CR-SUBMANIFOLDS OF A NEARLY TRANS-SASAKIAN MANIFOLD

IJMMS 31:3 2002) 167 175 PII. S0161171202109288 http://ijmms.hindawi.com 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. 2000 Mathematics Subject Classification: 53C40. 1. 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.

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.

CR-SUBMANIFOLDS OF A NEARLY TRANS-SASAKIAN... 169 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.

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 + 3.13) 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)

CR-SUBMANIFOLDS OF A NEARLY TRANS-SASAKIAN... 171 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 ) = 0 4.15) 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.

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 ) = 0 4.19) 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)

CR-SUBMANIFOLDS OF A NEARLY TRANS-SASAKIAN... 173 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)

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) ) = 0. 5.13) 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) = 0 5.14) 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.

CR-SUBMANIFOLDS OF A NEARLY TRANS-SASAKIAN... 175 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 = 0 5.21) for any Y,Z D. References [1] A. Bejancu, CR submanifolds of a Kaehler manifold. I, Proc. Amer. Math. Soc. 69 1978), no. 1, 135 142. [2] A. Bejancu and N. Papaghuic, CR-submanifolds of Kenmotsu manifold, Rend. Mat. 7 1984), no. 4, 607 622. [3] D. E. Blair, Contact Manifolds in Riemannian Geometry, Lecture Notes in Mathematics, vol. 509, Springer-Verlag, Berlin, 1976. [4] C. Gherghe, Harmonicity on nearly trans-sasaki manifolds, Demonstratio Math. 33 2000), no. 1, 151 157. [5] D. Janssens and L. Vanhecke, Almost contact structures and curvature tensors, Kodai Math. J. 4 1981), no. 1, 1 27. [6] M. Kobayashi, CR submanifolds of a Sasakian manifold, TensorN.S.) 35 1981), no. 3, 297 307. [7] J. A. Oubiña, New classes of almost contact metric structures, Publ. Math. Debrecen 32 1985), no. 3-4, 187 193. [8] M. H. Shahid, CR-submanifolds of a trans-sasakian manifold, Indian J. Pure Appl. Math. 22 1991), no. 12, 1007 1012. [9], CR-submanifolds of a trans-sasakian manifold. II, Indian J. Pure Appl. Math. 25 1994), no. 3, 299 307. [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. 15 1985), no. 1, 263 278. [11] K. Yano and M. Kon, Contact CR submanifolds, Kodai Math. J. 5 1982), no. 2, 238 252. Falleh R. Al-Solamy: Department of Mathematics, Faculty of Science, King Abdul Aziz University, P.O. Box 80015, Jeddah 21589, Saudi Arabia

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