Warped Product Submanifolds of LP-Sasakian Manifolds

(1)

Volume 2012, Article ID 868549,11pages doi:10.1155/2012/868549

Research Article

Warped Product Submanifolds of LP-Sasakian Manifolds

S. K. Hui,

¹

S. Uddin,

²

C. ¨ Ozel,

³

and A. A. Mustafa

²

1Nikhil Banga Sikshan Mahavidyalaya Bishnupur, Bankura, West Bengal 722 122, India

2Institute of Mathematical Sciences, Faculty of Science, University of Malaya, 50603 Kuala Lumpur, Malaysia

3Department of Mathematics, Abant Izzet Baysal University, 14268 Bolu, Turkey

Correspondence should be addressed to S. Uddin,[email protected] Received 21 February 2012; Accepted 20 April 2012

Academic Editor: Bo Yang

Copyrightq2012 S. K. Hui et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study of warped product submanifolds, especially warped product hemi-slant submanifolds of LP-Sasakian manifolds. We obtain the results on the nonexistance or existence of warped product hemi-slant submanifolds and give some examples of LP-Sasakian manifolds. The existence of warped product hemi-slant submanifolds of an LP-Sasakian manifold is also ensured by an interesting example.

1. Introduction

The notion of warped product manifolds was introduced by Bishop and O’Neill1, and later it was studied by many mathematicians and physicists. These manifolds are generalization of Riemannian product manifolds. The existence or nonexistence of warped product manifolds plays some important role in diﬀerential geometry as well as in physics.

On the analogy of Sasakian manifolds, in 1989, Matsumoto2introduced the notion of LP-Sasakian manifolds. The same notion is also introduced by Mihai and Ros¸ca3and obtained many interesting results. Later on, LP-Sasakian manifolds are also studied by sev- eral authors.

The notion of slant submanifolds in a complex manifold was introduced and studied by Chen 4, which is a natural generalization of both invariant and anti-invariant submanifolds. Chen4also found examples of slant submanifolds of complex Euclidean spaces C² andC⁴. Then, Lotta5has defined and studied the slant immersions of a Riemannian manifold into an almost contact metric manifold and proved some properties of such

(2)

immersions. Also, Cabrerizo et al. 6studied slant immersions of K-contact and Sasakian manifolds.

In 1994, Papaghuic7introduced the notion of semi-slant submanifolds of almost Hermitian manifolds. Then, Cabrerizo et. al8defined and investigated semi-slant submanifolds of Sasakian manifolds. The idea of hemi-slant submanifolds was introduced by Carriazo as a particular class of bi-slant submanifolds and he called them anti-slant submanifolds9.

Recently, these submanifolds were studied by Sahin for their warped products of K¨ahler manifolds 10. Recently, Uddin 11 studied warped product CR-submanifolds of LP- Sasakian manifolds.

The purpose of the present paper is to study the warped product hemi-slant submanifolds of LP-Sasakian manifolds. The paper is organized as follows.Section 2is concerned with some preliminaries.Section 3deals with the study of warped and doubly warped product submanifolds of LP-Sasakian manifolds. In Section 4, we define hemi-slant submanifolds of LP-contact manifolds and investigate their warped products. Section 5 consists some examples of LP-Sasakian manifolds and their warped products.

2. Preliminaries

Ann-dimensional smooth manifoldMis said to be an LP-Sasakian manifold3if it admits a 1,1tensor fieldφ, a unit timelike contravariant vector fieldξ, an 1-formη, andaLorentzian metricg, which satisfy

ηξ −1, gX, ξ ηX, φ²XXηXξ, 2.1 g

φX, φY

gX, Y ηXηY, ∇XξφX, 2.2

∇Xφ

Y gX, YξηYX2ηXηYξ, 2.3

where ∇ denotes the operator of covariant diﬀerentiation with respect to the Lorentzian metricg. It can be easily seen that, in an LP-Sasakian manifold, the following relations hold:

φξ0, η φX

0, rankφn−1. 2.4

Again, we put

ΩX, Y g X, φY

2.5

for any vector fieldsX,Y tangent toM. The tensor fieldΩX, Yis a symmetric0,2tensor field2. Also, since the vector fieldηis closed in an LP-Sasakian manifold, we have2

∇Xη

Y ΩX, Y, ΩX, ξ 0, 2.6

for any vector fieldsXandYtangent toM.

(3)

LetNbe a submanifold of an LP-Sasakian manifoldMwith induced metricgand let

∇and∇^⊥be the induced connections on the tangent bundleTNand the normal bundleT^⊥N ofN, respectively. Then, the Gauss and Weingarten formulae are given by

∇XY ∇XY hX, Y, 2.7

∇XV −AVX∇^⊥_XV, 2.8

for allX,Y ∈TNandV ∈T^⊥N, wherehandA_V are second fundamental form and the shape operatorcorresponding to the normal vector fieldV, respectively, for the immersion ofN intoM. The second fundamental formhand the shape operatorAV are related by12

ghX, Y, V gAVX, Y 2.9

for anyX, Y ∈TNandV ∈T^⊥N For anyX∈TN, we may write

φXEXFX, 2.10

whereEXis the tangential component andFXis the normal component ofφX.

Also, for anyV ∈T^⊥N, we have

φV BV CV, 2.11

whereBV andCV are the tangential and normal components ofφV, respectively. The covariant derivatives of the tensor fieldsEandFare defined as

∇XE

Y ∇XEY−E∇XY, 2.12

∇XF

Y ∇^⊥_XFY−F∇XY 2.13

for anyX, Y ∈TN.

Throughout the paper, we considerξto be tangent toN. The submanifoldNis said to be invariant ifFis identically zero, that is,φX∈TNfor anyX ∈TN. On the other hand,N is said to anti-invariant ifEis identically zero, that is,φX∈T^⊥Nfor anyX∈TN.

Furthermore, for a submanifold tangent to the structure vector fieldξ, there is another class of submanifolds which is called a slant submanifold. For each nonzero vectorXtangent toNatx∈N, the angleθX, 0≤θXπ/2betweenφXandEXis called the slant angle or wirtinger angle. If the slant angle is constant then the submanifold is called aslant submanifold.

Invariant and anti-invariant submanifolds are particular classes of slant submanifolds with slant angleθ 0 andθ π/2, respectively. A slant submanifold is said to be proper slant if the slant angleθlies strictly between 0 andπ/2, that is, 0< θ < π/26.

(4)

Theorem 2.1see13. LetN be a submanifold of a Lorentzian almost paracontact manifoldM such thatξis tangent toN. Then,Nis slant submanifold if and only if there exists a constantλ ∈ 0,1such that

E²λ

Iη⊗ξ

. 2.14

Furthermore, ifθis the slant angle ofN, thenλcos²θ. Also from2.14, we have gEX, EY cos²θ

gX, Y ηXηY

, 2.15

gFX, FY sin²θ

gX, Y ηXηY

2.16

for anyX,Ytangent toN.

The study of semi-slant submanifolds of almost Hermitian manifolds was introduced by Papaghuic7, which was extended to almost contact manifold by Cabrerizo et al.8.

The submanifoldNis called semi-slant submanifold ofMif there exist an orthogonal direct decomposition ofTNas

TND₁⊕D₂⊕ {ξ}, 2.17

whereD₁is an invariant distribution, that is,φD1 D₁andD₂is slant with slant angleθ /0.

The orthogonal complement ofFD2in the normal bundleT^⊥Nis an invariant subbundle of T^⊥Nand is denoted byμ. Thus, we have for a semi-slant submanifold

T^⊥NFD₂⊕μ. 2.18

For an LP-contact manifold this study is extended by Y ¨uksel et al.13.

3. Warped and Doubly Warped Products

The notion of warped product manifolds was introduced by Bishop and O’Neill1. They defined the warped product manifolds as follows.

Definition 3.1. Let N1, g₁ and N2, g₂ be two semi-Riemannian manifolds and f be a positive diﬀerentiable function onN1. Then, the warped product ofN1andN2is a manifold, denoted byN₁×fN₂ N1×N₂, g, where

g g₁f²g₂. 3.1

A warped product manifoldN1×fN2is said to be trivial if the warping functionfis constant.

More explicitely, if the vector fieldsXandYare tangent toN₁×fN₂atx, y, then gX, Y g1π1∗X, π1∗Y f²xg2π2∗X, π2∗Y, 3.2

(5)

whereπi i1,2are the canonical projections ofN1×N2ontoN1andN2, respectively, and

∗stands for the derivative map.

LetN N₁×fN₂ be a warped product manifold, which means thatN₁ and N₂ are totally geodesic and totally umbilical submanifolds ofN, respectively.

For the warped product manifolds, we have the following result for later use1.

Proposition 3.2. LetNN₁×fN₂be a warped product manifold. Then, I∇XY ∈TN1is the lift of∇XY onN1,

II∇UX∇XU XlnfU, III∇UV ∇_UV −gU, V∇lnf,

for anyX, Y ∈TN₁andU, V ∈TN₂, where∇and∇ denote the Levi-Civita connections onNand N2, respectively.

Doubly warped product manifolds were introduced as a generalization of warped product manifolds by ¨Unal14. A doubly warped product manifold ofN1andN2, denoted asf2N1×f1N2is endowed with a metricgdefined as

gf₂²g1f₁²g2, 3.3

wheref₁andf₂are positive diﬀerentiable functions onN₁andN₂, respectively.

In this case formulaIIofProposition 3.2is generalized as

∇XZ

Xlnf₁ Z

Zlnf₂

X 3.4

for eachXinTN1andZinTN215.

One has the following theorem for doubly warped product submanifolds of an LP- Sasakian manifold11.

Theorem 3.3. LetNf2N1×f1N2be a doubly warped product submanifold of an LP-Sasakian mani- foldMwhereN1andN2are submanifolds ofM. Then,f2is constant andN2is anti-invariant if the structure vector fieldξis tangent toN₁, andf₁is constant andN₁is anti-invariant ifξis tangent to N₂.

The following corollaries are immediate consequences of the above theorem.

Corollary 3.4. There does not exist a proper doubly warped product submanifold in LP-Sasakian manifolds.

Corollary 3.5. There does not exist a warped product submanifoldN1×fN2of an LP-Sasakian mani- foldMsuch thatξis tangent toN₂.

From the above theorem andCorollary 3.5, we have only the remaining case is to study the warped product submanifoldN₁×fN₂with structure vector fieldξis tangent toN₁.

(6)

4. Warped Product Hemi-Slant Submanifolds

In this section, first we define hemi-slant submanifolds of an LP-contact manifold and then we will discuss their warped products.

Definition 4.1. A submanifoldNof an LP-contact manifoldMis said to be a hemi-slant submanifold if there exist two orthogonal complementary distributionsD₁andD₂satisfying:

iTND1⊕D2⊕ ξ,

iiD1is a slant distribution with slant angleθ /π/2, iiiD2is anti-invariant, that is,φD2 ⊆T^⊥N.

If μ is φ-invariant subspace of the normal bundle T^⊥N, then in case of hemi-slant submanifold, the normal bundleT^⊥Ncan be decomposed as

T^⊥NFD₁⊕FD₂⊕μ. 4.1

Now, we discuss the warped product hemi-slant submanifolds of an LP-Sasakian manifoldM. IfNN1×fN2be a warped product hemi-slant submanifold of an LP-Sasakian manifold Mand N_θ and N_⊥ are slant and anti-invariant submanifolds of an LP-Sasakian manifoldM, respectively then their warped product hemi-slant submanifolds may be given by one of the following forms:

iN_⊥×fNθ,

iiNθ×fN_⊥.

In the following theorem, we start with the casei.

Theorem 4.2. There does not exist a proper warped product hemi-slant submanifoldN N_⊥×fNθ

of an LP-Sasakian manifoldMsuch thatξis tangent toN_θ, whereN_⊥andN_θare anti-invariant and proper slant submanifolds ofM, respectively.

Proof. Let N N_⊥×fNθ be a proper warped product hemi-slant submanifold of an LP- Sasakian manifoldMsuch thatξis tangent toN_θ. Then, for anyX ∈ TN_θ andU ∈ TN_⊥, we have

∇Xφ

U∇XφU−φ∇XU. 4.2

By virtue of2.3and2.7–2.11, it follows from4.2that ηUX −A_FUX∇^⊥_XFU−E∇XU

−F∇XU−BhX, U−ChX, U. 4.3

UsingProposition 3.2IIin4.3and then equating the tangential components, we get ηUXAFUX

Ulnf

EXBhX, U. 4.4

(7)

Taking the inner product withEX in 4.4and using the fact that X andEX are mutually orthogonal vector fields, then we have

gAFUX, EX Ulnf

gEX, EX gBhX, U, EX 0. 4.5

Using2.9and2.15, we get

− Ulnf

cos²θX²ghX, EX, FU−ghX, U, FEX. 4.6 ReplacingXbyEXin4.6and using2.14, we obtain

− Ulnf

cos²θX²−ghX, EX, FU ghEX, U, EX. 4.7 Adding4.6and4.7, we get

Ulnf

cos²θX²0. 4.8

SinceN_θis proper slant andXis nonnull,4.8yieldsUlnf0, which shows thatf is constant and consequently the theorem is proved.

The second case is dealt with the following theorem.

Theorem 4.3. LetN Nθ×fN_⊥ be a warped product hemi-slant submanifold of an LP-Sasakian manifoldM such thatN_θ is a proper slant submanifold tangent toξ and N_⊥ is an anti-invariant submanifold ofM. Then,∇XFUlies in the invariant normal subbundleμ, for eachX ∈TNθand U∈TN_⊥.

Proof. Consider N N_θ×fN_⊥ be a warped product hemi-slant submanifold of an LP- Sasakian manifold Msuch that Nθ is a proper slant submanifold tangent to ξ and N_⊥ is an anti-invariant submanifold ofM. Then, for anyX ∈TN_θandU∈TN_⊥, we have

∇XφUφ∇XU. 4.9

Using2.7and2.8, we obtain

−AFUX∇^⊥_XFUφ∇XUhX, U. 4.10

By virtue of2.10,2.11andProposition 3.2II, it follows from4.10that

−AFUX∇^⊥_XFU Xlnf

EU Xlnf

FU

BhX, U ChX, U. 4.11

Equating the normal components, we obtain

∇^⊥_XFU Xlnf

FUChX, U. 4.12

(8)

Taking the inner product of withFW1, for anyW1∈TN_⊥in4.13, we get g

∇^⊥_XFU, FW₁

Xlnf

gFU, FW1 gChX, U, FW1

Xlnf g

φU, φW1

g

φhX, U, φW1

Xlnf

gU, W1.

4.13

Also for anyX∈TNθandU∈TN_⊥, we have ∇XF

U∇^⊥_XFU− Xlnf

FU. 4.14

Taking the inner productFW1for anyW1 ∈TN_⊥in4.14and using2.1and2.2, we derive g

∇XF

U, FW₁ g

∇^⊥_XFU, FW₁

− Xlnf

gU, W1. 4.15

By virtue of4.13, the above equation yields

g

∇XF

U, FW₁

0, for anyX ∈TN_θ, U, W₁ ∈TN_⊥. 4.16

Similarly, if anyW₂∈TN_θ, then from2.13, we obtain g

∇XF

U, φW₂ g

∇^⊥_XFU, φW₂

−g

F∇XU, φW₂

. 4.17

Since the product of tangential component with normal is zero and N_θ is a proper slant submanifold, we may conclude from4.17that

g

∇XF

U, φW₂

0 for anyX, W₂∈TN_θ, U∈TN_⊥. 4.18

From4.16and4.18, it follows that∇XFU∈μand hence the proof is complete.

5. Examples on LP-Sasakian Manifolds

Example 5.1. We consider a 3-dimensional manifold M {x, y, z ∈ R³ : z > 0}, where x, y, zare the standard coordinates inR³. Let{E1, E₂, E₃}be a linearly independent global frame onMgiven by

E1e^z ∂

∂x, E2e^z−ax ∂

∂y, E3 − ∂

∂z, 5.1

where ais a nonzero constant such that a /1. Let g be the Lorentzian metric defined by gE1, E₃ gE2, E₃ gE1, E₂ 0, gE1, E₁ gE2, E₂ 1, gE3, E₃ −1. Letη be

(9)

the 1-form defined byηU gU, E3for anyU∈TM. Letθbe the1,1tensor field defined byηE₁ −E1,φE₂ −E2, andφE₃ 0. Then, using the linearity ofφandgwe haveηE3

−1,φ²UUηUE3, andgφU, φW gU, W ηUηWfor anyU, W ∈TM. Thus for E3ξ,φ, ξ, η, gdefines a Lorentzian paracontact structure onM.

Let∇be the Levi-Civita connection with respect to the Lorentzian metricg. Then, we have

E1, E₂ −ae^zE₂, E1, E₃ −E1, E2, E₃ −E2. 5.2

Using Koszul formula for the Lorentzian metric g, we can easily calculate

∇E1E₁−E3, ∇E1E₂0, ∇E1E₃−E1,

∇E2E1ae^zE2, ∇E2E2−ae^zE1−E3, ∇E2E3 −E2,

∇E3E₁0, ∇E3E₂0, ∇E3E₃0.

5.3

From the above computations, it can be easily seen that forE₃ξ,φ, ξ, η, gis an LP-Sasakian structure onM. Consequently,M³φ, ξ, η, gis an LP-Sasakian manifold.

Example 5.2 see 16. LetR⁵ be the 5-dimensional real number space with a coordinate systemx, y, z, t, s. Define

ηds−ydx−tdz, ξ ∂

∂s, gη⊗η−dx²−

dy₂

−dz²−dt², φ

∂

∂x − ∂

∂x−y∂

∂s, φ ∂

∂y − ∂

∂y, φ

∂

∂z − ∂

∂z −t ∂

∂s, φ ∂

∂t −∂

∂t, φ ∂

∂s 0,

5.4

the structureφ, η, ξ, gbecomes an LP-Sasakian structure inR⁵.

Example 5.3. Consider a 4-dimensional submanifold N of R⁷ with the cordinate system x1, x₂, . . . , x₆, tand the structure is defined as

φ ∂

∂x_i ∂

∂x_i, i1,2,3, φ

∂

∂x_j

∂

∂x_j,

j 4,5,6 ,

ηdt, ξ−∂

∂t, φ ∂

∂t 0, g dx_i²dx²_jη⊗η.

5.5

(10)

Hence, the structureφ, ξ, η, gis an LP-contact structure onR⁷. Now, for anyα ∈ 0, π/2 and nonzerouandv, we define the submanifoldNas follows:

ωu, v, α, t 2u, v, ucosα,−vsinα, usinα, vcosα, t. 5.6 Then, the tangent spaceTNis spanned by the vectors:

e1 ∂

∂x1 cosα ∂

∂x3 sinα ∂

∂x5, e2 ∂

∂x₂ −sinα ∂

∂x₄ cosα ∂

∂x₆, e3 −usinα ∂

∂x₃ −vcosα ∂

∂x₄ ucosα ∂

∂x₅ −vsinα ∂

∂x₆, e₄ − ∂

∂t.

5.7

Then the distributions D_θ span{e1, e₂, e₄} is a slant distribution tangent to ξ e₄ and D^⊥ span{e3} is an anti-invariant distribution, respectively. Let us denote byNθ and N_⊥ their integral submanifolds, then the metricgonNis given by

g2

du²dv²

u²v²

dα². 5.8

Hence, the submanifoldN N_θ×fN_⊥ is a hemi-slant-warped product submanifold of R⁷ with the warping functionf

u²v².

References

1 R. L. Bishop and B. O’Neill, “Manifolds of negative curvature,” Transactions of the American Math- ematical Society, vol. 145, pp. 1–49, 1969.

2 K. Matsumoto, “On Lorentzian paracontact manifolds,” Bulletin of Yamagata University, vol. 12, no. 2, pp. 151–156, 1989.

3 I. Mihai and R. Ros¸ca, “On lorentzianP-Sasakian manifolds,” in Classical Analysis, pp. 155–169, World Scientific Publisher, 1992.

4 B. Y. Chen, Geometry of Slant Submanifolds, Katholieke Universiteit Leuven, 1990.

5 A. Lotta, “Slant submanifolds in contact geometry,” Bulletin Mathématique de la Société des Sciences Mathématiques de Roumanie, vol. 39, pp. 183–198, 1996.

6 J. L. Cabrerizo, A. Carriazo, L. M. Fern´andez, and M. Fern´andez, “Slant submanifolds in Sasakian manifolds,” Glasgow Mathematical Journal, vol. 42, no. 1, pp. 125–138, 2000.

7 N. Papaghiuc, “Semi-slant submanifolds of a Kaehlerian manifold,” Analele S¸tiint¸ifice Ale Universit˘at¸ii

“Alexandru Ioan Cuza” Din Ias¸i, vol. 40, no. 1, pp. 55–61, 1994.

8 J. L. Cabrerizo, A. Carriazo, L. M. Fern´andez, and M. Fern´andez, “Semi-slant submanifolds of a Sasakian manifold,” Geometriae Dedicata, vol. 78, no. 2, pp. 183–199, 1999.

9 A. Carriazo, “Bi-slant immersions,” in Proceedings of The International Construction Risk Assessment Model (ICRAM ’00), pp. 88–97, Kharagpur, India, 2000.

10 B. Sahin, “Warped product submanifolds of Kaehler manifolds with a slant factor,” Annales Polonici Mathematici, vol. 95, no. 3, pp. 207–226, 2009.

11 S. Uddin, “Warped product CR-submanifolds in Lorentzian para Sasakian manifolds,” Serdica Mathematical Journal, vol. 36, no. 3, pp. 237–246, 2010.

(11)

12 K. Yano and M. Kon, Structures on Manifolds, World Scientific Publishing, Singapore, 1984.

13 P. Y ¨uksel, S. Kilic, and S. Keles, “Slant and semi-slant submanifolds of a Lorentzian almost paracontact manifold,” In press,http://arxiv.org/abs/1101.3156.

14 B. ¨Unal, “Doubly warped products,” Diﬀerential Geometry and Its Applications, vol. 15, no. 3, pp. 253–

263, 2001.

15 M. I. Munteanu, “A note on doubly warped product contact CR-submanifolds in trans-Sasakian manifolds,” Acta Mathematica Hungarica, vol. 116, no. 1-2, pp. 121–126, 2007.

16 K. Matsumoto, I. Mihai, and R. Ros¸ca, “ξ-null geodesic gradient vector fields on a Lorentzian para- Sasakian manifold,” Journal of the Korean Mathematical Society, vol. 32, no. 1, pp. 17–31, 1995.

(12)

Submit your manuscripts at http://www.hindawi.com

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Mathematics

^{Journal of}

Hindawi Publishing Corporation http://www.hindawi.com

Differential Equations

International Journal of

Volume 2014

Applied Mathematics^{Journal of}

Mathematical PhysicsAdvances in

Complex Analysis

^{Journal of}

Optimization

^{Journal of}

Combinatorics

Journal of

Function Spaces

Abstract and Applied Analysis

International Journal of Mathematics and Mathematical Sciences

The Scientific World Journal

Discrete Dynamics in Nature and Society

Discrete Mathematics

^{Journal of}

Warped Product Submanifolds of LP-Sasakian Manifolds

Research Article