2000 Mathematics Subject Classification: 53C40, 53B25.In the present note we study totally umbilical pseudo-slant submanifolds of a nearly cosymplectic manifold. We have obtained a classification theorem for totally umbilical pseudo-slant submanifolds of a nearly cosymplectic manifold

Khan, M. A.

Singh, Khushwant

Uddin, Siraj

Bulgarian Digital Mathematics Library at IMI-BAS

Serdica Math. J. 36 (2010), 137–148TOTALLY UMBILICAL PSEUDO-SLANT SUBMANIFOLDSOF A NEARLY COSYMPLECTIC MANIFOLDM. A. Khan, Siraj Uddin, Khushwant SinghCommunicated by O. MusˇkarovAbstract. In the present note we study totally umbilical pseudo-slantsubmanifolds of a nearly cosymplectic manifold. We have obtained a classi-fication theorem for totally umbilical pseudo-slant submanifolds of a nearlycosymplectic manifold.1. Introduction. Slant immersions in complex geometry were definedby B. Y. Chen as a natural generalization of both holomorphic and totally realimmersions [4, 5]. Recently, A. Lotta has introduced the notion of slant immersionof a Riemannian manifold into an almost contact metric manifold [7] and slantsubmanifold in Sasakian manifolds have been studied by J. L. Cabrerizo et al.[2].N. Papaghiuc [8] introduced the notion of semi-slant submanifold of an al-most Hermitian manifolds, which is in fact a generalization of CR-submanifolds.2010 Mathematics Subject Classification: 53C40, 53B25.Key words: Slant submanifold, pseudo-slant submanifold, totally umbilical, nearly cosym-plectic.138 M. A. Khan, Siraj Uddin, Khushwant SinghRecently, Carriazo [3] defined and studied bi-slant immersion in almost Her-mitian manifolds and simultaneously gave the notion of pseudo-slant submani-folds. Later on V. A. Khan and M. A. Khan [6] studied pseudo-slant submanifoldsof a Sasakian manifold. The purpose of the present paper is to study totally um-bilical pseudo-slant submanifolds of a nearly cosymplectic manifold.In section 2, we review and collect some necessary results. In section 3,we work out integrability condition of distributions involved in this setting andin the last section 4, we obtained a classification theorem for totally umbilicalpseudo-slant submanifold of a nearly cosymplectic manifold.2. Preliminaries. In this section, we give some terminology and nota-tions used throughout this paper. We recall some necessary facts and formulasfrom the theory of almost contact metric manifolds and their submanifolds.Let M¯ be an odd-dimensional manifold. An almost contact structure onM is an quadruple of tensor fields (φ, ξ, η, g), where φ is an endomorphism, ξ isa vector field, η is a one form and g is a Riemannian metric, respectively, suchthat(1) φ2 = −I + η ⊗ ξ, φξ = 0, η(φ) = 0, η(ξ) = 1,(2) g(φX,φY ) = g(X,Y )− η(X)η(Y ), η(X) = g(X, ξ)for any vector field X,Y ∈ TM¯ . It is easy to see thatg(φX, Y ) = −g(X,φY ).A (2n+1)-dimensional manifold M¯ together with an almost contact met-ric structure (φ, ξ, η, g) is said to be an almost contact metric manifold. Thefundamental 2-form Φ of M¯ is defined byΦ(X,Y ) = g(X,φY ), ∀X,Y ∈ TM¯.The almost contact metric manifold M¯ is called a nearly cosymplecticmanifold if it satisfy the following condition [1](3) (∇¯Xφ)Y + (∇¯Y φ)X = 0.Now, let M be a submanifold immersed in M¯ . The Riemannian metricinduced on M is denoted by the same symbol g. Let TM and T⊥M be the LieTotally umbilical pseudo-slant submanifolds. . . 139algebras of vector fields tangential and normal to M , respectively and ∇ be theinduced Levi-Civita connection on TM , then the Gauss and Weingarten formulasare given by(4) (a) ∇¯XY = ∇XY + h(X,Y ), (b) ∇¯XV = −AVX +∇⊥XVfor any X,Y ∈ TM and V ∈ T⊥M, where ∇⊥ is the connection on the normalbundle T⊥M , h is the second fundamental form and AV is the Weingarten mapassociated with V asg(AVX,Y ) = g(h(X,Y ), V ).The mean curvature vector H on M is given byH =1nn∑i=1h(ei, ei)where n is the dimension of M and {e1, e2, · · · , en} is a local orthonormal frameof vector fields on M .A submanifold M of a Riemannian manifold M¯ is called totally umbilicalif(5) h(X,Y ) = g(X,Y )H.If H = 0 then the submanifold is said to be totally geodesic submanifold in M¯ .For any x ∈M and X ∈ TxM , we write(6) φX = TX +NX,where TX ∈ TxM and NX ∈ T⊥x M. Similarly, for V ∈ T⊥x M, we have(7) φV = tV + fV,where tV (resp. fV ) is the tangential component (resp. normal component) ofφV. The covariant derivative of the morphisms T,N, t and f are defined respec-tively as(8) (∇¯XT )Y = ∇XTY − T∇XY(9) (∇¯XN)Y = ∇⊥XNY −N∇XY(10) (∇¯X t)V = ∇XtN − t∇⊥XN140 M. A. Khan, Siraj Uddin, Khushwant Singh(11) (∇¯Xf)V = ∇⊥XfV − f∇⊥Xξfor any X,Y ∈ TM and V ∈ T⊥M .Now, for any X,Y ∈ TM , let us denote the tangential and normal partsof (∇¯Xφ)Y by PXY and QXY respectively. Then by an easy computation, weobtain the following formulae(12) PXY = (∇¯XT )Y −ANYX − th(X,Y ),(13) QXY = (∇¯XN)Y + h(X,TY )− fh(X,Y ).Similarly, for V ∈ T⊥M , denoting the tangential and normal parts of(∇¯XJ)V by PXV and QXV respectively, we find that(14) PXV = (∇¯Xt)V + TAVX −AfVX,(15) QXV = (∇¯Xf)V + h(tV,X) +NAVX.Now, from equation (3) we have(16) (a) PXY + PYX = 0 (b) QXY +QYX = 0A submanifold M of an almost contact metric manifold M¯ is said to bea slant submanifold if for any x ∈M and X ∈ TxM − 〈ξ〉, the angle between φXand TxM is constant. The constant angle θ ∈ [0, pi/2] is then called slant angleof M in M¯ . The tangent bundle TM of M is decomposed asTM = D ⊕ 〈ξ〉where the orthogonal complementary distribution D of 〈ξ〉 is known as the slantdistribution on M .We say that M is a pseudo-slant submanifold of an almost contact metricmanifold M¯ , if there exist two orthogonal distributions D⊥ and Dθ on M suchthat(i) TM = D⊥ ⊕Dθ ⊕ 〈ξ〉.(ii) The distribution D⊥ is anti-invariant i.e., φD⊥ ⊆ T⊥M .(iii) The distribution Dθ is slant with slant angle θ 6= pi/2.Totally umbilical pseudo-slant submanifolds. . . 141From the above definition it is clear that if θ = 0, the pseudo-slant sub-manifold become semi-invariant submanifold.If µ is the invariant subspace of the normal bundle T⊥M then, in thecase of pseudo-slant submanifold, the normal bundle T⊥M can be decomposedas follows(17) T⊥M = µ⊕ND⊥ ⊕NDθ.We have the following result in the setting of almost contact manifoldsgiven by Cabrerizo et.al [2].Theorem 1 [2]. Let M be a submanifold of an almost contact metricmanifold M¯ . Then, M is slant submanifold of M¯ if and only if there exists aconstant λ ∈ [0, 1] such that(18) T 2 = λ(−I + η ⊗ ξ).Moreover, if θ is the slant angle of M , then λ = cos2 θ.Hence, for a slant submanifold, we have(19) g(TX, TX) = cos2 θ(g(X,Y )− η(X)η(Y ))(20) g(NX,NY ) = sin2 θ(g(X,Y )− η(X)η(Y ))for any X,Y ∈ TM .3. Integrability conditions of involved distributions. In thissection we shall see the integrability of involved distributions.Proposition 2. Let M be a pseudo-slant submanifold of a nearly cosym-plectic manifold then the anti-invariant distribution D⊥⊕ 〈ξ〉 is integrable if andonly ifg(AφXY −AφYX,φZ) = 2g((∇¯Xφ)Y, φZ)for all X,Y ∈ D⊥ ⊕ 〈ξ〉 and Z ∈ Dθ.P r o o f. For any X,Y ∈ D⊥ ⊕ 〈ξ〉, we have(21) ∇¯XφY = −AφYX +∇⊥XφY.142 M. A. Khan, Siraj Uddin, Khushwant SinghAlso, we know that(22) ∇¯XφY = (∇¯Xφ)Y + φ∇¯XY = (∇¯Xφ)Y + φ∇XY + φh(X,Y )From equations (21) and (22), we obtain−AφYX +∇⊥XφY = (∇¯Xφ)Y + φ∇XY + φh(X,Y ),or(23) (∇¯Xφ)Y + φ∇XY + φh(X,Y ) +AφYX −∇⊥XφY = 0.Interchanging X and Y , we get(24) (∇¯Y φ)X + φ∇YX + φh(Y,X) +AφXY −∇⊥Y φX = 0.Then from equations (23) and (24), we obtain2(∇¯Xφ)Y + φ[X,Y ]−AφXY +AφYX −∇⊥XφY +∇⊥Y φX = 0.That isφ[X,Y ] = AφXY −AφYX +∇⊥XφY −∇⊥Y φX − 2(∇¯Xφ)Y.Taking the inner product with φZ, where Z ∈ Dθg(φ[X,Y ], φZ) = g(AφXY −AφYX,φZ)− 2g((∇¯Xφ)Y, φZ).Using (2), the above equation takes the form(25) g([X,Y ], Z) = g(AφXY −AφYX,φZ)− 2g((∇¯Xφ)Y, φZ) + η([X,Y ])η(Z).Thus the assertion follows from (25). Proposition 3. Let M be a pseudo-slant submanifold of a nearly cosym-plectic manifold then the slant distribution Dθ ⊕ 〈ξ〉 is integrable if and only ifh(Z, TW ) − h(W,TZ)−∇⊥WNZ +∇⊥ZNW + 2g((∇¯Wφ)Z ∈ NDθfor any Z,W ∈ Dθ ⊕ 〈ξ〉.P r o o f. For any Z,W ∈ Dθ ⊕ 〈ξ〉 and X ∈ D⊥, we haveφ[Z,W ] = φ∇¯ZW − φ∇¯WZ.Totally umbilical pseudo-slant submanifolds. . . 143Then from (8), we obtainφ[Z,W ] = ∇¯ZφW − (∇¯Zφ)W + (∇¯Wφ)Z − ∇¯WφZ.Using (6), we getφ[Z,W ] = 2(∇¯Wφ)Z + ∇¯ZTW + ∇¯ZNW − ∇¯WTZ − ∇¯WNZ.Thus by Gauss and Wiengarten formulae we obtainφ[Z,W ] = 2(∇¯Wφ)Z +∇ZTW + h(Z, TW )−ANWZ +∇⊥ZNW−∇WTZ − h(W,TZ) +ANZW −∇⊥WNZ.Taking the inner product with φX, for any X ∈ D⊥g(φ[Z,W ], φX) = g(2(∇¯Wφ)Z, φX)+g(h(Z, TW ) − h(W,TZ) −∇⊥WNZ +∇⊥ZNW,φX)Using (2), we getg([Z,W ],X) = 2g((∇¯Wφ)Z, φX) + η([Z,W ])η(X)+g(h(Z, TW ) − h(W,TZ) −∇⊥WNZ +∇⊥ZNW,φX)Since ξ is tangential to Dθ then the second term of right hand side of the aboveequation is zero, hence we obtain(26) g([Z,W ],X) = 2g((∇¯Wφ)Z, φX)+g(h(Z, TW ) − h(W,TZ)−∇⊥WNZ +∇⊥ZNW,φX).Thus the assertion follows from the above equation. 4. Totally umbilical pseudo-slant submanifolds. Through outthe section we consider M as a totally umbilical pseudo-slant submanifold ofnearly cosymplectic manifold M¯ . We shall prove that the totally umbilicalpseudo-slant submanifold is totally geodesic under some conditions. Finally, weobtain a characterization result for totally umbilical pseudo-slant submanifold ofnearly cosymplectic manifold.Theorem 4. Let M be a totally umbilical pseudo-slant submanifold of anearly cosymplectic manifold, then the following conditions are equivalent144 M. A. Khan, Siraj Uddin, Khushwant Singh(a) M has a nearly cosymplectic character (T, g) on slant distribution, i.e.(∇¯XT )X = 0, for all X ∈ Dθ ⊕ 〈ξ〉.(b) The mean curvature vector H ∈ µ.P r o o f. For any X ∈ Dθ ⊕ 〈ξ〉h(X,TX) = 0andh(X,TY ) + h(Y, TX) = 0.From (4) the above equation reduced to0 = φ(∇XY + h(X,Y )) +ANYX −∇⊥XNY −∇XTY+φ(∇YX + h(X,Y )) +ANXY −∇⊥YNX −∇Y TX.Using equation (6) and equating the tangential components0 = T (∇XY +∇YX)−∇XTY −∇Y TX + 2th(X,Y ) +ANYX +ANXY.In particular,0 = T∇XX −∇XTX + g(X,Y )tH +ANXX.AsM is totally umbilical then the term ANXX becomes Xg(H,NX), then aboveequation reduced to(27) 0 = (∇¯XT )X + g(X,Y )tH +Xg(H,NX).Thus the assertion follows from above equation. Theorem 5. Let M be a totally umbilical pseudo-slant submanifold ofnearly cosymplectic manifold M¯ with parallelism of the endomorphism f . ThenM is totally geodesic submanifold of M¯ .P r o o f. For any X,Y ∈ Dθ〈ξ〉, we have∇¯XφY = (∇¯Xφ)Y + φ∇¯XY.Now, by equations (4), (5) and (6) the above equation become∇XφY+g(X,TY )H−ANYX+∇⊥XNY = PXY+QXY+T∇XY+N∇XY+g(X,Y )φH.Totally umbilical pseudo-slant submanifolds. . . 145Taking the inner product with φH, we obtaing(∇⊥XNY, φH) = g(QXY, φH) + g(N∇XY, φH) + g(X,Y )g(H,H).The above equation takes the form(28) g(∇¯XNY, φH) = g(QXY, φH) + g(X,Y )g(H,H).Similarly, we have(29) g(∇¯YNX,φH) = g(QYX,φH) + g(X,Y )g(H,H).Adding equations (28) and (29) and applying (16) (b) which yields(30) g(∇¯XNY + ∇¯YNX,φH) = 2g(X,Y )g(H,H).Now for any X ∈ Dθ ⊕ 〈ξ〉, we have∇¯XφH = (∇¯Xφ)H + φ∇¯XH.That is−AφHX +∇⊥XφH = PXH +QXH − TAHX −NAHX + t∇⊥XH + f∇⊥XH.Taking the inner product with NY for any Y ∈ Dθ ⊕ 〈ξ〉 and in view of the factthat f∇⊥XH ∈ µ the above equation takes the form(31) g(∇⊥XφH,NY ) = −g(NAHX,NY ) + g(QXH,NY ).Then from (20), we getg(∇⊥XφH,NY ) = − sin2 θ(g(X,Y )g(H,H) − η(AHX)η(Y )) + g(QXH,NY ).Since ∇¯ is a metric connection and NY and φH are orthogonal, we derive(32) g(∇¯XNY, φH) = sin2 θg(X,Y )g(H,H) − g(QXH,NY ).Similarly, we obtain(33) g(∇¯YNX,φH) = sin2 θg(X,Y )g(H,H) − g(QYH,NX).Thus from (32) and (33), we get(34)g(∇¯XNY+∇¯YNX,φH) = 2 sin2 θg(X,Y )g(H,H)−g(QXH,NY )−g(QYH,NX).146 M. A. Khan, Siraj Uddin, Khushwant SinghFrom equations (30) and (34), we derive2g(X,Y )g(H,H) = 2 sin2 θg(X,Y )g(H,H) − g(QXH,NY )− g(QYH,NX).That is,2 cos2 θg(X,Y )g(H,H) + g(QXH,NY ) + g(QYH,NX) = 0.In view of the equation (15), Theorem 4 and the assumption that f is parallelthe above equation reduces to(35) g(X,Y )g(H,H) = 0.Since g is Riemannian metric then from equation (35), it follows that H = 0, i.e.,M is totally geodesic submanifold. Hence theorem is proved. Let us consider M as a pseudo-slant submanifold of nearly cosymplecticmanifold M¯ , then for any U ∈ TM , we have (∇¯Uφ)U = 0, using this fact, we get(36) (∇¯Zφ)Z = 0for any Z ∈ D⊥. Therefore the tangential and normal parts of the above equationare PZZ = 0 and QZZ = 0, respectively. From equation (12) and tangentialcomponents of equation (36), we obtainPZZ = 0 = (∇¯ZT )Z −ANZZ − th(Z,Z)or,(∇¯ZT )Z = ANZZ + th(Z,Z).That isT∇ZZ = −g(H,NZ)Z − |Z|2tH.Taking the inner product with W ∈ D⊥, we get(37) g(H,NZ)g(Z,W ) + |Z|2g(tH,W ) = 0.Thus the equation (37) has solution if one of the following holds:(a) dimD⊥ = 1,(b) H ∈ µ.Totally umbilical pseudo-slant submanifolds. . . 147Now, we are in position to prove our main result.Theorem 6. Let M be a totally umbilical pseudo-slant submanifold ofa nearly cosymplectic manifold M¯ with slant and anti-invariant distributions Dθand D⊥, respectively. Then at least one of the following statement is true(i) M is totally real submanifold,(ii) M has nearly cosymplectic character (T, g) on the slant distribution Dθ,(iii) M is totally geodesic when f is parallel,(iv) dimD⊥ = 1.P r o o f. If Dθ = {0}, then by definition M is totally real which is case(i). If Dθ 6= {0} and H ∈ µ, then by Theorem 4, (T, g) has nearly cosymplecticcharacter on Dθ, if moreover H ∈ µ and f is parallel then by Theorem 5, Mis totally geodesic. Finally, if H 6∈ µ, then equation (37) has solution if eitherdimD⊥ = 1 which is the case (iv), this proves the theorem completely. Acknowledgement. The authors are thankful to referees and professorAngelina Chin Yan Mui for their valuable suggestions and comments.REFERENCES[1] D. E. Blair. Contact manifolds in Riemannian geometry. Lecture Notes inMathematics, vol. 509, Berlin–Heidelberg–New York, Springer-Verlag. 1976.[2] J. L. Cabrerizo, A. Carriazo, L. M. Ferna´ndez, M. Ferna´ndez.Slant submanifolds in Sasakian manifolds. Glasg. Math. J. 42, 1 (2000),125–138.[3] A. Carriazo. New developments in slant submanifolds theory. Narosa Pub-lishing House, New Delhi, India, 2002.[4] B. Y. Chen. Geometry of slant submanifolds. Katholieke Universiteit Leu-ven, 1990.[5] B. Y. Chen. Slant immersions. Bull. Austral. Math. Soc. 41 (1990),135–147.148 M. A. Khan, Siraj Uddin, Khushwant Singh[6] V. A. Khan, M. A. Khan. Pseudo-Slant Submanifolds of a SasakianManifold. Indian J. Pure Appl. Math. 38 (2007), 31–42.[7] A. Lotta. Slant Submanifolds in Contact Geometry. Bull. Math. Soc.Romania 39 (1996), 183–198.[8] N. Papaghuic. Semi-slant submanifolds of Kahlerian manifold. An. S¸tiint¸.Univ. Al. I. Cuza Ias¸i Sect¸. I a Mat. vol. XL, 1 (1994), 55–61.M. A. Khan, Khushwant SinghSchool of Mathematics and Computer ApplicationsThapar UniversityPatiala-147 004, Indiae-mails: meraj79@gmail.com, khushwantchahil@gmail.comSiraj UddinInstitute of Mathematical SciencesFaculty of SciencesUniversity of Malaya50603 Kuala Lumpur, Malaysiae-mail: siraj.ch@gmail.com Received May 5, 2010

Totally Umbilical Pseudo-Slant Submanifolds of a Nearly Cosymplectic Manifold

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Totally Umbilical Pseudo-Slant Submanifolds of a Nearly Cosymplectic Manifold

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