Warped Product Semi-Invariant Submanifolds in Almost Paracontact Riemannian Manifolds

Volume 2009, Article ID 621625,16pages doi:10.1155/2009/621625

Research Article

Warped Product Semi-Invariant Submanifolds in Almost Paracontact Riemannian Manifolds

Mehmet Atc¸eken

Department of Mathematics, Faculty of Arts and Sciences, Gaziosmanpas¸a University, 60250 Tokat, Turkey

Correspondence should be addressed to Mehmet Atc¸eken,[email protected] Received 2 December 2008; Revised 5 May 2009; Accepted 14 July 2009

Recommended by Fernando Lobo Pereira

We show that there exist no proper warped product semi-invariant submanifolds in almost paracontact Riemannian manifolds such that totally geodesic submanifold and totally umbilical submanifold of the warped product are invariant and anti-invariant, respectively. Therefore, we consider warped product semi-invariant submanifolds in the formN N_⊥×fNT by reversing two factor manifoldsNT andN_⊥. We prove several fundamental properties of warped product semi-invariant submanifolds in an almost paracontact Riemannian manifold and establish a general inequality for an arbitrary warped product semi-invariant submanifold. After then, we investigate warped product semi-invariant submanifolds in a general almost paracontact Riemannian manifold which satisfy the equality case of the inequality.

Copyrightq2009 Mehmet Atc¸eken. 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.

1. Introduction

It is well known that the notion of warped products plays some important role in differential geometry as well as physics. The geometry of warped product was introduced by Bishop and O’Neill1. Many geometers studied different objects/structures on manifolds endowed with an warped product metricsee2–6.

Recently, Chen has introduced the notion of CR-warped product in Kaehlerian manifolds and showed that there exist no proper warped product CR-submanifolds in the formN N_⊥×fNT in Kaehlerian manifolds. Therefore, he considered warped product CR- submanifolds in the formN N_T×fN_⊥ which is called CR-warped product, whereN_T is an invariant submanifold, andN_⊥ is an anti-invariant submanifold of Kaehlerian manifold M see 2, 7, 8. Analogue results have been obtained for Sasakian ambient as the odd dimensional version of Kaehlerian manifold by Hasegawa and Mihai in3and Munteanu in9.

Almost paracontact manifolds and almost paracontact Riemannian manifolds were defined and studied by S¸ato 10. After then, many authors studied invariant and

anti-invariant submanifolds of the almost paracontact Riemannian manifold M with the structure F, g, ξ, η, when ξ is tangent to the submanifold, and ξ is not tangent to the submanifold11.

We note that CR-warped products in Kaehlerian manifold are corresponding warped product semi-invariant submanifolds in almost paracontact Riemannian manifolds. In this paper, we showed that there exist no warped product semi-invariant submanifolds in the form N NT×fN_⊥ in contrast to Kaehlerian manifoldsseeTheorem 3.1. So, from now on we consider warped product semi-invariant submanifolds in the formN N_⊥×fNT, whereN_⊥is an anti-invariant submanifold, andN_Tis an invariant submanifold of an almost paracontact Riemannian manifoldMby reversing the two factor manifoldsNTandN_⊥,and it simply will be called warped product semi-invariant submanifold in the rest of this paper seeExample 3.3andTheorem 3.4.

2. Preliminaries

Although there are many papers concerning the geometry of semi-invariant submanifolds of almost paracontact Riemannian manifoldssee11–13, there is no paper concerning the geometry of warped product semi-invariant submanifolds of almost paracontact Riemannian manifolds in literature so far. So the purpose of the present paper is to study warped product semi-invariant submanifolds in almost paracontact Riemannian manifolds. We first review basic formulas and definitions for almost paracontact Riemannian manifolds and their submanifolds, which we shall use for later.

LetMbe anm1-dimensional differentiable manifold. If there exist onMa1,1 type tensor fieldF, a vector fieldξ,and 1-formηsatisfying

F²I−η⊗ξ, ηξ 1, 2.1

thenMis said to be an almost paracontact manifold, where⊗,the symbol, denotes the tensor product. In the almost paracontact manifold, the following relations hold good:

Fξ0, η◦F 0, rankF m. 2.2

An almost paracontact manifoldMis said to be an almost paracontact metric manifold if Riemannian metricgonMsatisfies

gFX, FY gX, Y−ηXηY, ηX gX, ξ 2.3 for allX, Y ∈ ΓTM 14, whereΓTM denotes the differentiable vector field set onM.

From2.2and2.3, we can easily derive the relation

gFX, Y gX, FY. 2.4

An almost paracontact metric manifold is said to be an almost paracontact Riemannian manifold withF, g, ξ, η-connection if∇F 0 and∇η 0, where∇denotes the connection onM.SinceF²I−η⊗ξ,the vector fieldξis also parallel with respect to∇11,13.

In the rest of this paper, let us suppose thatMis an almost paracontact Riemannian manifold with structureF, g, ξ, η-connection.

LetM be an almost paracontact Riemannian manifold, and letN be a Riemannian manifold isometrically immersed in M. For each x ∈ N, we denote by Dx the maximal invariant subspace of the tangent spaceTxN ofN. If the dimension ofDx is the same for allxinN, thenD_xgives an invariant distributionDonN.

A submanifold N in an almost paracontact Riemannian manifold is called semi- invariant submanifold if there exists onN a differentiable invariant distribution D whose orthogonal complementaryD^⊥is an anti-invariant distribution, that is,FD^⊥⊂TN^⊥, where TN^⊥ denotes the orthogonal vector bundle ofTNinTM.A semi-invariant submanifold is called anti-invariant resp., invariant submanifold if dimDx 0 resp., dimD^⊥_x 0.

It is called proper semi-invariant submanifold if it is neither invariant nor anti-invariant submanifold.

A semi-invariant submanifoldNof an almost paracontact Riemannian manifoldMis called a Riemannian product if the invariant distributionDand anti-invariant distribution D^⊥ are totally geodesic distributions in N. The geometry notion of the semi-invariant submanifolds has been studied by many geometers in the various type manifolds. Authors researched the fundamental properties of such submanifoldssee references.

Let N₁ and N₂ be two Riemannian manifolds with Riemannian metrics g₁ and g₂, respectively, andf a differentiable function onN1.Consider the product manifoldN1×N2

with its projectionπ₁:N₁×N2 → N₁andπ₂:N₁×N2 → N₂. The warped product manifold NN₁×fN₂is the manifoldN₁×N₂equipped with the Riemannian metric tensor such that gX, Y g1π1∗X, π_1∗Y f²π1xg2π2∗X, π_2∗Y 2.5 for anyX, Y ∈ΓTN, where∗is the symbol for the tangent map. Thus we haveg g1f²g2, wherefis called the warping function of the warped product. The warped product manifold N N₁×fN₂ is characterized by the fact thatN₁ and N₂ are totally geodesic and totally umbilical submanifolds ofN, respectively. Hence Riemannian products are special classes of the warped products4.

In this paper, we define and study a new class of warped product semi-invariant submanifolds in an almost paracontact Riemannian manifolds, namely, we investigate the class of warped product semi-invariant submanifolds, and we establish the fundamental theory of such submanifolds.

Now, let N be an isometrically immersed submanifold in an almost paracontact Riemannian manifoldM.We denote by∇and∇the Levi-Civita connections onNandM, respectively. Then the Gauss and Weingarten formulas are, respectively, defined by

∇XY ∇XY hX, Y,

∇XV −AVX∇^⊥_XV

2.6

for anyX, Y ∈ΓTN,V ∈ΓTN^⊥, where∇^⊥is the connection in the normal bundleTN^⊥, h is the second fundamental form ofN,andAVis the shape operator. The second fundamental formhand the shape operatorAare related by

gAVX, Y ghX, Y, V. 2.7

Now, letNbe a differentiable manifold, and we suppose thatNis an isometrically immersed submanifold in almost paracontact Riemannian manifoldM. We denote bygthe metric tensor ofMas well as that induced onN.For any vector fieldXtangent toN, we put

FXtXnX, 2.8

wheretXandnXdenote the tangential and normal components ofFX, respectively. For any vector fieldV normal toN, we also put

FV BV CV, 2.9

whereBV and CV denote the tangential and normal components ofFV, respectively. The submanifoldNis said to be invariant ifnis identically zero, that is,FTN TN. On the other hand,Nis said to be anti-invariant submanifold iftis identically zero, that is,FTN⊂ TN^⊥.

We note that for any invariant submanifoldNof an almost paracontact Riemannian manifold M, ifξ is normal toN, then the induced structure from the almost paracontact structure on N is an almost product Riemannian structure whenevertis nontrivial. Ifξ is tangent toN,then the induced structure onNis an almost paracontact Riemannian structure.

Furthermore, we say that N is a semi-invariant submanifold if there exist two orthogonal distributionsD1andD2such that

1TNsplits into the orthogonal direct sumTND1⊕D2; 2the distributionD1is invariant, that is,FD1⊆D1; 3the distributionD2is anti-invariant, that is,FD2⊆TN^⊥.

Given any submanifoldN of an almost paracontact Riemannian manifold M, from 2.4and2.8we have

gtX, Y gX, tY, gnX, V gX, BV 2.10

for anyX, Y ∈ΓTN,V ∈ΓTN^⊥.

From now on we suppose that the vector fieldξis tangent toN.

We recall the following general lemma from1for later use.

Lemma 2.1. LetNN1×fN2be a warped product manifold with warping functionf, then one has 1∇XY ∈ΓTN1for eachX, Y ∈ΓTN1,

2∇XZ∇ZXXlnfZ, for each X∈ΓTN1, Z∈ΓTN2, 3∇ZW∇^N_Z²W−gZ, Wgradf/f, for eachZ, W∈ΓTN2, where∇and∇^N2denote the Levi-Civita connections onNandN₂, respectively.

LetNbe a semi-invariant submaniold of an almost paracontact Riemannian manifold M. We denote by the invariant distributionD and anti-invariant distributionD^⊥. We also

denote the orthogonal complementary ofFD^⊥inTN^⊥byν, then we have direct sum

TN^⊥ F D^⊥

⊕ν. 2.11

We can easily see thatνis an invariant subbundle with respect toF.

3. Warped Product Semi-Invariant Submanifolds in an Almost Paracontact Riemannian Manifold

Useful characterizations of warped product semi-invariant submanifolds in almost paracon- tact Riemannian manifolds are given the following theorems.

Theorem 3.1. If N N_T×fN_⊥ is a warped product semi-invariant submanifold of an almost paracontact Riemannian manifoldMsuch thatNT is an invariant submanifold andN_⊥is an anti- invariant submanifold ofM, thenNis a usual Riemannian product.

Proof. Letξbe normal toN.Taking into account thathis symmetric and using2.3,2.6, 2.7, and consideringLemma 2.12, forX∈ΓTNTandZ, W∈ΓTN⊥, we have

g∇XZ, W g∇ZX, W g

∇ZX, W g

F∇ZX, FW η

∇ZX ηW, Xln

f

gZ, W g

∇ZFX, FW

ghZ, FX, FW g

∇FXZ, FW g

∇FXFZ, W

−gAFZFX, W −ghFX, W, FZ −g

∇WFX, FZ −g

∇WX, Z

−Xln f

gW, Z,

3.1

which implies thatXlnf 0.

Ifξis tangent toN, thenξcan be written as follows:

ξξ₁ξ₂, 3.2

whereξ1∈ΓTNTandξ2 ∈ΓTN⊥. Since∇Xξ0, from the Gauss formulae, we have

hX, ξ 0, ∇Xξ0 3.3

for anyX∈ΓTN. ConsideringLemma 2.12, we get

∇Zξ1ξ1

lnf

Z0, ∇Xξ2X lnf

ξ20,

∇ξ2ξ1∇ξ1ξ2ξ1

lnf

ξ20 3.4

for anyX∈ΓTNTandZ∈ΓTN⊥. Ifξ2is identically zero, then fromLemma 2.1we have

∇Zξ₁∇ξ1Zξ₁ lnf

Z0, ∇Xξ₁∈ΓTN_T. 3.5

It follows that the warping function f is a constant andN is usual Riemannian product.

Hence the proof is complete.

If the warping function is constant, then the metric on the “second” factor could be modified by an homothety, and hence, the warped product becomes a direct product.

Now, we give two examples for almost paracontact Riemannian manifold and their submanifolds in the form N N_⊥×fN_T to illustrate our results. Firstly, we construct an almost paracontact metric structure onR²ⁿ¹ see Example 3.2and after give an example which is concerning its submanifoldseeExample 3.3.

Example 3.2. Let

R²ⁿ¹

x1, x2, . . . , xn, y1, y2, . . . , yn, t

|xi, yi, t∈R, i1,2, . . . , n

. 3.6

The almost paracontact Riemannian structureF, g, ξ, ηis defined onR²ⁿ¹in the following way:

F ∂

∂x_i ∂

∂y_i, F ∂

∂y_i ∂

∂x_i, F ∂

∂t 0, ξ ∂

∂t, ηdt. 3.7

IfZλi∂/∂xi μi∂/∂yi ν∂/∂t∈TR²ⁿ¹, then we have

gZ, Z ⁿ

i1

λ²_i ⁿ

i1

μ²_i ν². 3.8

From this definition, it follows that

gZ, ξ ηZ ν, gFZ, FZ gZ, Z−η²Z, Fξ0, ηξ 1 3.9

for an arbitrary vector field Z. Thus R²ⁿ¹, F, g, ξ, η becomes an almost paracontact Riemannian manifold, where g and {∂/∂xi, ∂/∂y_i, ∂/∂t} denote usual inner product and standard basis ofTR²ⁿ¹, respectively.

Example 3.3. LetNbe a submanifold inR⁵with coordinatesx1, x2, y1, y2, tgiven by

x1vcosθ, x2 vsinθ, y1vcosβ, y2vsinβ, t√

2u. 3.10

It is easy to check that the tangent bundle ofNis spanned by the vectors

Z₁cosθ ∂

∂x1 sinθ ∂

∂x2 cosβ ∂

∂y1 sinβ ∂

∂y2

, Z₂−vsinθ ∂

∂x₁ vcosθ ∂

∂x₂, Z₃−vsinβ ∂

∂y₁ vcosβ ∂

∂y₂, Z₄√

2∂

∂t.

3.11

We define the almost paracontact Riemannian structure ofR⁵by

F ∂

∂xi − ∂

∂xi

, F

∂

∂yi ∂

∂yi

, F

∂

∂t 0, η 1

√2dt. 3.12

Then with respect to the almost paracontact Riemannian structureFofR⁵, the spaceFTN becomes

FZ₁ −cosθ ∂

∂x1 −sinθ ∂

∂x2 cosβ ∂

∂y1 sinβ ∂

∂y2

, FZ₂ vsinθ ∂

∂x₁ −vcosθ ∂

∂x₂, FZ₃ −vsinβ ∂

∂y₁ vcosβ ∂

∂y₂, FZ₄ 0.

3.13

SinceFZ₁ andFZ₄ are orthogonal toNand FZ₂,FZ₃ are tangent toN,D andD^⊥ can be taken subspace sp{Z1, Z₄}and subspace sp{Z2, Z₃}, respectively, whereξcan be taken asZ₄ forFZ40 andηZ4 1. Furthermore, the metric tensor ofNis given by

g 2

du²dv² v²

dθ²dβ²

2gN⊥v²gNT. 3.14

Thus N is a warped product semi-invariant submanifold with dimensional 5 of almost paracontact manifoldR⁵with warping functionfv².

Now, letNN_⊥×fNTbe a warped product semi-invariant submanifold of an almost paracontact Riemannian manifoldM, whereN_⊥is an anti-invariant submanifold, andN_Tis

an invariant submanifold ofM. If we denote the Levi-Civita connections onMandNby∇ and∇, respectively, by using2.6and2.8, we have

∇XFY F∇XY,

∇XtYhX, tY−AnYX∇^⊥_XnY t∇XY n∇XY BhX, Y ChX, Y 3.15 for anyX, Y ∈ΓTN. Taking into account the tangential and normal components of3.15, respectively, we have

∇XtY A_nYXBhX, Y, 3.16

∇XnY ChX, Y−hX, tY, 3.17

where the derivatives oftandnare, respectively, defined by

∇XtY ∇XtY−t∇XY, ∇XnY ∇^⊥_XnY−n∇XY. 3.18 Next, we are going to investigate the geometric properties of the leaves of the warped product semi-invariant submanifolds in an almost paracontact Riemannian manifold.

Theorem 3.4. Let N be a warped product semi-invariant submanifold of an almost paracontact Riemannian manifoldM.Then the invariant distributionDand the anti invariant distributionD^⊥ are always integrable.

Proof. From3.16and consideringLemma 2.11, we have

∇XFUF∇XU, Xln

f

tUhX, tU Xln f

tUBhX, U ChX, U 3.19 for anyX ∈ ΓD^⊥andU ∈ ΓD. From the tangential and normal components of3.19, respectively, we arrive at

BhX, U 0, 3.20

hX, tU ChX, U. 3.21

By using3.16and3.20we get

A_nXU−X lnf

tU. 3.22

Furthermore, by using the Gauss-Weingarten formulas and taking into account that D^⊥ is totally geodesic inNand it is anti-invariant inM, by direct calculations, it is easily to see that

AnYX −BhX, Y, 3.23

which is also equivalent to

A_nYXA_nXY 3.24

for anyX, Y ∈ΓD^⊥. Moreover, using2.4and2.7and making use ofAbeing self-adjoint, we obtain

gAnXY, Z ghY, Z, nX g

∇ZY, FX g

∇ZFY, X −gAnYZ, X −gAnYX, Z,

3.25

which gives us

A_nXY −AnYX 3.26

for anyX, Y ∈ΓD^⊥andZ∈ΓTN. Thus from3.24and3.26, we arrive at

A_nXY 0, BhX, Y 0 3.27

for any X, Y ∈ ΓD^⊥. Furthermore, by using 2.6, 2.8, and 2.9 and considering Lemma 2.13, we have

hU, tV ∇UtV F∇UV FhV, U F

∇_UV −gV, Ugradf

f BhV, U ChV, U t

∇_UV

−gV, Un

gradf

f BhV, U ChV, U

3.28

for anyV, U ∈ΓD, where∇denote the Levi-Civita connection onD.Taking into account the normal and tangential components of3.28, respectively, we have

hU, tV −gU, Vn

gradf

f ChU, V, 3.29

∇_UtV−gtV, Ugradf f t

∇_UV

BhV, U. 3.30

From3.29, we can easily see that

hU, tV hV, tU 3.31

for anyU, V ∈ΓD. Finally, by using3.17and3.31, we have nV, U n∇VU− ∇UV ∇^⊥_VnU−∇VnU− ∇^⊥_UnV ∇UnV

∇UnV−∇VnUChU, V−hU, tV−ChV, U hV, tU 0 3.32

for anyV, U∈ΓD, that is,V, U∈ΓD.

In the same way, making use of3.16and3.27for anyX, Y ∈ ΓD^⊥, we conclude that

tX, Y t∇XY− ∇YX

∇XtY−∇XtY − ∇YtX ∇YtX ∇YtX−∇XtY AnXY −AnYX0

3.33

that is,X, Y∈ΓD^⊥. So we obtain the desired result.

Since the distributionsDandD^⊥are integrable, we denote the integral manifolds ofD andD^⊥byN_TandN_⊥, respectively.

Now, the following theorem characterizeswarped product or Riemannian product semi-invariant submanifolds in almost paracontact manifolds.

Theorem 3.5. LetNbe a submanifold of an almost paracontact Riemannian manifoldM. ThenNis a semi-invariant submanifold if and onlynt0.

Proof. Let us assume that N is a semi-invariant submanifold of an almost paracontact Riemannian manifoldMand byQandP; we denote the projection operators on subspaces ΓD^⊥andΓD, respectively, then we have

PQI, P²P, Q²Q, P QQP 0. 3.34

Moreover, by using2.1,2.8, and2.9, ifξis tangent toN, then we get

X−ηXξt²XBnX, ntXCnX0, 3.35

tBV BCV 0, nBV C²V V 3.36

for anyX∈ΓTNandV ∈ΓTN^⊥. On the other hand, ifξis normal toN, then3.35and 3.36become, respectively,

Xt²XBnX, ntXCnX0, V −ηVξnBV C²V, tBVBCV 0.

3.37

From2.8, we have

FXFP XFQX,

tXnXtP XtQXnP XnQX 3.38

for anyX∈ΓTN. From the tangential and normal components, we have

tXtP XtQX, nXnP XnQX. 3.39

SinceDis invariant andD^⊥is anti-invariant, we get

nP0, Qt0. 3.40

We have

tP t 3.41

by virtue ofQI−P. Now by using the right-hand side to the second equation of3.35and using3.40and3.41, we conclude that

nt0, 3.42

which is also equivalent to

Cn0. 3.43

Conversely, for a submanifoldNof an almost paracontact Riemannian manifoldM, we suppose thatnt 0. For any vector fields tangentXtoNandV normal toN, by using 2.4and3.43, we have

gX, FV gFX, V, gX, BV gnX, V, gX, FBV gFnX, V,

gX, tBV gCnX, V 0

3.44

for allX ∈ΓTN. So we havegtBV, X 0. SinceX, tBV ∈ΓTN, it impliestB0 which is also equivalent toBC0 from3.36. SinceFξ 0, we gettξnξ0. So, from3.35and 3.36, we conclude

t³t, C³ C. 3.45

Now, if we put

Pt², QI−P, 3.46

then we can derive that P Q I, P² P, Q² Q, andP Q QP 0 which show thatQandPare orthogonal complementary projection operators and define complementary distributionsD^⊥ and D, respectively, where D and D^⊥ denote the distributions which are belong to subspacesTNTandTN_⊥, respectively. From3.42,3.45, and3.46we can derive

tP t, tQ0, QtP 0, nP 0. 3.47

These equations show that the distributionDis an invariant and the distributionD^⊥ is an anti-invariant. The proof is complete.

Theorem 3.6. LetNbe a semi-invariant submanifold of an almost paracontact Riemannian manifold M. ThenNis a warped product semi-invariant submanifold if and only if the shape operator ofN satisfies

AFXZ−X μ

FZ, X∈Γ D^⊥

, Z∈ΓD 3.48

for some functionμonNsatisfyingWμ 0,W∈ΓD.

Proof. We suppose that N is a warped product semi-invariant submanifold in an almost paracontact Riemannian manifoldM. Then from3.22, we have

A_FXZ−X lnf

FZ 3.49

for anyX ∈ΓD^⊥andZ ∈ΓD. Sincef is the only function onN_⊥, we can easily see that Wlnf 0 for allW ∈ΓD.

Conversely, let us assume that N is a semi-invariant submanifold in an almost paracontact Riemannian manifoldMsatisfying

AFXZX μ

FZ, X ∈Γ D^⊥

, Z∈ΓD 3.50

for some functionμonNsatisyingWμ 0 for allW∈ΓD. Since the ambient spaceMis an almost paracontact Riemannian manifold and making use of2.4and3.27, we arrive at

g∇XY, FZ g

∇XY, FZ g

∇XFY, Z

−gAFYX, Z 0 3.51

for anyX, Y ∈ΓD^⊥andZ∈D. Thus the anti-invariant distributionD^⊥is totally geodesic inN. In the same way, making use of∇being Levi-Civita connection and3.22, we have

g∇ZW, X g

∇ZW, X −g

∇ZX, W −g

∇ZFX, FW gAFXZ, FW X

μ

gZ, W

3.52

for anyZ, W ∈ ΓDandX ∈ ΓD^⊥, whereμ ln1/f. Since the invariant distribution D of semi-invariant submanifoldNis always integrable Theorem 3.4and Wμ 0, for eachW ∈ΓTNT, which implies that the integral manifold ofDis an extrinsic sphere inN, that is, it is a totally umbilical submanifold and its mean curvature vector field is non-zero and parallel, thus we know thatNis a Riemannian warped productN_⊥×fNT, whereN_⊥and N_Tdenote the integral manifolds of the distributions ofD^⊥ andD, respectively, andfis the warping function. So we obtain the desired result.

In the rest of this section, we are going to obtain an inequality for the squared norm of the second fundamental form by means of the warping function for warped product semi-invariant submanifolds of an almost paracontact Riemannian manifold. Now, we recall that semi-invariant N is said to be mixed geodesic resp., D-geodesic and D^⊥-geodesic submanifold if the second fundamental formhof N satisfieshX, Z 0,X ∈ ΓDand Z∈ΓD^⊥ resp.,hX, Y 0,X, Y ∈ΓDandhZ, W 0,Z, W∈ΓD^⊥.

Now, we are going to give the following lemma for later use.

Lemma 3.7. Let N N_⊥×fN_T be a warped product semi-invariant submanifold of an almost paracontact Riemannian manifoldM. Then one has

1ghD^⊥, D^⊥, FD^⊥ 0,

2ghZ, W, FX −XlnfgtZ, W,Z, W∈ΓD,X∈ΓD^⊥, 3ghX, Z, FY 0, for anyX, Y ∈ΓD^⊥andZ∈ΓD,

4ghD, FD, FD^⊥ 0 if and only ifNN_⊥×fNTis a usual Riemannian product, where DandD^⊥denote the leaves ofNTandN_⊥, respectively.

Proof. 1For anyX, Y, Z∈ΓD^⊥, by using2.4and3.27and considering that the ambient space is an almost paracontact Riemannian manifold, we have

ghX, Y, FZ g

∇XY, FZ g

∇XFY, Z

−gAnYX, Z 0. 3.53

2Making use of∇being Levi-Civita connection andLemma 2.12.2, we get ghZ, W, FX g

∇WFZ, X −g

∇WX, tZ

−Xlng·gW, tZ 3.54

for anyZ, W∈ΓD,X∈ΓD^⊥. 3In the same way, we have

ghX, Z, FY g

∇XZ, FY

g∇XtZ, Y XlnfgtZ, Y 0 3.55

for anyX, Y ∈ΓD^⊥andZ∈ΓD.

4ConsideringLemma 2.13we derive ghW, FZ, FX g

∇FZW, FX g

∇FZFW, X

−gtZ, tWX lnf

3.56

for anyZ, W∈ΓDandX∈ΓD^⊥.

Theorem 3.8. Let N N_⊥×fNT be a warped product semi-invariant submanifold of an almost paracontact Riemannian manifoldM.Then one has the following.

1The squared norm of the second fundamental form ofNinMsatisfies

h²≥ 1

f²gradf²Trt², 3.57

where Trtdenote the trace of mappingt.

2If the equality sign of 3.57holds identically, thenN_⊥is a totally geodesic,NTis a totally umbilical submanifolds ofM,andNis a mixed geodesic submanifold inM.Furthermore, N is a minimal submanifold Mif and only if Trt 0 orN N_⊥×fN_T is a usual Riemannian product.

Proof. Let {e1, e₂, . . . , e_p, e¹, e², . . . , e^q, N₁, N₂, . . . , N_s, ξ} be an orthonormal basis of an almost paracontact Riemannian manifold M such that {e1, e₂, . . . , e_p} is tangent to ΓTN⊥,{e¹, e², . . . , e^q}is tangent toΓTNT,and{N1, N2, . . . , Ns}is tangent toΓν. Taking into accountLemma 3.7and the basic linear algebra rules, by direct calculations, we have

hX, Y g

hX, Y, Nj

N_j, 1≤j ≤s, hZ, W −eilnftZ, WFeig

hZ, W, Nj

N_j, 1≤i≤p, hX, Z g

hX, Z, Nj

Nj

3.58

for allX, Y∈ΓTN_⊥andZ, W∈ΓTNT. Since

h² p i,j1

s 1

g h

e_i, e_j , N₂

2 p

i1

q k1

s 1

g h

e_i, e^k , N₂

q r,k1

p i1

eilnf₂ g

te^k, e^r2

q r,k1

s 1

g h

e^k, e^r , N

2

,

3.59

here by direct calculations, we get ei

lnf₂ 1

f²gradf², Trt q k1

g te^k, e^k

. 3.60

So we conclude that

h²≥ 1

f²gradf²Trt², 3.61

which proves our assertion.

Now we assume that the equality case of3.57holds identically, then from 3.58, respectively, we obtain

h

D^⊥, D^⊥

0, hD, D∈Γ F

D^⊥

, 3.62

h D^⊥, D

0. 3.63 Since N_⊥ is totally geodesic submanifold in N, the first condition in 3.62 implies that N_⊥ is totally geodesic submanifold inM. Moreover,Lemma 2.13shows thatNT is totally umbilical submanifold inN. Therefore, the second condition in3.62implies thatN_Tis also totally umbilical submanifold inM. On the other hand, 3.20and 3.63 imply thatN is mixed geodesic submanifold inM.

Conclusion 3.9. The geometry of the warped products in Riemannian manifolds is totally different from the geometry of the warped products in complex manifolds. Namely, in the complex manifolds, there exists no proper warped product CR-submanifold in the formN N_⊥×fN_Tsee2,8while there exists no proper warped product semi-invariant submanifold in the formNN_T×fN_⊥in Riemannian manifoldsseeTheorem 3.1. The first condition in 3.62implies that warped product CR-submanifold is minimal in complex manifolds while it does not imply that warped product semi-invariant submanifold is minimal in Riemannian product manifolds.

Acknowledgment

The author would like to thank the referees for valuable suggestions and comments, which have improved the present paper.

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