Semi-Slant Warped Product Submanifolds of a Kenmotsu Manifold

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Volume 2012, Article ID 708191,10pages doi:10.1155/2012/708191

Research Article

Semi-Slant Warped Product Submanifolds of a Kenmotsu Manifold

Falleh R. Al-Solamy

¹

and Meraj Ali Khan

²

1Department of Mathematics, King Abdulaziz University, P. O. Box 80015, Jeddah 21589, Saudi Arabia

2Department of Mathematics, University of Tabuk, P. O. Box 741, Tabuk, Saudi Arabia

Correspondence should be addressed to Meraj Ali Khan,[email protected] Received 1 February 2012; Revised 29 March 2012; Accepted 19 April 2012 Academic Editor: Kwok W. Wong

Copyrightq2012 F. R. Al-Solamy and M. A. Khan. 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 semi-slant warped product submanifolds of a Kenmotsu manifold. We obtain a characterization for warped product submanifolds in terms of warping function and shape operator and finally proved an inequality for squared norm of second fundamental form.

1. Introduction

In1Tanno classified the connected almost contact metric manifold whose automorphism group has maximum dimension, there are three classes:

ahomogeneous normal contact Riemannian manifolds with constantφholomorphic sectional curvature if the sectional curvature of the plane section contains ξ, say CX, ξ>0;

bglobal Riemannian product of a line or a circle and Kaehlerian manifold with constant holomorphic sectional curvature,CX, ξ 0;

ca warped product spaceR×fCⁿ, ifCX, ξ<0.

Manifolds of classaare characterized by some tensorial equations, it has a Sasakian structure and manifolds of class b are characterized by some tensor equations called Cosymplectic manifolds. Kenmotsu 2obtained some tensorial equations to Characterize manifolds of classc, these manifolds are called Kenmotsu manifolds.

The notion of semi-slant submanifolds of almost Hermitian manifolds was introduced by Papaghiuc3after that cabrerizo et al.4defined and studied semi-slant submanifolds in the setting of almost contact manifolds.

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Bishop and O’Neill 5 introduced the notion of warped product manifolds. These manifolds are generalization of Riemannian product manifolds and occur naturally. Recently, many important physical applications of warped product manifolds have been discovered, giving motivation to study of these spaces with diﬀerential geometric point of view. For instance, it has been accomplished that warped product manifolds provide an excellent setting to model space time near black hole or bodies with large gravitational fields c.f., 5–7. Due to wide applications of these manifolds in physics as well as engineering this becomes a fascinating and interesting topic for research, and many articles are available in literaturec.f.,3,8,9.

Recently, Atc¸eken 10 proved that the warped product submanifolds of type N_θ×fN_T andN_θ×fN_⊥ of a Kenmotsu manifoldM do not exist where the manifoldsN_θ and N_T resp. N_⊥ are proper slant and invariant resp., anti-invariant submanifolds of Kenmotsu manifoldM, respectively. After that Siraj-Uddin et al.11investigated warped product of the typesN_T×fN_θandN_⊥×fN_θ and obtained some interesting results. In this continuation we obtain a characterization and an inequality for squared norm of second fundamental form.

2. Preliminaries

A 2n1 dimensionalC^∞manifoldMis said to have an almost contact structure if there exist onMa tensor fieldφ of type1,1, a vector fieldξ, and 1-formη satisfying the following properties:

φ² −Iη⊗ξ, φξ 0, η◦φ0, ηξ 1. 2.1

There always exists a Riemannian metricgon an almost contact manifoldMsatisfying the following conditions:

g

φX, φY

gX, Y−ηXηY, ηX gX, ξ, 2.2

whereX, Yare vector fields onM.

An almost contact metric structureφ, ξ, η, gis said to be Kenmotsu manifold, if it satisfies the following tensorial equation2:

∇Xφ Y g

φX, Y

ξ−ηYφX, 2.3

for anyX, Y ∈TM, whereTM is the tangent bundle ofMand ∇denotes the Riemannian connection of the metricg. Moreover, for a Kenmotsu manifold

∇XξX−ηXξ. 2.4

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LetMbe a submanifold of an almost contact metric manifoldMwith induced metric g and if∇and ∇^⊥ are the induced connection on the tangent bundleTM and the normal bundleT^⊥MofM, respectively, then Gauss and Weingarten formulae are given by

∇XY ∇XY hX, Y,

∇XN−ANX∇^⊥_XN,

2.5

for eachX, Y ∈TMandN ∈ T^⊥M, wherehandA_Nare the second fundamental form and the shape operator, respectively, for the immersion ofMintoMand they are related as

ghX, Y, N gANX, Y, 2.6 wheregdenotes the Riemannian metric onMas well as onM.

For anyX∈TM, we write

φXP XFX, 2.7

whereP Xis the tangential component andFXis the normal component ofφX.

Similarly, for anyN∈T^⊥M, we write

φNtNfN, 2.8

wheretNis the tangential component andfNis the normal component ofφN. The covariant derivatives of the tensor fieldPandFare defined as

∇XP

Y ∇XP Y−P∇XY, ∇XF

Y ∇^⊥_XFY−F∇XY.

2.9

From2.3,2.5,2.7and2.8we have ∇XP

Y A_FYXthX, Y−gX, P Yξ−ηYP X, 2.10 ∇XF

Y fhX, Y−hX, P Y−ηYFX. 2.11 Definition 2.1see12. A submanifoldMof an almost contact metric manifoldMis said to be slant submanifold if for anyx ∈ MandX ∈ TxM− ξ is constant. The constant angle θ ∈ 0, π/2 is then called slant angle of Min M. Ifθ 0 the submanifold is invariant submanifold, ifθ π/2 then it is anti-invariant submanifold, ifθ /0,π/2 then it is proper slant submanifold.

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For slant submanifolds of contact manifolds Cabrerizo et al.13proved the following lemma.

Lemma 2.2. LetMbe a submanifold of an almost contact manifoldM, such thatξ∈TMthenMis slant submanifold if and only if there exists a constantλ∈0,1such that

P²λ

I−η⊗ξ

, 2.12

whereλ−cos²θ.

Thus, one has the following consequences of above formulae:

gP X, P Y cos²θ

gX, Y−ηXηY

, 2.13

gFX, FY sin²θ

gX, Y−ηXηY

. 2.14

A submanifold M of M is said to be semi-slant submanifold of an almost contact manifold M if there exist two orthogonal complementary distributions D_T and D_θ on M such that

iTMD_T⊕D_θ⊕ ξ ,

iithe distributionD_Tis invariant that is,φD_T ⊆D_T, iiithe distributionD_θis slant with slant angleθ /0.

It is straight forward to see that semi-invariant submanifolds and slant submanifolds are semi-slant submanifolds withθπ/2 andDT {0}, respectively.

Ifμis invariant subspace underφof the normal bundleT^⊥M, then in the case of semi- slant submanifold, the normal bundleT^⊥Mcan be decomposed as

T^⊥Mμ⊕FD_θ. 2.15

A semi-slant submanifoldMis called a semi-slant product if the distributionsD_Tand D_θare parallel onM. In this caseMis foliated by the leaves of these distributions.

As a generalization of the product manifolds and in particular of a semi-slant product submanifold, one can consider warped product of manifolds which are defined as

Definition 2.3. LetB, gBandF, gFbe two Riemannian manifolds with Riemannian metric gBandgF, respectively, andfbe a positive diﬀerentiable function onB. The warped product ofBandFis the Riemannian manifoldB×F, g, where

g gBf²gF. 2.16

For a warped product manifoldN1×fN2, we denote byD1andD2 the distributions defined by the vectors tangent to the leaves and fibers, respectively. In other words,D₁ is obtained by the tangent vectors ofN₁via the horizontal lift, andD₂is obtained by the tangent vectors ofN2via vertical lift. In case of semi-slant warped product submanifoldsD1andD2

are replaced byD_TandD_θ, respectively.

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The warped product manifoldB×F, gis denoted byB×fF. IfXis the tangent vector field toMB×fFatp, qthen

X²dπ1X²f² p

dπ2X². 2.17

Bishop and O’Neill5proved the following.

Theorem 2.4. LetMB×fFbe warped product manifolds. IfX,Y ∈TBandV,W∈TFthen i∇XY ∈TB,

ii∇XV ∇VX Xf/fV, iii∇VW −gV, W/f∇f.

∇fis the gradient offand is defined as g

∇f, X

Xf, 2.18

for allX∈TM.

Corollary 2.5. On a warped product manifoldMN₁×fN₂, the following statements hold:

iN1is totally geodesic inM, iiN2is totally umbilical inM.

Throughout, one denotes by NT and Nθ an invariant and a slant submanifold, respectively, of an almost contact metric manifoldM.

Khan et al.14proved the following corollary.

Corollary 2.6. LetMbe a Kenmotsu manifold and N1 andN2 be any Riemannian submanifolds ofM, then there do not exist a warped product submanifoldM N₁×fN₂ ofMsuch that ξ is tangential toN2.

Thus, one assumes that the structure vector fieldξ is tangential toN1 of a warped product submanifoldN1×fN2ofM.

In this paper we will consider the warped product of the typeN_θ×fN_TandN_T×fN_θ. The warped product of the typeN_θ×fN_Tis called warped product semi-slant submanifolds;

this type of warped product is studied by Atc¸eken10, they proved that the warped product N_θ×fN_T does not exist. The warped product of the type N_T×fN_θ is called semi-slant warped product; these submanifolds were studied by Siraj-Uddin et al.11and they proved the following Lemma

Lemma 2.7. LetMN_T×fN_θbe warped product semi-slant submanifold of a Kenmotsu manifold Msuch thatξis tangent toNT, whereNTandNθare invariant and proper slant submanifolds ofM.

then

ighX, Z, FP Z ghX, P Z, FZ {Xlnf−ηX}cos²θZ², iighX, Z, FZ −P XlnfZ²,

for anyX ∈TN_TandZ∈TN_θ.

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ReplacingXbyP Xin partiiof above lemma one has

ghP X, Z, FZ XlnfZ². 2.19

3. Semi-Slant Warped Product Submanifolds

Throughout this section we will study the warped product of the typeN_T×fN_θ, for these submanifolds byTheorem 2.4we have

∇XZ∇ZXXlnfZ, 3.1

for anyX∈TNTandZ∈TNθ.

Lemma 3.1. Let M N_T×fN_θ be a semi-slant warped product submanifolds of a Kenmotsu manifoldM, then

ghX, Y, FZ 0, 3.2

for anyX, Y ∈TNTandZ∈TNθ.

Proof. AsN_T is totally geodesic inMthen∇XPY ∈TN_Tand therefore by formula2.10:

∇XP

Y thX, Y−gX, P Yξ−ηYP X, 3.3

taking inner product withZ∈TN_θwe get3.2.

Now we have the following Characterization.

Theorem 3.2. A semi-slant submanifoldM of a Kenmotsu manifoldMwith integrable invariant distributionD_T⊕ ξ and integrable slant distributionD_θis locally a semi-slant warped product if and only if∇ZP Z∈Dθand there exists aC^∞- functionαonMwithZα0,

A_FZX

Xα−ηX

P Z−P XαZ, 3.4

for allX∈DT⊕ ξ andZ∈Dθ. Proof. From2.10and3.1we have

AFZXthX, Z 0. 3.5

Similarly,

P XlnfZ−XlnfP ZthX, Z−ηXP Z, 3.6

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from3.5and3.6, we get

A_FZX XlnfP Z−P XlnfZ−ηXP Z, 3.7 taking inner product withW∈TN_θ, we have

gAFZX, W

Xlnf−ηX

gP Z, W−P XlnfgZ, W. 3.8 FromLemma 3.1and3.8we get the desired result.

Conversely, letMbe a semi-slant submanifold ofMsatisfying the hypothesis of the theorem, then for anyX, Y ∈DT⊕ ξ andZ∈Dθ

ghX, Y, FZ 0, 3.9

that meanshX, Y∈μ. Then from2.11

−F∇XY fhX, Y−hX, P Y. 3.10 SincehX, Y ∈ μ, then we haveF∇XY 0, that is, ∇XY ∈ D_T ⊕ ξ . Hence, each leaf of DT⊕ ξ is totally geodesic inM.

Further, suppose Nθ be a leaf of Dθ and hθ be second fundamental form of the immersion ofN_θinM, then for anyX∈D_T⊕ ξ andZ∈D_θ, we have

g

hθZ, Z, φX g

∇ZZ, φX

, 3.11

using2.7and2.5, the above equation yields g

hθZ, Z, φX

g∇ZP Z, X gAFZZ, X, 3.12

applying3.4, we get

g

h_θZ, Z, φX

−P XlnfgZ, Z. 3.13

ReplacingXbyP X, the above equation gives

hθZ, Z ∇αgZ, Z. 3.14

From above equation it is easy to derive

hθZ, W ∇αgZ, W, 3.15 that is,Nθis totally umbilical and asZα0, for allZ∈Dθ, ∇μis defined onNT, this mean that mean curvature vector ofN_θis parallel, that is, the leaves ofD_θare extrinsic spheres in

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M. Hence by virtue of result of15which says that if the tangent bundle of a Riemannian manifoldMsplits into an orthogonal sumTME₀⊕E₁of nontrivial vector subbundles such thatE₁is spherical and its orthogonal complementE₀is autoparallel, then the manifoldMis locally isometric to a warped productM0×fM1, we can sayMis locally semi-slant warped product submanifoldNT×fNθ, where the warping functionfe^α.

Let us denote byD_T andD_θthe tangent bundles onN_T andN_θ, respectively, and let {X0 ξ, X1, . . . , Xp, X_p1 φX1, . . . , X2p φXp}and{Z1, . . . , Zq, Z_q1 P Z1, . . . , Z2q P Zq} be local orthonormal frames of vector fields onNTandNθ, respectively, with 2pand 2qbeing real dimension. SincehX, ξ 0 for allX ∈TM, then the second fundamental form can be written as

h² ^2p

i,j1

g h

Xi, Xj

, h Xi, Xj

^2p

i1

2q r1

ghXi, Zr, hXi, Zr

2q r,s1

ghZr, Z_s, hZr, Z_s.

3.16

Now, on a semi-slant warped product submanifold of a Kenmotsu manifold, we prove the following.

Theorem 3.3. Let M NT×fNθ be a semi-slant warped product submanifold of a Kenmotsu manifoldMwithNTandNθinvariant and slant submanifolds, respectively, ofM. IfηX≥2Xlnf for allX∈TNT, then the squared norm of the second fundamental formhsatisfies

h²≥4qcsc²θ 1cos⁴θ∇lnf², 3.17

where∇lnfis the gradient of lnfand 2qis the dimensionN_θ. Proof. In view of the decomposition2.15, we may write

hU, V hFDθU, V hμU, V, 3.18 for eachU, V ∈TM, wherehFDθU, V∈FDθandhμU, V∈μwith

h_FD_θU, V 2q r1

h^rU, VFZr, 3.19

h^rU, V csc²θghU, V, FZr, 3.20

for eachU, V ∈TM. In view of above formulae we have

ghFDθP Xi, Z_r, hFDθP Xi, Z_r gh^rP Xi, Z_rFZr, h^rP Xi, Z_rFZr

gh^sP Xi, ZrFZr, h^sP Xi, ZrFZr. 3.21

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Now using2.14and2.19

ghFDθP Xi, Z_r, hFDθP Xi, Z_r h^rP Xi, Z_rXilnfsin²θ

s /r

h^sP Xi, Z_r². 3.22

In view of3.20and2.19, we get

ghFDθP Xi, Z_r, hFDθP Xi, Z_r csc²θ

X_ilnf₂

sin²θ

s /r

h^sP Xi, Z_r². 3.23

Summing overi1, . . . ,2pandr1, . . . ,2qthe above equation yields 2p

i1

2q r1

ghFDθP Xi, Z_r, hFDθP Xi, Z_r 4qcsc²θ∇lnf² sin²θ

2p i1

2q r,s1,r /s

h^sP Xi, Z_r².

3.24

Since we have choose the orthonormal frame of vector fields onDθ as{Z1, . . . , Zq, Z_q1 P Z1, . . . , Z2qP Zq}, then the second term in the right-hand side of3.24is written as

csc²θ 2p

i1

_q

r1

ghP Xi, Zr, FP Zr₂

ghP Xi, P Zr, FZr₂

q r1

q s1,s /r

ghP Xi, Z_r, FP Zs2

ghP Xi, P Z_r, FZs2 .

3.25

From partiofLemma 2.7, the first two terms of above equation can be written as

csc²θ 2p i1

2q

X_ilnf−ηXi₂ cos⁴θ

. 3.26

In account of to hypothesisηXi≥2X_ilnfthe above expression is greater than equal to the following term:

4qcsc²θ∇lnf²cos⁴θ. 3.27

Using above inequality into3.24, we have

ghFDθP Xi, Z_r, hFDθP Xi, Z_r≥4qcsc²θ 1cos⁴θ∇lnf². 3.28 The inequality3.17follows from3.16and3.28.

The equality holds ifhDT, DT 0,hDθ, Dθ 0,hP X, Zis orthogonal toFZand FP Zfor allX∈D_TandZ∈D_θandηX 2Xlnf.

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4. Conclusion

In this paper we study nontrivial warped product submanifolds of a Kenmotsu manifold and in this study there emerge natural problems of finding the estimates of the squared norm of second fundamental form and to find the relation between shape operator and warping function. This study predict the geometric behavior of underlying warped product submanifolds. Further, as it is known that the warping function of a warped product manifold is a solution of some partial diﬀerential equationsc.f.,8and most of physical phenomenon is described by partial diﬀerential equations. We hope that our study may find applications in physics as well as in engineering.

Acknowledgment

The work is supported by Deanship of Scientific research, University of Tabuk, Saudi Arabia.

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