in Riemannian Geometry, Lecture Notes in Mathematics, vol. 509, Springer-Verlag, Berlin, [4] C. Gherghe, Harmonicity on nearly trans-sasaki manifolds, Demonstratio Math ), no. 1, [5] D. Janssens and L. Vanhecke, Almost contact structures and curvature tensors, Kodai Math. J ), no. 1, [6] M. Kobayashi, CR submanifolds of a Sasakian manifold, TensorN.S.) ), no. 3, [7] J. A. Oubiña, New classes of almost contact metric structures, Publ. Math. Debrecen ), no. 3-4, [8] M. H. Shahid, CR-submanifolds of a trans-sasakian manifold, Indian J. Pure Appl. Math ), no. 12, [9], CR-submanifolds of a trans-sasakian manifold. II, Indian J. Pure Appl. Math ), no. 3, [10] M. H. Shahid, A. Sharfuddin, and S. A. Husain, CR-submanifolds of a Sasakian manifold, Univ. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat ), no. 1, [11] K. Yano and M. Kon, Contact CR submanifolds, Kodai Math. J ), no. 2, Falleh R. Al-Solamy: Department of Mathematics, Faculty of Science, King Abdul Aziz University, P.O. Box 80015, Jeddah 21589, Saudi Arabia

10 Mathematical Problems in Engineering Special Issue on Modeling Experimental Nonlinear Dynamics and Chaotic Scenarios Call for Papers Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system. Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision. In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points. Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from Qualitative Theory of Differential Equations, allowing more precise analysis and synthesis, in order to produce new vital products and services. Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers. This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems. Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment. Authors should follow the Mathematical Problems in Engineering manuscript format described at Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at mts.hindawi.com/ according to the following timetable: Guest Editors José Roberto Castilho Piqueira, Telecommunication and Control Engineering Department, Polytechnic School, The University of São Paulo, São Paulo, Brazil; piqueira@lac.usp.br Elbert E. Neher Macau, Laboratório Associado de Matemática Aplicada e Computação LAC), Instituto Nacional de Pesquisas Espaciais INPE), São Josè dos Campos, São Paulo, Brazil ; elbert@lac.inpe.br Celso Grebogi, Department of Physics, King s College, University of Aberdeen, Aberdeen AB24 3UE, UK; grebogi@abdn.ac.uk Manuscript Due February 1, 2009 First Round of Reviews May 1, 2009 Publication Date August 1, 2009 Hindawi Publishing Corporation