2000 Mathematics Subject Classification: 53C40, 53C25.In the present note, it is proved that there donot exist warped product semi-slant submanifolds in a Sasakian manifold other than contact CR-warped product submanifolds and thus the results obtained in [8] are generalized

Al-Solamy, Falleh R.

Khan, Viqar Azam

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Serdica Math. J. 34 (2008), 597–606WARPED PRODUCT SEMI-SLANT SUBMANIFOLDS OF ASASAKIAN MANIFOLDFalleh R. Al-Solamy, Viqar Azam KhanCommunicated by O. MusˇkarovAbstract. In the present note, it is proved that there donot exist warpedproduct semi-slant submanifolds in a Sasakian manifold other than contactCR-warped product submanifolds and thus the results obtained in [8] aregeneralized.1. Introduction. Bishop and O’Neill [1] introduced the notion ofwarped product manifolds and obtained results which reveal important geometricproperties of these manifolds. Many physical applications of these manifolds arerecently discovered e.g., the space around a black hole or a massive star is mod-eled on warped product manifolds. To be more precise, the best relativistic modelof the Schwarzchild space time describing the neighbourhoods of stars or blackholes is given as a warped product (c.f. [10]). Recently B. Y. Chen [5] initiatedthe study of warped product CR-submanifolds of Kaehler manifolds. B. Sahin2000 Mathematics Subject Classification: 53C40, 53C25.Key words: Sasakian manifolds, semi-slant warped product submanifold, contact CR-submanifold, warped product manifold.598 Falleh R. Al-Solamy, Viqar Azam Khan[11], extending the study of Chen, proved that there donot exist proper warpedproduct semi-slant submanifolds in a Kaehler manifold. Hesegawa and Mihai[7] initiated the study of contact CR-warped product submanifolds in Sasakianmanifolds. They obtained a sharp relationship between the warping function fof a warped product CR-submanifold and the squared norm of the second funda-mental form. K. Matsumoto and Mihai [9] studied warped product submanifoldsin a Sasakian space form. They also worked out variuous inequalities regardingthe squared norm of mean curvature vector of warped products in a Sasakianspace form and derived some important applications. In view of the physical ap-plications of these manifolds, the question of existance or non existanc of warpedproduct submanifolds assumes significant in general. In the present note, we haveadressed the same problem by studying warped product semi-slant submanifoldof a Sasakian manifold.2. Preliminaries. Let M be a (2m + 1)-dimensional almost contactmanifold i.e., a manifold endowed with an almost contact structure (φ, ξ, η) whereφ is a (1, 1) tensor field, ξ is a vector field and η is a 1-form such thatφ2 = −I + η ⊗ ξ, φ(ξ) = 0, η(ξ) = 1, η ◦ φ = 0.The almost contact structure is said to be normal if on the product manifoldM ×R, the induced almost complex structure J defined byJ(U, λddt)=(φU − λξ, η(U)ddt),is integrable, where U is tangent to M , t is the coordinate function on R and λis a smooth function on M × R. The condition for an almost contact structureto be normal is equivalent to the vanishing of the torsion tensor [φ, φ] + 2dη ⊗ ξwhere [φ, φ] denotes the Nijenhuis tensor of φ.On an almost contact manifold, there exists a Riemannian metric g whichis compatible with the contact structure (φ, ξ, η) i.e.,(2.1) g(φU, φV ) = g(U, V )− η(U)η(V ),from which it can be observed that(2.2) g(U, ξ) = η(U),Warped Product Semi-Slant Submanifolds 599for any U, V ∈ T (M). In this case, the Riemannian manifold (M, g) is called analmost contact metric manifold. An almost contact metric structure is called acontact metric structure if dη = Φ where Φ is the fundamental 2-form defined byΦ(U, V ) = g(U, φV ).A normal contact metric manifold is called a Sasakian manifold. It is known thatan almost contact metric manifold is Sasakian if and only if(2.3) (∇Uφ)V = −g(U, V )ξ + η(V )U,and,(2.4) ∇Uξ = φU,where ∇ is the Riemannian connection on M .Let M be a submanifold of an almost contact metric manifold M . Thenthe induced Riemannian metric on M is denoted by the same symbol g and theinduced connection by ∇. If TM and TM denote the tangent bundle on M andon M respectively, then the Gauss and Weingarten formulae are written as:(2.5) ∇UV = ∇UV + h(U, V ),(2.6) ∇UN = −ANU +∇⊥UN,for U, V ∈ TM and N ∈ T⊥M , where ∇⊥ denotes the induced connection in thenormal bundle T⊥M . AN and h denote the shape operator of the immersion andthe second fundamental respectively. The two are related as :(2.7) g(ANU, V ) = g(h(U, V ), N),For any U ∈ TM , we write(2.8) φU = TU + FU,where TU and FU are respectively the tangential and normal components of φU .Similarly, for N ∈ T⊥M , we decompose φN into tangential and normal parts as:(2.9) φN = BN + CN.The covariant derivatives of T and F are defined as:(2.10) (∇UT )V = ∇UTV − T∇UV,600 Falleh R. Al-Solamy, Viqar Azam Khan(2.11) (∇UF )V = ∇⊥UFV − F∇UV,for any U, V ∈ TM .Making use of equations (2.3)–(2.6) and (2.8)–(2.11), we obtain that(2.12) (∇UT )V = AFV U + Bh(U, V )− g(U, V )ξ + η(V )U,and,(2.13) (∇UF )V = Ch(U, V )− h(U, TV ).A submanifold M of an almost contact metric manifold M tangential tothe structure vector field ξ is called an invariant submanifold if φ-preserves alltangent spaces of M , that is φTx(M) ⊂ Tx(M) for every x ∈ M . A submanifoldM tangent to ξ is called an anti-invariant submanifold if φ maps a tangent spaceof M into the normal space, i.e., φTx(M) ⊂ T⊥x (M) for all x ∈ M where T⊥x (M)denotes the normal space at x ∈ M . The above definitions have been generalizedin several ways:(i) A submanifold M tangent to ξ, is called a contact CR-Submanifold if thereexists a differentiable distribution D : x −→ Dx ⊂ Tx(M) such that Dis invariant with respect to φ and the complementry distribution D⊥ isanti-invariant with respect to φ.(ii) A submanifold M tangent to ξ is called a slant submanifold if for all nonzero vector U tangent to M and independent with ξ, the angle θ(U) betweenφU and Tx(M) is a constant i.e., it doesnot depend on the choice of x ∈ Mand U ∈ Tx(M) [8].(iii) A submanifold M is called a semi-slant submanifold if it is endowed withtwo orthogonal complementry distributions D and D0 where D is invariantwith respect to φ and D0 is slant i.e., θ(U) between φU and D0x is constantfor each U ∈ D0x and x ∈ M [3].It is clear that invariant and anti-invariant submanifolds are CR-Submani-folds (resp., slant submanifolds) with D⊥ = {0} (resp. θ = 0) and D = {0} (resp.θ = pi/2). It is also clear that contact CR-submanifolds and slant submanifoldsare particular semi-slant submanifolds with θ = pi/2 and D = {0} respectively.A submanifold M of an almost contact metric manifold M is a slantsubmanifold if and only if(2.14) T 2 = −λ(−I + η ⊗ ξ),Warped Product Semi-Slant Submanifolds 601for some real number λ ∈ [0, 1]. Furthermore, in such case if θ is the slant angleof M , then λ = cos2 θ [4]. Hence,(2.15) g(TU, TV ) = cos2 θ(g(U, V )− η(U)η(V )),(2.16) g(FU,FV ) = sin2 θ(g(U, V )− η(U)η(V )),for any U, V ∈ TM .Let (M1, g1) and (M2, g2) be two Riemannian manifolds and f a positivedifferentiable function on M1. Then the warped product M1 ×f M2 is a productmanifold M1 ×M2 endowed with a Riemannian metric g given byg = g1 + f2g2.More explicitly, if U is a tangent vector on M = M1 ×f M2 then‖U‖2 = ‖dpi1U‖2 + f2(p)‖dpi2U‖2,where pii(i = 1, 2) are the canonical projections of M onto M1 and M2 respectivelyand dpii are their differentials.Theorem 2.1 [1]. Let M1 ×f M2 be a warped product manifold. Thenfor any X,Y ∈ TM1 and Z,W ∈ TM2,(i) ∇XY ∈ TM1,(ii) ∇XZ = ∇ZX =(Xff)Z,(iii) nor(∇ZW ) = −g(Z,W )f∇f ,where nor(∇ZW ) is the component of ∇ZW in TM1 and ∇f is the gradient off defined byg(∇f, U) = Uf.Corollary 2.1. On a warped product manifold M = M1 ×f M2,(i) M1 is totally geodesic in M .(ii) M2 is totally umbilical in M .602 Falleh R. Al-Solamy, Viqar Azam Khan3. Semi-Slant Submanifolds as Warped Products. Let M bea Sasakian manifold. Throughout the section, we denote by MT an invariantsubmanifold of M and Mθ, a slant submanifold of M with slant angle θ. Our aimin this section is to study warped product submanifolds MT×f Mθ and Mθ×f MT .We further assume that the structure vector field ξ is tangential to the underlyingwarped product submanifolds.From equations (2.4) and (2.5), we have(3.1) ∇Xξ = φX,(3.2) h(X, ξ) = 0,for any X ∈ TMT whereas for Z ∈ TMθ, we get(3.3) ∇Zξ = TZ.(3.4) h(Z, ξ) = FZ.Matsumoto and Mihai [9] provedTheorem 3.1 [9]. If M = M1 ×f M2 is a warped product submanifoldof a Sasakian manifold M where M1 and M2 are any submanifolds of M with ξtangential to M2. Then M is a Riemannian product.Hence, the possible non trivial semi-slant warped product submanifoldsare Mθ ×f MT and MT ×f Mθ with ξ tangential to Mθ and MT respectively.For θ = pi/2 the above warped products are known as warped product con-tact CR-submanifold and contact CR-warped product submanifolds respectively.Hesegawa and Mihai [7] showed the the warped product contact CR-submanifoldswith ξ tangential to MT are non existant.Theorem 3.2. Let M be a (2m + 1)-dimensional Sasakian manifold.Then there do not exist warped product submanifolds Mθ ×f MT on M such thatMθ is a slant submanifold tangent to ξ and MT is an invariant submanifold of M .P r o o f. Let M = Mθ ×f MT be a submanifold of M with ξ tangential toMθ. Then by Theorem 2.1,(3.5) ∇XZ = ∇ZX = (Z ln f)X,Warped Product Semi-Slant Submanifolds 603for any X ∈ TMT and Z ∈ TMθ. In particular for Z = ξ, we get ξf = 0. Nowby equations (3.1) and (3.5), it follows thatφX = ∇Xξ = (ξ ln f)X = 0.Thus MT can not exist and the Theorem is proved.Now, we obtain same important relations for later useLemma 3.1. Let M = MT ×f Mθ be a warped product semi-slant sub-manifold of a Sasakian manifold M such that MT is an invariant submanifoldtangent to ξ and Mθ a slant submanifold of M with slant angle θ 6= 0, then(i) g(h(X,Y ), FZ) = 0.(ii) g(h(X,W ), FZ) = g(h(X,Z), FW ).(iii) g(h(φX,Z), FW ) = (X ln f)g(Z,W ).for any X,Y ∈ TMT and Z,W ∈ TMθP r o o f. By Corollary 2.1, MT is totally geodesic in M and therefore byformula (2.10), (∇XT )Y ∈ TMT for all X,Y ∈ TMT . Using this fact in equation(2.12), we deduce thatg(Bh(X,Y ), Z) = 0,for all X,Y ∈ TMT and Z ∈ TMθ. The above equation is equivalent tog(h(X,Y ), FZ) = 0.This prove (i). Now, by Theorem (2.1),(3.6) ∇XZ = ∇ZX = (X ln f)Z.Taking account of (3.6) in (2.10) and (2.12), it follows thatAFZX + Bh(X,Z) = 0,which proves that(3.7) g(h(X,W ), FZ) = g(h(X,Z), FW ).Further, by equations (2.10) and (3.6), we obtain that(3.8) (∇ZT )X = (TX ln f)Z − (X ln f)TZ,604 Falleh R. Al-Solamy, Viqar Azam Khanwhereas by formula (2.12)(3.9) (∇ZT )X = Bh(X,Z) + η(X)Z.From the above two equations, it follows thatg(Bh(φX,Z),W ) = −(X ln f)g(Z,W )− (TX ln f)g(TZ,W ),which on making use of (3.7) yields(3.10) g(h(φX,Z), FW ) = (X ln f)g(Z,W ),for X ∈ TMT and Z,W ∈ TMθ. This proves statement (iii).Theorem 3.3. There does not exist a warped product semi-slant subman-ifold in a Sasakian manifold other than a contact CR-warped product submanifold.P r o o f. Consider a warped product submanifold M = MT ×f Mθ of aSasakian manifold M with ξ tangential to MT . On replacing X by φX in equation(3.10), we getη(X)g(h(Z, ξ), FW ) − g(h(X,Z), FW ) = (φX ln f)g(Z,W ).The above relation on making use of formula (3.4) and (2.16) yields(3.11) g(h(X,Z), FW ) = (η(X) − csc2 θ(φX ln f))g(FZ,FW ),for any X ∈ TMT and Z,W ∈ TMθ.If we denote by D0 the tangent bundle of Mθ and ν the orthogonal com-plement of FD0 i.e.,T⊥M = FD0 ⊕ ν,then we may writeh(X,Z) = hFD0(X,Z) + hν(X,Z),where hFD0(X,Z) ∈ FD0 and hν(X,Z) ∈ ν. In view of the above decomposition,equation (3.11) yields(3.12) hFD0(X,Z) = (η(X) − csc2 θ(φX ln f))FZ.Warped Product Semi-Slant Submanifolds 605On the other hand by equations (3.8) and (3.9), we have(TX ln f)Z − (X ln f)TZ = Bh(X,Z) + η(X)Z.Taking product with TZ and using the formula (2.15), the above equation impliesthat(3.13) g(h(X,Z), FTZ) = cos2 θ(X ln f)g(Z,Z).From equations (3.12) and (3.13), it follows that either θ = pi2or X(ln f) = 0.This proves the assertion.REFERE NCES[1] R. L. Bishop, B. O’Neill. Manifolds of negative curvature. Trans. Amer.Math. Soc. 145 (1969), 1–49.[2] D. E. Blair. Contact manifolds in Riemannian geometry. Lecture Note inMath. Vol. 509, Springer Verlag, Berlin, 1976.[3] J. L. Cabrerizo, A. Carriazo, L. M. Fernandez, M. Fernandez.Semi-slant submanifolds of a Sasakian manifolds. Geom. Dedicata 78 (1999),183–199.[4] J. L. Cabrerizo, A. Carriazo, L. M. Fernandez, M. Fernandez.Slant submanifolds in Sasakian manifolds. Glasgow Math. J. 42, 1 (2000),125–138.[5] B. Y. Chen. Geometry of warped product CR-Submanifolds in Kaehlermanifolds. Monatsh. Math. 133 (2001), 177–195.[6] B. Y. Chen. Geometry of warped products as Riemannian submanifoldsand related problems. Soochow J. Math. 28, 2 (2002), 125–156.[7] I. Hesegawa, I. Mihai. Contact CR-warped product submanifolds inSasakian manifolds. Geom. Dedicata 102 (2003), 143–150.[8] A. Lotta. Slant submanifolds in contact geometry. Bull. Math. Soc. Sci.Math. Roum., Nouv. Ser. 39, 1–4 (1996), 183-198.606 Falleh R. Al-Solamy, Viqar Azam Khan[9] K. Matsumoto, I. Mihai. Warped product submanifolds in Sasakian spaceform. SUT J. Math. 38, 2 (2002), 135–144.[10] B. O’Neill. Semi-Riemannian geometry with applications to relativity.Academic Press, New York, 1983.[11] B. Sahin. Non existance of warped product semi-slant submanifolds ofKaehler manifolds. Geom. Dedicata 115, 1 (2005), 195–202.Department of MathematicsKing AbdulAziz UniversityP. O. Box 80015, Jeddah 21589Saudi Arabia.e-mail: falleh@hotmail.com, viqar@gmail.comReceived May 12, 2007Revised November 8, 2007

Warped Product Semi-Slant Submanifolds of a Sasakian Manifold

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Warped Product Semi-Slant Submanifolds of a Sasakian Manifold

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