2000 Mathematics Subject Classification: 53C15, 53C42.In this paper, we research some fundamental properties of contact CR-Submanifolds of a Kenmotsu manifold. We show that the anti-invariant distribution is always integrable and give a necessary and sufficient condition for the invariant distribution to be integrable. After then, properties of the induced structures on submanifold by almost contact metric structure on the ambient manifold are categorized. Finally, we give some results for contact CR-product and totally umbilical contact CR-submanifold in a Kenmotsu manifold and Kenmotsu space form

Atçeken, Mehmet

Bulgarian Digital Mathematics Library at IMI-BAS

Serdica Math. J. 37 (2011), 67–78CONTACT CR-SUBMANIFOLDS OF KENMOTSUMANIFOLDSMehmet Atc¸ekenCommunicated by O. MushkarovAbstract. In this paper, we research some fundamental properties of con-tact CR-Submanifolds of a Kenmotsu manifold. We show that the anti-invariant distribution is always integrable and give a necessary and sufficientcondition for the invariant distribution to be integrable. After then, prop-erties of the induced structures on submanifold by almost contact metricstructure on the ambient manifold are categorized. Finally, we give some re-sults for contact CR-product and totally umbilical contact CR-submanifoldin a Kenmotsu manifold and Kenmotsu space form.1. Introduction. The geometry of semi-invariant submanifolds of aKenmotsu manifold has been defined and ivestigated by K. Kenmotsu and M.Kobayashi [4, 6]. Furthermore, many geometers contributed to study of sev-eral classes of different manifolds with endowed Riemannian metric tensor[seereferences]. In present paper deal with the geometry of leaves of contact CR-submanifolds of a Kenmotsu manifold.2010 Mathematics Subject Classification: 53C15, 53C42.Key words: Kenmotsu manifold, contact CR-submanifold and contact CR-product.68 Mehmet Atc¸ekenIn particular, we obtain necessary and sufficient conditions for the con-tact CR-submanifold to be CR-product, D-geodesic and D⊥-geodesic. Further-more, we get an inequality for the squared norm of the second fundamentalform in terms of the dimensional of distributions which are involved definitionof contact CR-submanifold in a Kenmotsu manifold. Finally, we discuss contactCR-products and totally umbilical contact CR-submanifolds(CR-products) in aKenmotsu space form M¯(c).2. Preliminaries. In this section, we give some notations used through-out this paper. We recall some necessary facts and formulas from the theory ofKenmotsu manifolds and their submanifolds.A (2m + 1)-dimensional Riemannian manifold (M¯, g) is said to be analmost contact metric manifold if it admits an endomorphism φ of its tangentbundle TM¯ , a vector field ξ and a 1-form η, satisfyingφ2X = −X + η(X)ξ, φξ = 0, η(ξ) = 1, η(φX) = 0(1)andg(φX,φY ) = g(X,Y )− η(X)η(Y ), η(X) = g(X, ξ)(2)for any vector fields X,Y tangent to M¯ . Furthermore, an almost contact metricmanifold is called a Kenmotsu manifold if φ and ξ satisfy(∇¯Xφ)Y = g(φX, Y )ξ − η(Y )φX and ∇¯Xξ = −φ2X = X − η(X)ξ,(3)where ∇¯ denotes the Levi-Civita connection on M¯ [2].Now, let M¯ be a 2n+ 1-dimensional Kenmotsu manifolds with structuretensors (φ, ξ, η, g) and M be an m-dimensional isometrically immersed submani-fold in M¯ . Moreover, we denote the Levi-Civita connections by ∇¯ and ∇, respec-tively. Then the Gauss and Weingarten formula’s for M in M¯ are, respectively,given by∇¯XY = ∇XY + h(X,Y )(4)∇¯XV = −AVX +∇⊥XV(5)for any vector fields X,Y tangent to M and vector V normal to M , where ∇⊥ isthe normal connection on T⊥M , h and A denote the second fundamental formand shape operator of M in M¯ , respectively. The A and h are related byg(h(X,Y ), V ) = g(AVX,Y ).(6)Contact CR-submanifolds of Kenmotsu manifolds 69We denote the Riemannian curvature tensor of the induced connection ∇ by R.Then the Gauss and Codazzi equations are, respectively, given by(R¯(X,Y )Z)⊤ = R(X,Y )Z +Ah(X,Z)Y −Ah(Y,Z)X(7)and(R¯(X,Y )Z)⊥ = (∇¯Xh)(Y,Z)− (∇¯Y h)(X,Z)(8)for any vector fields X,Y,Z tangent to M , where the covariant derivative of h isdefined by(∇¯Xh)(Y,Z) = ∇⊥Xh(Y,Z) − h(∇XY,Z)− h(Y,∇XZ)(9)for any vector fields X,Y,Z tangent toM , where (R¯(X,Y )Z)⊥ and (R¯(X,Y )Z)⊤denote the normal and tangent components of R¯(X,Y )Z, respectively.For any vector field X tangent to M , we setφX = fX + ωX,(10)where fX and ωX are the tangential and normal components of φX, respectively.Then f is an endomorphism of the TM and ω is a normal-bundle valued 1-formof TM . For the same reason, any vector field V normal to M , we setφV = BV + CV,(11)where BV and CV are the tangential and normal components of φV , respectively.Then B is an endomorphism of the normal bundle T⊥M of TM and C is atangent-bundle valued 1-form of T⊥M .Let R¯ be the curvature tensor of the connection ∇¯. The sectional cur-vature of a φ-section is called a φ-sectional curvature. A Kenmotsu manifoldwith constant φ-sectional curvature c is said to be a Kenmotsu space form andis denoted by M¯(c). The curvature tensor R¯ of a Kenmotsu space form M¯ (c) isgivenR¯(X,Y )Z =c− 34{g(Y,Z)X − g(X,Z)Y }+c+ 14{η(X)η(Z)Y − η(Y )η(Z)X+ η(Y )g(X,Z)ξ − η(X)g(Y,Z)ξ + g(X,φZ)φY − g(Y, φZ)φX+ 2g(X,φY )φZ}(12)for any vector fields X,Y,Z tangent to M¯ [1].70 Mehmet Atc¸eken3. Contact CR-submanifolds in Kenmotsu manifolds. In thissection, we shall define contact CR-submanifolds in a Kenmotsu manifold andresearch fundamental properties of their from theory of submanifold.LetM be an isometrically immersed in an almost contact metric manifoldM¯ , then for every x ∈M , there exists a maximal invariant subspace denoted byDx of the tangent space TxM of M . If the dimension of Dx is the same for allvalue of x ∈M , then Dx gives an invariant distribution D on M .Definition 3.1. A submanifold M of a Kenmotsu manifold M¯ is calledcontact CR-submanifold if there exists onM a differentiable invariant distributionD whose orthogonal complementary distribution D⊥ is anti-invariant, i.e.,1.) TM = D ⊕D⊥, ξ ∈ D2.) φ(Dx) = Dx,3.) φ(D⊥x ) ⊆ (T⊥x M),for any x ∈ M . A contact CR-submanifold is called anti-invariant(or, totallyreal) if Dx = {0} and invariant(or, holomorphic) if D⊥x = {0}, respectively, forany x ∈ M . It is called proper contact CR-submanifold if neither Dx = {0} norD⊥x = {0}.Next, let M be a contact CR-submanifold of a Kenmotsu manifold M¯ .Then from the (1), (10) and (11), we can write as the following way;f2X +BωX = −X + η(X)ξ, ωfX + CωX = 0,(13)fBV +BCV = 0 and ωBV + C2V = −V(14)for any vector fields X tangent to M and V normal to M .Proposition 3.1. Let M be a contact CR-submanifold of a Kenmotsumanifold M¯ . For any vector field X tangent to M belong to D (resp. D⊥) isnecessary and sufficient that ωX = 0 (resp. fX = 0).Furthermore, taking account of (2) and Proposition 3.1, we havef2X = −X + η(X)ξ(15)for any vector field X in D. Moreoverg(fX, fY ) = g(X,Y )− η(X)η(Y )(16)for any vector fields X,Y in D.Contact CR-submanifolds of Kenmotsu manifolds 71Proposition 3.2. Let M be a contact CR-submanifold of a Kenmotsumanifold M . Then the invariant distribution D has an almost contact metricstructure (f, ξ, η, g) and so dim(D)=odd.We denote by ν the orthogonal distribution φD⊥ in T⊥M . Then we haveT⊥M = φD⊥ ⊕ ν and φD⊥⊥ν.(17)From (17), it can easily to see that ν is an invariant distribution withrespect to φ and it has an almost complex structure C and so ν is even dimen-sional.Let M be a contact CR-submanifold of a Kenmotsu manifold M¯ . Byusing (4), (5), (10) and (11) we have(∇Xf)Y = AωYX +Bh(X,Y ) + g(fX, Y )ξ − η(Y )fX(18)and(∇Xω)Y = Ch(X,Y )− h(X, fY )− η(Y )ωX,(19)where the covariant derivatives of f and ω are defined by(∇Xf)Y = ∇XfY − f(∇XY ) and (∇Xω)Y = ∇⊥XωY − ω(∇XY )for any vector fields X,Y tangent to M .Theorem 3.1. Let M be a contact CR-submanifold of a Kenmotsu man-ifold M¯ . Then the anti-invariant distribution D⊥ is always integrable.P r o o f. For any vector fields Z,W tangent to D⊥, by using (18), we havef(∇ZW ) = −Bh(Z,W )−AφWZwhich is also equivalent tof [Z,W ] = AφZW −AφWZ.(20)On the other hand, we obtaing(AφWZ,U) = g(h(U,Z), φW ) = −g(φ(∇¯UZ),W ) = −g(∇¯UφZ − (∇¯Uφ)Z,W )= g(AφZW,U) + g(−η(Z)φU + g(φU,Z)ξ,W ) = g(AφZW,U)(21)for any U ∈ Γ(TM). It implies thatAφZW = AφWZ,(22)72 Mehmet Atc¸ekenfor any vector fields Z,W ∈ Γ(D⊥). By combining of (20) and (22), we getf [Z,W ] = 0, that is, [Z,W ] ∈ Γ(D⊥) which proves our assertion. Theorem 3.2. Let M be a contact CR-submanifold of a Kenmotsu man-ifold M¯ . Then the invariant distribution D is integrable if and only if the secondfundamental form of M satisfiesh(X,φY ) = h(φX, Y )(23)for any X,Y ∈ Γ(D).P r o o f. For any vector fields X,Y in D, making use of (3) and (4), wehaveφ[X,Y ] = φ(∇XY −∇YX) = φ(∇¯XY − ∇¯YX) = ∇¯XφY − (∇¯Xφ)Y− ∇¯Y φX + (∇¯Y φ)X = ∇XφY −∇Y φX + h(X,φY )− h(Y, φX)− η(X)φY + η(Y )φX + g(φY,X)ξ − g(φX, Y )ξ.(24)From the normal components of (24), we concludeω[X,Y ] = h(X,φY )− h(φX, Y ).(25)Thus D is integrable if and only if (23) is satisfied. Proposition 3.3. Let M be a contact CR-submanifold of a Kenmotsumanifold M¯ . The invariant distribution D is integrable if 1-form ω is parallel.P r o o f. If 1-form ω is parallel, then from (19) we have Ch(X,Y ) =h(X, fY ) for any X,Y ∈ Γ(D). It implies (23). LetM be a contact CR-submanifold of a Kenmotsu manifold M¯ . For anyvector fields U tangent to M and V normal to M , by using (3), (4), (5) and (11)we have(∇¯Uφ)V = ∇¯UφV − φ(∇¯UV )g(φU, V )ξ = (∇¯UB)V + (∇¯UC)V + h(U,BV ) + ωAV U+ −ACV U + fAV U.(26)The tangential and normal components of (26), respectively, we have(∇¯UB)V = g(φU, V )ξ +ACV U − fAV U(27)Contact CR-submanifolds of Kenmotsu manifolds 73and(∇¯UC)V = −h(U,BV )− ωAV U,(28)where the covariant derivatives of B and C are, respectively, defined by(∇¯UB)V = ∇UBV −B(∇⊥UV ) and (∇¯UC)V = ∇⊥UCV − C(∇⊥UV ).Furthermore, for any X,Y ∈ Γ(D), the Equation (18) reduces(∇Xf)Y = Bh(X,Y ) + g(fX, Y )ξ − η(Y )fX.(29)Thus we have the following Proposition.Proposition 3.4. Let M be a contact CR-submanifold of a Kenmotsumanifold M¯ . Then induced structure (f, ξ, η, g) is an almost contact metric struc-ture on D if and only if Bh(X,Y ) = 0 for any X,Y ∈ Γ(D).Lemma 3.1. Let M be a contact CR-submanifold of a Kenmotsu man-ifold M¯ . M is anti-invariant submanifold of Kenmotsu manifold M¯ if and onlyif the endomorphism f is parallel.P r o o f. If f is parallel, then from (29) we haveBh(X,Y ) + g(fX, Y )ξ − η(Y )fX = 0(30)for any X,Y ∈ Γ(D). Taking Y = ξ in (30), we get Bh(X, ξ)+g(fX, ξ)−fX = 0.Since h(X, ξ) = 0 and fξ = 0, we conclude fX = 0 which implies M is anti-invariant submanifold.Conversely, we suppose thatM is anti-invariant. Then form (29), we have(∇Xf)Y = −f(∇XY ) = Bh(X,Y ),for any X,Y ∈ Γ(TM). Because M is anti-invariant, f(∇XY ) ∈ Γ(TM⊥) andBh(X,Y ) ∈ Γ(TM), we conclude −f(∇XY ) = Bh(X,Y ) = 0. Thus we get thedesired result. Theorem 3.3. Let M be a contact CR-submanifold of a Kenmotsu man-ifold M¯ . Then the anti-invariant distribution D⊥ is totally geodesic in M if andonly if h(X,Z) ∈ Γ(ν) for any X ∈ Γ(D) and Z ∈ Γ(D⊥).74 Mehmet Atc¸ekenP r o o f. For any Z,W ∈ Γ(D⊥) and X ∈ Γ(D), we haveg(∇ZW,φX) = g(∇¯ZW,φX) = −g(∇¯ZφX,W )= −g((∇¯Zφ)X + φ∇¯ZX,W )= −g(−η(X)φZ + g(φZ,X)ξ,W ) + g(∇¯ZX,φW )= g(h(X,Z), φW )Thus ∇ZW ∈ Γ(D⊥) if and only if h(X,Z) ∈ Γ(ν). Theorem 3.4. Let M be a contact CR-submanifold of a Kenmotsu man-ifold M¯ . Then the invariant distribution D is totally geodesic in M if and only ifthe second fundamental form of M satisfies h(X,Y ) ∈ Γ(ν) for any X,Y ∈ Γ(D).P r o o f.g(∇XφY,Z) = g(∇¯XφY,Z) = g((∇¯Xφ)Y + φ(∇¯XY ), Z)= g(g(φX, Y )ξ − η(Y )φX,Z) − (∇¯XY, φZ)= g(φX, Y )η(Z) − η(Y )g(φX,Z) − g(h(X,Y ), φZ)= −g(h(X,Y ), φZ)(31)for any X,Y ∈ Γ(D) and Z ∈ Γ(D⊥). Thus∇XY ∈ Γ(D) if and only if h(X,Y ) ∈Γ(ν). This completes of the proof. 4. Contact CR-products in a Kenmotsu manifold. In thissection we shall define a contact CR-product in Kenmotsu manifolds, give anecessary and sufficient condition that a contact CR-submanifold to be a contactCR-product and we research contact CR products and totally umbilical contactCR-(submanifolds)products in a Kenmotsu space form M¯(c).Definition 4.1. A contact CR-submanifold M of a Kenmotsu manifoldM¯ is called a contact CR-product ifMT and M⊥ are totally geodesic submanifoldsof M , where MT and M⊥ denote the integral manifolds of the leaves of D andD⊥, respectively.Next we shall prove the following Theorem.Theorem 4.1. Let M be a contact CR-submanifold of a Kenmotsu man-ifold M¯ . M is a contact CR-product if and only if the shape operator of MsatisfiesAφD⊥D = 0.(32)Contact CR-submanifolds of Kenmotsu manifolds 75P r o o f. We suppose that M be a contact CR-product in a Kenmotsumanifold M¯ . From Theorem 3.3 and Theorem 3.4, respectively, we have AφZX ∈Γ(D) and AφZX ∈ Γ(D⊥) for any X ∈ Γ(D) and Z ∈ Γ(D⊥). It imply that (32)is satisfied.Conversely, (32) is satisfied. Then Theorem 3.3 and Theorem 3.4 tell usthat MT and M⊥ are totally geodesic submanifolds in M . Thus M is a contactCR-product. Hence the theorem is proved completely. Now, let M be a contact CR-product of a Kemotsu space form M¯(c), weshall calculate bisectional sectional curvature of Kenmotsu manifold M¯(c). Byusing (8), (9) and considering Theorem 3.3 and Theorem 3.4, we have−HB(X,Z) = g(R(X,φX)Z, φZ) = g((∇¯Xh)(φX,Z) − (∇¯φXh)(X,Z), φZ)= g(∇⊥Xh(φX,Z) − h(∇XφX,Z)− h(∇XZ, φX), φZ)−g(∇⊥φXh(X,Z) − h(∇φXX,Z)− h(∇φXZ,X), φZ)= Xg(h(φX,Z), φZ) − g(∇¯XφZ, h(φX,Z)) − φXg(h(X,Z), φZ)+g(∇¯φXφZ, h(X,Z))= g((∇¯φXφ)Z + φ(∇¯φXZ), h(X,Z)) − g((∇¯Xφ)Z+φ(∇¯XZ), h(φX,Z))= g(−η(Z)φ2X + g(φ2X,Z)ξ + φ(∇¯φXZ), h(X,Z))−g(−η(Z)φX + g(φX,Z)ξ + φ(∇¯XZ), h(φX,Z))= g(φh(φX,Z), h(X,Z)) − g(φh(X,Z), h(φX,Z))= 2g(φh(φX,Z), h(X,Z))= −2g(∇¯ZφX,φh(X,Z)) = −2g((∇¯Zφ)X + φ(∇¯XZ), φh(X,Z))= −2g(−η(X)φZ + g(φZ,X)ξ + φh(X,Z), φh(X,Z))= −2g(h(X,Z), h(X,Z)) = −2‖h(X,Z)‖2,for any X ∈ Γ(D) and Z ∈ Γ(D⊥). So we getHB(X,Z) = 2‖h(X,Z)‖2 .(33)Thus we have following the Theorem.76 Mehmet Atc¸ekenTheorem 4.2. Let M be a contact CR-product of a Kenmotsu spaceform M¯(c) with constant φ-holomorphic sectional curvature c. Then there do notexist contact CR-products in a Kenmotsu space form M¯(c) such that c < −1.P r o o f. We suppose that M is a contact CR-product in Kenmotsu spaceform M¯(c). Then from (3) and (4), we know h(Z, ξ) = 0. By using (12) and (33),we haveg(R(X,φX)φZ,Z) =(c+ 12){g(X,X) − η2(X)}g(Z,Z)=(c+ 12)g(φX,φX)g(Z,Z) = 2‖h(X,Z)‖2,for any X ∈ Γ(D) and Z ∈ Γ(D⊥). So we have‖h(X,Z)‖2 =(c+ 14)g(Z,Z)g(φX,φX),(34)for any X ∈ Γ(D) and Z ∈ Γ(D⊥). This equality is impossible for c < −1. Thisproves our assertion. Theorem 4.3. Let M be a contact CR-product in Kenmotsu space formM¯(c). Then we have‖ h ‖2≥{c+ 12}pq,(35)where dimD = 2p + 1 and dimD⊥ = q.P r o o f. Let {e1, e2, . . . , e2p, ξ, e1, e2, . . . , eq} be an orthonormal basis ofΓ(TM) such that e1, e2, . . . , e2p, ξ are tangent to Γ(D) and e1, e2, . . . , eq are tan-gent to D⊥. Then norm of the second fundamental form ‖ h ‖2 is defined by‖ h ‖2 =p∑i,j=1g(h(ei, ej), h(ei, ej)) +q∑r,s=1g(h(er , es), h(er , es))+ 2p∑i=1q∑r=1g(h(ei, er), h(ei, er))Taking X = e1, e2, . . . , e2p, ξ and Z = e1, e2, . . . , eq in (34), then we obtain‖ h ‖2>(c+ 12)pq 2Contact CR-submanifolds of Kenmotsu manifolds 77Contact CR-submanifold M is called a totally umbilical contact CR-submanifold if its second fundamental form h satisfies h(X,Y ) = g(X,Y )H,for any X,Y ∈ Γ(TM), where H denote the mean curvature vector of M .Theorem 4.4. Let M be a contact CR-submanifold of a Kenmotsu spaceform M¯(c). There exist no totally umbilical contact CR-submanifolds in a Ken-motsu space form M¯(c) such that c 6= −1.P r o o f. We suppose thatM is a totally umbilical contact CR-submanifoldin Kenmotsu space form M¯(c). Then by using (8), we obtaing(R¯(φX,X)Z, φZ) = g(X,Z)g(∇⊥φXH,φZ)− g(φX,Z)g(∇⊥XH,φZ)= 0,(36)for any X ∈ Γ(D) and Z ∈ Γ(D⊥). Since the ambient space M¯ is a Kenmotsuspace form, from (12) we inferg(R¯(φX,X)Z, φZ) =c+ 12{g(φX,φX)g(Z,Z)}.(37)Thus from (36) and (37), we obtain the desired result. Theorem 4.5. Let M be a totally umbilical contact CR-submanifold ofa Kenmotsu manifold M¯ . Then at least one of the following is true;i.) M is totally geodesicorii.) dim(D⊥) > 1P r o o f. By direct calculations, we haveAφVX = −AV φX(38)for any X ∈ Γ(D) and V ∈ Γ(ν). Since M is a totally umbilical contact CR-submanifold and by using (38), we haveg(AφCHX,X) = −g(ACHφX,X)g(X,X)g(H,φCH) = −g(X,φX)g(CH,H) = 0which is equivalent to CH = 0. On the other hand, by using (22) we haveg(AφBHZ,W ) = g(AφZBH,W )g(Z,W )g(H,φBH) = g(BH,W )g(H,φZ)g(Z,W )g(BH,BH) = g(BH,W )g(BH,Z)78 Mehmet Atc¸ekenfor any Z,W ∈ Γ(D⊥). This is implies that BH is either identical zero, or BHand Z are linearly dependent. If BH = 0, then totally umbilical contact CR-submanifold is totally geodesic otherwise, the anti-invariant distribution D⊥ isone dimensional. This completes of the proof of the theorem. REFERENCES[1] B. B. Sinha, A. K. Srivastava. Semi-Invariant Submanifolds of A Ken-motsu Manifold with Constant φ-Holomorphic Sectional Curvature. IndianJ. Pure Appl. Math. 23, 11 (1992), 783–789.[2] K. Kenmotsu. A class of almost contact Riemannian Manifold. TohokuMath. J. 24 (1972), 93–103.[3] K. Matsumoto. On Contact CR-Submanifolds of Sasakian Manifolds. In-ternat. J. Math. Math. Sci. 6, 2 (1983), 313–326.[4] M. Kobayahsi. Semi-Invariant Submanifolds of a certain class of almostcontact manifolds. Tensor (N.S), 43 (1986), 28–36.[5] M. Shoeb. CR-Submanifolds of A Normal Almost Paracontact Manifold.Invertis Journal of Science and Technology 1, 3 (2008), 190–194.[6] N. Papaghuic. Semi-invariant submanifolds in a Kenmotsu manifold. Rend.Mat. Appl. (7) 3 (1983), 607–622.[7] I. Hasegawa, I. Mihai. Contact CR-Warped Product Submanifolds inSasakian Manifolds. Geom. Dedicata 102 (2003), 143–150.[8] W. Yong. Totally Geodesic Invariant Submanifolds of Contact Metric Man-ifold. Chinese Quart. J. Math. 13, 4 (1998).[9] V. A. Khan, K. A. Khan, Sirajuddin. Contact CR-warped product sub-manifolds of Kenmotsu manifolds. Thai J. Math. 6, 1 (2008), 139–145.Gaziosmanpasa UniversityFaculty of Arts and SciencesDepartment of Mathematics60250 Tokat, Turkeye-mail: matceken@gop.edu.trReceived March 18, 2011Revised June 7, 2011

Contact CR-Submanifolds of Kenmotsu Manifolds

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Contact CR-Submanifolds of Kenmotsu Manifolds

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