Geometry of warped product immersions of Kenmotsu space forms and its applications to slant immersions

Journal of Geometry and Physics 114 (2017) 276–290

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Geometry of warped product immersions of Kenmotsu space forms and its applications to slant immersions Akram Ali a , Pişcoran Laurian-Ioan b, * a

Institute of Mathematical Sciences, Faculty of Science, University of Malaya, 50603, Kuala Lumpur, Malaysia Department of Mathematics and Computer Science, Victoriei 76, North University Center of Baia Mare Technical University of Cluj Napoca, 430122, Baia Mare, Romania b

article

info

Article history: Received 1 October 2016 Received in revised form 29 November 2016 Accepted 6 December 2016 Available online 15 December 2016 MSC: primary 53C40 secondary 53C20 53C42 53B25 53Z05 53C40 58J60 Keywords: Warped products Semi-slant immersions Connected Compact Riemannian submanifolds Kenmotsu space forms Hamiltonian Kinetic energy

a b s t r a c t In this paper, some relations among the second fundamental form which is an extrinsic invariant, Laplacian of the warping function and constant sectional curvature of a warped product semi-slant submanifold of a Kenmotsu space form and its totally geodesic and totally umbilical submanifolds are described from the exploitation of the Gauss equation instead of the Codazzi equation in the sense of Chen’s studies in (2003). These relations provide us an approach to the classifications of equalities by the following case studied of Hasegawa and Mihai (2003). These are exemplified by the classifications of the totally geodesic and totally umbilical submanifolds. Moreover, we provide some applications of the inequality case by using the harmonicity of the smooth warping functions. In particular, we prove the triviality of connected, compact warped product semi-slant manifolds isometrically immersed into a Kenmotsu space form using Hamiltonian, Hessian, and the Kinetic energy of the warped function. Further, we generalize some results for contact CRwarped products in a Kenmotsu space form. © 2016 Elsevier B.V. All rights reserved.

1. Introduction and motivations The study of warped product submanifolds of different ambient manifold with constant sectional curvature has been an active field of research in Riemannian submanifolds theory for years. Especially, Chen in [1,2] inaugurated the idea of CR-warped product submanifolds in Kaehler manifolds and he constructed the first inequality for the second fundamental form which is related to the squared norm of warping function. So far, these types of inequalities represent one of the most fundamental problems regarding the warped product submanifolds. Therefore, many authors have discussed a lot of interesting geometric properties of these warped product immersions types (for examples see [3–14] and [15–20] for more details). The beautiful theory of the warped product submanifolds has rich applications in both Riemannian geometry and semi-Riemannian geometry. Until now, the concept of warped product has been playing a crucial role in the theory of general relativity which provides the best mathematical model of the universe. The warped product model was successfully applied in general relativity and semi-Riemannian geometry in the direction to build basic cosmological models such as the Robertson-Walker spacetime, the Friedmann cosmological model and the standard static spacetime [21].

*

Corresponding author. E-mail addresses: [email protected] (A. Ali), [email protected] (P. Laurian-Ioan).

http://dx.doi.org/10.1016/j.geomphys.2016.12.001 0393-0440/© 2016 Elsevier B.V. All rights reserved.

A. Ali, P. Laurian-Ioan / Journal of Geometry and Physics 114 (2017) 276–290

277

It is well known that the branch of warped product manifolds comes into light after initiated by Bishop and O’Neill [22] as a simple generalization of Riemannian product manifolds. According to them, these manifolds are defined as follows: Definition 1.1. Let (N1 , g1 ) and (N2 , g2 ) be two Riemannian manifolds and f : N1 → (0, ∞), a positive differentiable function on N1 . Consider the product manifold N1 × N2 with its canonical projections γ1 : N1 × N2 → N1 , γ2 : N1 × N2 → N2 and the projection maps given by γ1 (t , s) = t and γ2 (t , s) = s for every l = (t , s) ∈ N1 × N2 . The warped product M = N1 ×f N2 is the product manifold N1 × N2 equipped with the Riemannian structure such that

∥X ∥2 = ∥γ1 ∗ (U)∥2 + f 2 (γ1 (t))∥π2 ∗ (X )∥2 ,

(1.1)

for any tangent vector X ∈ X (Tt M), where ∗ is the symbol of differential maps and g = g1 + f 2 g2 is the Riemannian metric on M. Thus the function f is called a warping function of M. The following lemma can be seen as a direct consequence of the warped product manifolds: Lemma 1.1 ([22]). Let M = N1 ×f N2 be a warped product manifold. For any X , Y ∈ X (TN1 ) and Z , W ∈ X (TN2 ), we have (i) ∇X Y ∈ X (TN1 ), (ii) ∇Z X = ∇X Z = (X ln f )Z , (iii) ∇Z W = ∇ ′Z W − g(Z , W )∇ ln f , where ∇ is the Levi-Civita connection on M and ∇ ln f is the gradient of ln f which is defined as g(∇ ln f , U) = U ln f . Now we will give the following definition based on the above lemma, Definition 1.2. A warped product manifold M = N1 ×f N2 is said to be trivial if the warping function f is constant or in other words, a warped product manifold M = N1 ×f N2 is called simply a Riemannian product if f is constant function on M. Definition 1.3. If M = N1 ×f N2 is a warped product manifold, then N1 (N2 ) are called totally geodesics (or totally umbilical submanifold) of M, respectively. By the isometrically embedding theorem of J. F. Nash [23] we know that every Riemannian manifold can be isometrically immersed into an Euclidean space with sufficiently high dimension. Afterward, followed the concept of Nolker [24], Chen p in [7] developed a sharp inequality under the name of another general inequality in a CR-warped product NTh ×f N⊥ in a complex space form M m (4c) with a holomorphic constant sectional curvature 4c by means of Codazzi equation satisfying the relation

( ) ∥σ ∥2 ≤ 2p ∥∇ ln f ∥2 + ∆(ln f ) + 4hc , where h = dimC NT , p = dim N⊥ and σ is the second fundamental form. Therefore, it is called Chen’s second inequality of the second fundamental form. Inspired by these studies, other geometers in [8,10,11,25] obtained some sharp inequalities for the squared norm of the second fundamental form, which is an extrinsic invariant, in terms of the warping function for the contact CR-warped products isometrically immersed in both a Sasakian space form and a Kenmotsu space form using the same techniques by taking equation of Codazzi. Some classifications of contact CR-warped products in spheres which satisfy the equality cases identically are given. After the slant immersion concept was introduced in [26,27] and corresponding slant curve studied in [28,29], this impose significant restrictions on the geometry of its generalization. Therefore, the warped product semi-slant submanifold might be regarded as the simplest generalization of CR-warped product submanifolds. For instance, the warped product semislant submanifold does not admits non-trivial warped product in some ambient manifolds (see [30,31]). Moreover, in [6], M. Atceken studied the non existence of warped product semi-slant submanifolds of a Kenmotsu manifold for which the structure vector ξ is tangent to fiber. Meanwhile, the existence of warped product semi-slant submanifold of a Kenmotsu manifold of the form M = NT ×f Nθ or M = Nθ ×f NT , in the case that the structure vector field ξ is tangent to NT or Nθ respectively, has been proved in [32,33]. They have obtained lots of examples on the existence of the warped product semislant products in a Kenmotsu manifold and derived many general inequalities for the second fundamental form in terms of warping functions. Consequently, there was a difficulty to describe the differential geometric properties and to establish a relation between the squared norm of the second fundamental form and warping functions in terms of slant immersions by using Codazzi. Applying a new method under assumption of the Gauss equation instead of the Codazzi equation, we establish a sharp general inequality for the warped product semi-slant submanifolds which are isometrically immersed into a Kenmotsu space form as a generalization of the contact CR-warped products. To do these, we consider a non-trivial warped product semi-slant n n submanifold of the type M n = NT 1 ×f Nθ 2 in a Kenmotsu manifold and obtain the following result. n

n

˜ 2m+1 (c) is an isometric immersion of a warped product semi-slant Theorem 1.1. Assume that χ : M n = NT 1 ×f Nθ 2 → M n1 n2 2m + 1 ˜ NT ×f Nθ into a Kenmotsu space form M (c). Then

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(i) The squared norm of the second fundamental form of M n is defined as

( 2

c−3

2

∥h∥ ≥ 2n2 ∥∇ ln f ∥ +

4

n1 −

c+1 4

) − ∆(ln f ) ,

(1.2)

n

n

where n1 and n2 are the dimensions of invariant NT 1 and slant submanifold Nθ 2 , respectively. n n ˜ 2m+1 (c). (ii) The equality sign holds in (1.2) if and only if NT 1 is totally geodesic and Nθ 2 is a totally umbilical submanifold in M n 2m + 1 ˜ Moreover, M is a minimal submanifold in M (c). Another goal of our was to study the geometry of the second fundamental form of a compact oriented Riemannian ˆ submanifold M isometrically immersed into a Kenmotsu space form M(c), when M is a product manifold and the first ˜ is a warped product. Some authors in [4,5,10,11,13] proved the triviality fundamental form of the immersion i : M → M(c) results for the CR-warped products in the different ambient manifolds with constant sectional curvatures by following to [8]. One of the main problem concerning warped product semi-slant into a Kenmotsu space form is how we can give nice conditions for a warped product submanifold to be a Riemannian product submanifold. As a direct application of Theorem 1.1, by considering the equality case, we prove the following interesting theorem which shows that every compact oriented warped product semi-slant submanifold into a Kenmotsu space form is trivial with necessary and sufficient condition such that: n

n

˜ 2m+1 (c) is an isometric immersion of a compact orientable proper warped Theorem 1.2. Assume that χ : M n = NT 1 ×f Nθ 2 −→ M n2 n1 ˜ 2m+1 (c). Then M n is simply a Riemannian product if product semi-slant submanifold NT ×f Nθ into a Kenmotsu space form M and only if n1 n2 ∑ ∑

( ∥hµ (ei , ej )∥ = n2 2

c−3 4

i=1 j=1

n1 −

c+1 4

) + 2cot θ + 1 , 2

(1.3)

where θ is a slant angle defined on TM and hµ is the component of the second fundamental form h in Γ (µ). The proof of Theorem 1.2 is based on the so called Bochner∫technique (see [34]). We will use Green theorem on a compact manifold M and given a smooth function f : M → R, one has M ∆fdV = 0. We can ∫ attribute the result of K. Yano and M. Kon from (see [34]), immediately follows: ∆f = −div (∇ f ) and from Green lemma M div (X )dV = 0. Similarly, by Theorem 1.1 and the Hessian of the warping function, we prove another result about the triviality of the warped product semi-slant submanifolds into a Kenmotsu space form by taking idea from [35,36] as follows. n n ˜ 2m+1 (c) be a warped product semi-slant submanifold into a Kenmotsu space form Theorem 1.3. Let χ : M n = NT 1 ×f Nθ 2 −→ M ˜ 2m+1 (c). If the inequality M

( 2

∥h∥ ≥ 2n2

c−3 4

n1 −

c+1

+1+

4

α ( ∑

H

ln f

(ei , ei ) + H

ln f

(ϕ ei , ϕ ei )

))

,

(1.4)

i=1 n

n

holds, where H ln f is Hessian of warping function ln f , then M n is simply a Riemannian product of NT 1 and Nθ 2 , respectively. Finally, we consider a similar problem for a connected, compact warped product semi-slant submanifold in a Kenmotsu space form with nonempty boundary. In this case there exists a trivial warped product semi-slant submanifold into a Kenmotsu space form which provides a necessary and sufficient condition in terms of kinetic energy function. As a new result, we will give now the following: n

n

Theorem 1.4. Assume that M n = NT 1 ×f Nθ 2 is a connected, compact warped product semi-slant submanifold in a Kenmotsu ˜ 2m+1 (c), then M n is a Riemannian product of NTn1 and Nθn2 if and only if the kinetic energy satisfies the following space form M equality E(ln f ) =

1 4

tan θ 2

(∫

(c + 3 Mn

4

n1 −

c−1 4

n2 n2 1 ∑∑

)

+ 2cot θ + 1 dV − 2

n2

)

)

∥hµ (ei , ej )∥ dV , 2

(1.5)

i=1 j=1

where E(ln f ) represents the kinetic energy of the warping function ln f and dV is volume element on M n . Many other results will be also presented in this paper, some of them as corollaries. 2. Preliminaries

˜ with an almost contact structure (ϕ, ξ , η) is called almost contact metric manifold A (2m + 1)-dimensional manifold M if the following properties are satisfied ϕ 2 = −I + η ⊗ ξ , η(ξ ) = 1, ϕ (ξ ) = 0, η ◦ ϕ = 0, η(X ) = g(X , ξ )

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279

and g(ϕ X , ϕ Y ) = g(X , Y ) − η(X )η(Y )

˜ where ϕ , ξ and η are called (1, 1) tensor field, a structure vector field and dual 1-form, respectively. for any X , Y ∈ Γ (T M), ˜X ϕ )Y = g(ϕ X , Y )ξ − η(X )ϕ Y , Furthermore, an almost contact metric manifold is said to be a Kenmotsu manifold (cf. [37]) if (∇ ˜ 2m+1 , where ∇ ˜X ξ = X − η(X )ξ , for any vector fields U , V on M ˜ denotes the Riemannian connection with respect to g and ∇ ˜ to denote Lie algebra of vector fields on a manifold M ˜ 2m+1 . For more studies on tangent and we shall use a symbol X (T M) bundle (see [38]). ˜ 2m+1 with induced metric g. If ∇ and ∇ ⊥ are the induced Let M n be a submanifold of an almost contact metric manifold M connections on the tangent bundle TM and the normal bundle T ⊥ M of M n , respectively. Thus the Gauss and Weingarten formulas are given by ˜U V = ∇U V + h(U , V ), (ii) ∇ ˜U N = −AN U + ∇ ⊥ (i) ∇ U N,

(2.1)

for each U , V ∈ X (TM) and N ∈ X (T ⊥ M), where h and AN are the second fundamental form and the shape operator ˜ 2m+1 , respectively. They are related as corresponding to the normal vector field N for the immersion of M n into M 2m+1 ˜ g(h(U , V ), N) = g(AN U , V ), where g is Riemannian metric on M as well as the metric induced on M. Now for any U ∈ X (TM) and N ∈ X (T ⊥ M), we have (i) ϕ U = PU + FU ,

(ii) ϕ N = tN + fN ,

(2.2)

where PU(tN) and FU(fN) are tangential and normal components of ϕ U(ϕ N), respectively. It is easy to observe the properties of Riemannian metric g and (1, 1)-tensor field P : for each U , V ∈ X (TM), we have (i) g(PU , V ) = −g(U , PV ) (ii) ∥P ∥2 =

n ∑

g 2 (Pei , ej ).

(2.3)

i,j=1

For a submanifold M n , the Gauss equation is defined as:

˜ R(U , V , Z , W ) = R(U , V , Z , W ) + g(h(U , Z ), h(V , W )) − g(h(U , W ), h(V , Z )),

(2.4)

˜ 2m+1 and M n , respectively. The mean curvature for any U , V , Z , W ∈ X (TM), where ˜ R and R are the curvature tensors on M vector H for an orthonormal frame {e1 , e2 , . . . , en } of tangent space TM on M n is defined by H=

1 n

n 1∑

trace(h) =

n

δ (ei , ei ),

(2.5)

i=1

where n = dimM. Also we set hrij = g(h(ei , ej ), er ) and ∥h∥2 =

n ∑

g(h(ei , ej ), h(ei , ej )).

(2.6)

i,j=1

˜ is given by The scalar curvature ρ for a submanifold M of an almost contact metric manifold M ∑

τ (TM) =

K (ei ∧ ej ),

(2.7)

1≤i̸ =j≤n

where K (ei ∧ ej ) is the sectional curvature of the plane section spanned by ei and ej . Let Gr be an r-plane section on TM and {e1 , e2 , . . . , er } any orthonormal basis of Gr . Then the scalar curvature ρ (Gr ) of Gr is given by

τ (Gr ) =

∑

K (ei ∧ ej ).

(2.8)

1≤i̸ =j≤r

⃗ f is given by Now, let f be a differential function defined on M n . Thus the gradient ∇ ⃗ f , X ) = Xf , (i) g(∇

⃗ f ∥2 = and (ii) ∥∇

n ∑ (

)2

ei (f ) .

(2.9)

i=1

From the above equations, the Hamiltonian in a local orthonormal frame is defined by H(df , x) =

n 1∑

2

j=1

df (ej )2 =

n 1∑

2

j=1

ej (f )2 =

1 2

∥∇ f ∥2 .

(2.10)

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Moreover, the Laplacian ∆f of f is also given by

∆f =

n n ∑ ∑ {(∇ei ei )f − ei (ei (f ))} = − g(∇ei grad f , ei ). i=1

(2.11)

i=1

Similarly, the Hessian of the function f is given by

∆f = −TraceH f = −

n ∑

H f (ei , ei ),

(2.12)

i=1

where H f is Hessian of function f . The compact manifold M n will be considered as without boundary such that ∂ M n = ∅. Thus, we have the following lemma (Hopf’s lemma). Lemma 2.1 ([39]). Let M n be a connected, compact Riemannian manifold and f a smooth function on M n such that ∆f ≥ 0(or ∆f ≤ 0). Then f is a constant function on M n . Moreover, for a compact orientable Riemannian manifold M n without boundary, then we have the following formula:

∫ Mn

∆f dV = 0,

(2.13)

such that dV denotes the volume of M n [34]. When M n is a manifold with boundary, the Hopf’s lemma becomes the uniqueness theorem for the Dirichlet problem. Thus we have the following result; Theorem 2.1 ([39]). Let M n be a connected, compact manifold and f a positive differentiable function on M n such that ∆f = 0, on M and f∂ M = 0. Then f = 0, where ∂ M is the boundary of M n . Additionally, let M n to be a compact Riemannian manifold and f to be a positive differentiable function on M n . Thus the kinetic energy function is defined as in [39] as follows: E(f ) =

1

∫

2

Mn

( ) ∥∇ f ∥2 dV.

(2.14)

The Euler–Lagrange equation which correspond to above function is defined as: Theorem 2.2 ([39]). The Euler–Lagrange equation for the Lagrangian function L =

∆f = 0.

1 2

∥f ∥2 is (2.15)

Some definitions related to submanifolds theory are:

˜ 2m+1 is said to be totally Definition 2.1. A submanifold M n isometrically immersed into an almost contact metric manifold M umbilical and totally geodesic if h(U , V ) = g(U , V )H and h(U , V ) = 0 for all U , V ∈ X (TM), respectively, where H is the ˜ 2m+1 . mean curvature vector of M n . Furthermore, if H = 0, then M n is minimal in M Definition 2.2. A submanifold M n tangent to ξ , is called a contact CR-submanifold if admits a pair of differentiable distributions D and D⊥ such that D is invariant and its orthogonal complementary distribution D⊥ is anti-invariant such that TM = D ⊕ D⊥ ⊕ ⟨ξ ⟩ with ϕ (Dx ) ⊆ Dx and ϕ (Dx⊥ ) ⊂ Tx⊥ M, for every x ∈ M.

˜ 2m+1 be a Kenmotsu manifold with an almost contact structure (ϕ, ξ , η) and M n be a submanifold Definition 2.3. Let M ˜ 2m+1 . Then M n is called invariant if ϕ (Tx M) ⊆ Tx M tangent to the structure vector field ξ which is isometrically immersed in M ⊥ and it is called anti-invariant if ϕ (Tx M) ⊂ Tx M for every x ∈ M where Tx M denotes the tangent space of M n at the point x. Moreover, M is called a slant submanifold if for all non zero vector U tangent to M at a point x, the angle of θ (U) between ϕ U and Tx M is constant such that, it does not depend on the choice of x ∈ M and U ∈ X (Tx M − ⟨ξ (x)⟩). Except invariant, anti-invariant and slant submanifolds, there are several other classes of submanifolds determined by the behavior of tangent space of the submanifold under the action of the (1, 1) tensor field ϕ of ambient manifold. Cabrerizo et al. [27] extended of the above definition into a characterization for a slant submanifold in a contact metric manifold.

˜ 2m+1 . Then M n is slant if and only if there exists a Lemma 2.2. Let M n be a submanifold of an almost contact metric manifold M constant λ ∈ [0, 1] such that P 2 = −λ I + η ⊗ ξ .

(

)

(2.16)

Furthermore, in such a case, we have λ = cos θ . 2

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281

˜ the following relations are consequences of Hence, for a slant submanifold M n of an almost contact metric manifold M, Lemma 2.2, (

)

g(PU , PV ) =cos θ g(U , V ) − η(U)η(V ) , 2

(

(2.17)

)

g(FU , FV ) =sin θ g(U , V ) − η(U)η(V ) , 2

(2.18)

for any U , V ∈ X (TM). In [40], Cabrerizio, et al. presented the semi-slant submanifolds as natural generalizations of the contact CR-submanifolds in the setting of contact manifolds. They defined these submanifolds as follows:

˜ 2m+1 is said to be a semi-slant submanifold if there Definition 2.4. A submanifold M n of an almost contact metric manifold M θ exist two orthogonal distributions D and D such that (i) TM = D ⊕ Dθ ⊕ ⟨ξ ⟩, where ⟨ξ (p)⟩ is a 1-dimensional distribution spanned by ξ (p). (ii) D is invariant, i.e., ϕ (D) ⊆ D, (iii) Dθ is a slant distribution with slant angle θ . Suppose that n1 and n2 denote the dimensions of invariant distribution D and slant distribution Dθ of semi-slant ˜ 2m+1 , respectively. Then we describe the following remarks and submanifold in an almost contact metric manifold M definitions. Definition 2.5. M n is invariant if n2 = 0 and slant if n1 = 0. Definition 2.6. If the slant angle θ =

π 2

, then M n is called a contact CR-submanifold.

Definition 2.7. If the slant angle θ ∈ (0,

π 2

), then M n is called a proper semi-slant submanifold.

Remark 2.1. Let µ be an invariant subspace under ϕ of normal bundle T ⊥ M. Then, in the case of semi-slant submanifold, the normal bundle T ⊥ M can be decomposed as T ⊥ M = F Dθ ⊕ µ. 3. General inequality for warped product semi-slant and its application In this section, we will discuss some geometric properties of the mean curvature for warped product pointwise semi-slant submanifolds and we will use these results to derive a general inequality. First, we study some geometric aspects about submanifolds. n n ˜ 2m+1 is an isometric immersion of warped product N1n1 ×f N2n2 into a Riemannian Assume that χ : M n = N1 1 ×f N2 2 → M ˜ with a constant section curvature c. If n1 , n2 and n are the dimensions of N1 , N2 , and N1n1 ×f N2n2 , respectively, manifold M n n then, for unit vector fields X and Z which are tangent to N1 1 and N2 2 , respectively, we have K (X ∧ Z ) = g(∇Z ∇X X − ∇X ∇Z X , Z ) =

1 f

{(∇X X )f − X 2 f }.

(3.1) n

If we consider the local orthonormal frame {e1 , e2 , . . . , en } such that e1 , e2 , . . . , en1 are tangent to N1 1 and en1 +1 , . . . , en are n tangent to N2 2 , we have

∆f f

=

n ∑

K (ei ∧ ej )

(3.2)

i=1

for each j = n1 + 1, . . . , n. Now we are able to prove the general characterization. To do this we need some preparatory lemmas and definitions. On the other hand, we know that, there are two types of warped product semi-slant submanifolds as: n

n

(i) Nθ 2 ×f NT 1 n

n

(ii) NT 1 ×f Nθ 2 . We recall the following result of Atceken in [6] for the classification of first case. n

n

˜ 2m+1 , Theorem 3.1 ([6]). There is no a proper warped product semi-slant submanifold M n = Nθ 2 ×f NT 1 in a Kenmotsu manifold M n2 n1 where Nθ is a proper slant submanifold and NT is an invariant submanifold tangent to ξ of M. The following result will help us to prove the equality case.

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A. Ali, P. Laurian-Ioan / Journal of Geometry and Physics 114 (2017) 276–290 n

n

˜ 2m+1 Theorem 3.2 ([33]). Assume that M n = NT 1 ×f Nθ 2 is a warped product semi-slant submanifold into a Kenmotsu manifold M n1 such that ξ is tangent to NT . Then ξ ln f = 1,

(3.3)

g(h(X , Z ), FPZ ) = g(h(X , PZ ), FZ ) =

(

)

(X ln f ) − η(Z ) cos2 θ∥Z ∥2 ,

(3.4)

g(h(X , Z ), FZ ) = −(ϕ X ln f )∥Z ∥2 ,

(3.5)

g(h(X , Y ), FZ ) = 0,

(3.6)

for any X , Y ∈ X (TNT ) and Z ∈ X (TNθ ). Note : Throughout the study of this paper, we consider only (ii) case for complete classifications of non-trivial warped product semi-slant submanifold of Kenmotsu manifolds when structure vector ξ is tangent to first factor of warped product manifolds. Thus we obtain the following important theorem such as: n n ˜ 2m+1 be an isometrically immersion of a warped product semi-slant NTn1 ×f Nθn2 Theorem 3.3. Let χ : M n = NT 1 ×f Nθ 2 → M n 2m+1 ˜ into a Kenmotsu manifold M such that ξ is tangent NT 1 . Then

(i) The squared norm of the second fundamental form of M is given by

(

)

∥h∥ ≥ 2 n2 ∥∇ ln f ∥ + ˜ τ (TM) − ˜ τ (TNT ) − ˜ τ (TNθ ) − n2 ∆(ln f ) , 2

2

(3.7)

where n2 is the dimension of slant submanifold Nθ . n n ˜ 2m+1 . Moreover, (ii) The equality in (3.7) holds if and only if NT 1 is totally geodesic and Nθ 2 is a totally umbilical submanifold in M ˜ 2m+1 . M n is a minimal submanifold in M Proof. The theorem can be directly derived with the help of the Theorem 4.4 in [9] by replacing a slant submanifold instead of a non-invariant Riemannian submanifold to change the fiber of the warped product submanifold. Remark 3.1. It is difficult to obtain Chen second inequality for the second fundamental form and its relation to the warping functions with in fact of slant immersions (Hokkaido Math. J., 32(2003), 415-444) by using Codazzi equation. Therefore, Theorem 3.3 is very useful to construct Chen’s type inequalities in terms of slant immersions. 3.1. Applications of Theorem 3.3 to Kenmotsu space forms As a direct application of the previous Theorem 3.3, we will establish some results on warped product semi-slant submanifold which is isometrically immersed into Kenmotsu space forms.

˜ 2m+1 is said to be a Kenmotsu space form M ˜ 2m+1 (c) with constant ϕ -sectional Proof of Theorem 1.1. A Kenmotsu manifold M ˜ curvature c if and only if the Riemannian curvature tensor R is given by (see [6]) ˜ R(X , Y , Z , W ) =

c−3

(

4

+

) g(Y , Z )g(X , W ) − g(X , Z )g(Y , W )

c+1 4

( η(X )η(Z )g(Y , W ) + η(W )η(Y )g(X , Z )

− η(Y )η(Z )g(X , W ) − η(X )g(Y , Z )η(W ) + g(ϕ Y , Z )g(ϕ X , W ) ) − g(ϕ X , Z )g(ϕ Y , W ) + 2g(X , ϕ Y )g(ϕ Z , W ) . Putting X = W = ei , Y = Z = ej in the above equation, we derive

( ) c+3 ˜ R(ei , ej , ej , ei ) = g(ei , ei )g(ej , ej ) − g(ei , ej )g(ej , ei ) 4

+

c+1 4

( η(ei )η(ej )g(ei , ej ) − η(ej )η(ej )g(ei , ei )

(3.8)

A. Ali, P. Laurian-Ioan / Journal of Geometry and Physics 114 (2017) 276–290

283

+ η(ei )η(ej )g(ei , ej ) − η(ei )η(ei )g(ej , ej )

) + g(ϕ ej , ej )g(ϕ ei , ei ) − g(ϕ ei , ej )g(ϕ ej , ei ) + 2g (ei , ϕ ej ) . 2

Summing up over the vector fields on TNT in the above equation, one obtains

{ 2˜ τ (TNT ) =

c−3 4

n1 (n1 − 1) −

c + 1( 4

} ) 2(n1 − 1) + 3∥P ∥2 .

As we know that ξ (p) is tangent to TNT for a n1 -dimensional invariant submanifold, then using (2.3)(ii), we find ∥P ∥2 = n1 − 1 and then we get 2˜ τ (TNT ) =

c−3 4

{n1 (n1 − 1)} +

c+1 4

(n1 − 1).

Similarly, for a slant submanifold, we have ∥P ∥2 = n2 cos2 θ , then with the help of (2.3)(ii) and (2.16), c+1 {n2 (n2 − 1)} + (3n2 cos2 θ ). 4 4 Summing up over basis vectors of TM such that 1 ≤ i ̸ = j ≤ n, we have

˜ τ (TNθ ) =

c−3

2˜ τ (TM) =

c+3 4

n(n − 1) +

c−1

(

) 3

4

∑

g (ϕ ei , ej ) − 2(n − 1) . 2

(3.9)

1≤i̸ =j≤n

˜ 2m+1 (c). According to [41], we will set the following Let M n be a proper semi-slant submanifold of Kenmotsu space form M frame e1 , e2 =ϕ e1 , . . . , e2α−1 , e2α = ϕ e2α−1 , e2α+1 , e2α+2 = sec θ Pe2α+1 , . . . e2α−1 , e2α = sec θ Pe2α−1 , . . ., · · · e2α+2β−1 , e2dα+2β = sec θ Pe2α−1 , e2α+2β , e2α+2β+1 = ξ . Obviously, we derive g 2 (ϕ ei , ei+1 ) = 1, for i ∈ {1, . . . , 2α − 1}

= cos2 θ fori ∈ {2α + 1, . . ., 2α + 2β − 1}. Then one obtains n ∑

g 2 (Pei , ej ) = 2(α + β cos2 θ ).

(3.10)

i,j=1

From (3.9) and (3.10), it follows that 2˜ ρ (TM) =

c−3 4

n(n − 1) +

c+1

(

) 6(α + β cos θ ) − 2(n − 1) . 2

4

(3.11)

Now using the above relations in Theorem 3.3, we get the required result (1.2). Moreover, the equality case holds according to Theorem 3.3 (ii) and this completes the proof of the theorem. The following results are direct consequences of Theorem 1.1,

˜ 2m+1 (c) be a Kenmotsu space form with c ≤ 3. Then there does not exists a warped product semi-slant Corollary 3.1. Let M n1 n2 n 2m+1 ˜ NT ×f Nθ into M (c) such that ln f is the eigenfunction of the Laplacian on NT 1 with respect to the eigenvalue λ > 0. ˜ 2m+1 (c) is a Kenmotsu space form with c ≤ 3. Then there does not exist a warped product semi-slant Corollary 3.2. Assume that M n1 n2 ˜ 2m+1 (c) such that ln f is a harmonic function on the invariant submanifold NTn1 . submanifold NT ×f Nθ into M We obtain the following corollary using Definition 2.6 which is generalized to results for contact CR-warped product submanifold into a Kenmotsu space form. n

n

˜ 2m+1 (c) is an isometrically immersion of a contact CR-warped product Corollary 3.3. Assume that χ : M n = NT 1 ×f N⊥2 → M n1 n2 2m ˜ +1 (c). Then submanifold NT ×f N⊥ into a Kenmotsu space form M

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(i) The squared norm of the second fundamental form of M n is given by

( 2

2

∥h∥ ≥ 2n2 ∥∇ ln f ∥ +

c−3 4

p−

c+1 4

) − ∆(ln f ) ,

(3.12)

n

where N⊥2 is the anti-invariant submanifold. n n (ii) The equality sign holds in (3.12) if and only if NT 2 is totally geodesic submanifold and N⊥1 is a totally umbilical submanifold ˜ 2m+1 (c). Moreover, M n is minimal in M(c) ˜ 2m+1 (c). in M 4. Applications to compact orientable warped product semi-slant submanifolds In this section, we consider compact Riemannian manifolds without boundary. Using integration theory on compact orientable manifolds, we obtain some characterizations. Proof of Theorem 1.2. Assuming that in the inequality (1.2), the equality case holds, one obtains

∥h(D, D)∥2 + ∥h(Dθ , Dθ )∥2 + 2∥h(D, Dθ )∥2 =

(c − 3) 2

n1 n2 −

(c + 1) 2

(

)

n2 + 2n2 ∥∇ ln f ∥2 − ∆(ln f ) .

(4.1)

n

˜ For the equality cases of Theorem 1.1(ii), we see that NT 1 is totally geodesic in M(c), then its second fundamental should n2 2m+1 ˜ be zero such that h(D, D) = 0 and Nθ is totally umbilical in M (c), then h(Z , W ) = g(Z , W )H. But M n is a minimal ˜ 2m+1 (c) and also M n is Dθ −minimal due to minimality of NTn1 . Hence, H = 0, which means that submanifold in M h(Z , W ) = 0. This implies that ∥h(Dθ , Dθ )∥2 = 0, thus from (4.1), one derives n1 (c − 3) 4

−

c+1 4

−

1 n2

∥h(D, Dθ )∥2 + ∥∇ ln f ∥2 = ∆(ln f ).

(4.2)

As from the hypothesis, M n is a compact orientable submanifold, then M n is closed and bounded. Hence, taking integration by the volume element dV over M n , using (2.13), we arrive at

∫ (

c−3

Mn

4

n1 n2 − n

c+1 4

)

∫ (

n2 dV = Mn

) θ

∥h(D, D )∥ − n2 ∥∇ ln f ∥ 2

2

dV.

(4.3)

n

Assume that M n = NT 1 ×f Nθ 2 is an n = n1 + n2 -dimensional warped product semi-slant submanifold of 2m + 1-dimensional n

n

n

n

˜ 2m+1 with NT 1 of dimension n1 = 2α+ 1 and Nθ 2 of dimension n2 = 2β , where Nθ 2 and NT 1 are integral Kenmotsu manifold M manifolds of Dθ and D ⊕ ξ , respectively, then we consider that {e1 , e2 , . . . , eα , eα+1 = ϕ e1 , . . . , e2α = ϕ eα , e2α+1 = ξ }, and {e2α+2 = e∗1 , . . . , e2α+1+β = e∗β , e2α+β+2 = e∗β+1 = sec θ Pe∗1 , . . ., ep+q = e∗q = sec θ Pe∗β } are orthonormal frames of TNT and TNθ , respectively. Thus, the respective orthonormal frames of the normal sub-bundles, F Dθ and µ are {en+1 = e¯ 1 = csc θ Fe∗1 , . . ., en+β = e¯ β = csc θ Fe∗1 , en+β+1 = e¯ β+1 = csc θ sec θ FPe∗1 , . . ., en+2β = e¯ 2β = csc θ sec θ FPe∗β }and {en+2β+1 , . . . , e2m+1 }. Next, we consider X = ei and Z = ej for 1 ≤ i ≤ n1 and 1 ≤ j ≤ n2 , respectively, then we define h(ei , ej ) =

n2 ∑

2m+1

g(h(ei , ej ), Fe∗r )Fe∗r +

r =1

∑

g(h(ei , ej ), τ el )τ el

l=1 n

n

for τ ∈ X (µ). Taking summation over the vector fields NT 1 and Nθ 2 and using an adapted frame for the semi-slant submanifold, we expressed as: n1 n1 ∑ ∑ i=1 j=1

g(h(ei , ej ), h(ei , ej )) = csc2 θ

β α ∑ ∑

g(h(ei , e∗j ), Fe∗k )2

i=1 j,k=1

+ csc2 θ sec2 θ

β α ∑ ∑

g(h(ei , Pe∗j ), Fe∗k )2

i=1 j,k=1

+ csc2 θ sec2 θ

β α ∑ ∑

g(h(ϕ ei , e∗j ), FPe∗k )2

i=1 j,k=1

+ csc2 θ sec2 θ

β α ∑ ∑ i=1 j,k=1

g(h(ϕ ei , e∗j ), FPe∗k )2

A. Ali, P. Laurian-Ioan / Journal of Geometry and Physics 114 (2017) 276–290

+ csc2 θ sec4 θ

β α ∑ ∑

285

g(h(ϕ ei , Pe∗j ), FPe∗k )2 .

i=1 j,k=1

+ csc2 θ sec2 θ

β α ∑ ∑

g(h(ϕ ei , Pe∗j ), Fe∗k )2

i=1 j,k=1

β α ∑ ∑

+ csc2 θ

g(h(ϕ ei , e∗j ), Fe∗k )2

i=1 j,k=1

+ csc2 θ sec4 θ

d2 α ∑ ∑

g(h(ei , Pe∗j ), FPe∗r )2

i=1 j,k=1

+

n1 n2 ∑ ∑

2m+1

∑

g(h(ei , ej ), er )2 .

i=1 j=1 r =n+n2 +1

Thus from Theorem 3.2 and using the fact that we have an orthonormal frame in the above equation, we get

∥h(D, Dθ )∥2 = 4cot2 θ

β 2α ∑ ∑

(ei ln f )2 g(e∗j , e∗k ) + 2

β 2α ∑ ∑

i=i j,k=1

∑

+

i=i j,k=1

n1

2m+1

(ϕ ei ln f )2 g(e∗j , e∗k )

n2

∑∑

g(h(ei , ej ), er )2 .

r =n+n1 +1 i=1 j=1

Adding and subtracting the same terms in the above equation satisfying equality (2.9)(ii), we get β

α+1 ∑ ( ) 2∑ ∥h(D, Dθ )∥2 = 2 1 + 2cot2 θ (ei ln f )2 g(e∗j , e∗k ) i=i j,k=1

− 2 1 + 2cot θ 2

(

β )∑

(ξ ln f )2 g(e∗j , e∗k )

j,k=1 n1 n2 ∑ ∑

2m+1

∑

+

g(h(ei , ej ), er )2 .

r =n+n2 +1 i=1 j=1

Hence, using (3.3), we find

∥h(D, Dθ )∥2 = n2 (1 + 2cot2 θ )∥∇ ln f ∥2 − n2 n1 n2 ∑ ∑ − 2n2 cot2 θ + ∥hµ (ei , ej )∥2 .

(4.4)

i=1 j=1

Now putting the above value in (4.3), we obtain

∫ (

c−3

Mn

4

n1 n2 −

c+1 4

) n2 + n2 + n2 cot θ dV = 2

∫ (∑ n1 n2 ∑ Mn

) ∥hµ (ei , ej )∥

2

dV

i=1 j=1

+ 2cot2 θ

(

∫ Mn

) ∥∇ ln f ∥2 dV.

(4.5)

If (1.3) is satisfied, then (4.5) implies that ln f is a constant function for a proper semi-slant submanifold M n . Thus, M n n n becomes a Riemannian product of the invariant and slant submanifolds NT 1 and Nθ 2 , respectively. Conversely, suppose that M n is simply a Riemannian product, then the warping function ln f must be constant, i.e., ∇ ln f = 0. Thus from relation (4.5) implies the equality (1.3). This completes the proof of the theorem. n

n

˜ 2m+1 (c) be an isometric immersion of a compact orientable warped product Theorem 4.1. Let χ : M n = NT 1 ×f Nθ 2 −→ M ˜ 2m+1 (c). Then M n is a trivial warped product if and only if semi-slant submanifold into a Kenmotsu space form M ∥ h∥ 2 ≥

c−3 2

n1 n2 −

c+1 2

n2 , n

n

where n1 and n2 are dimensions of NT 1 and Nθ 2 , respectively.

286

A. Ali, P. Laurian-Ioan / Journal of Geometry and Physics 114 (2017) 276–290

Proof. If the inequality holds in Theorem 1.1, thus simplification gives c−3 2

n2 n2 −

c+1 2

n2 + ∥∇ ln f ∥2 −

1 2n2

∥h∥2 ≤ ∆(ln f ).

(4.6)

From the integration theory of oriented Riemannian manifolds, we know that on a compact orientable Riemannian manifold M n without boundary, from (2.13) and (4.6), we will get

∫ (

c−3

Mn

2

n1 n2 −

c+1 2

1

2

n2 + ∥∇ ln f ∥ −

2n2

) 2

∥ h∥

∫ dV ≤ Mn

∆(ln f )dV = 0.

If the following inequality holds

∥h∥2 ≥

c−3 2

n1 n2 −

c+1 2

n2 .

Then

∫ Mn

(∥∇ ln f ∥2 )dV ≤ 0.

(4.7)

We know that the integration must be positive for a positive function. Hence, from (4.7), we get ∥∇ ln f ∥2 ≤ 0, but ∥∇ ln f ∥2 ≥ 0, which implies that ∇ ln f = 0, i.e., f is a constant function on M n . Thus M n becomes a Riemannian product n n manifold of NT 1 and Nθ 2 . The reciprocal follows easily and this completes the proof of the theorem. Note: Similarly, we can generalize some results for contact CR-warped product submanifolds into Kenmotsu space forms. If we are substituting the slant angle θ = π2 in Theorems 4.1 and 1.2 respectively (see Definition 2.6), then, we derive the following corollaries. n

n

˜ 2m+1 (c) be an isometric immersion of a compact orientable CR-warped product Corollary 4.1. Let χ : M n = NT 1 ×f N⊥2 −→ M ˜ 2m+1 (c). Then, M n is trivial CR-warped product if and only if submanifold into a Kenmotsu space form M ∥h∥2 ≥

c−3 2

n

n1 n2 −

c+1 2

n2 ,

n

where NT 1 and N⊥2 are invariant and anti-invariant submanifolds, respectively. n

n

˜ 2m+1 (c) is an isometric immersion of a compact orientable CR-warped Corollary 4.2. Assume that χ : M n = NT 1 ×f N⊥2 −→ M n n 2m+1 ˜ product submanifold into a Kenmotsu space form M (c) such that NT 1 is an invariant submanifold tangent to ξ and N⊥2 is an ˜ 2m+1 (c), then M n is simply a Riemannian product if and only if anti-invariant submanifold in M n1 n2 ∑ ∑

∥hµ (ei , ej )∥2 = n2

(c + 3 4

i=1 j=1

n1 −

c−1 4

) +1 ,

where hµ is the second fundamental form of component in Γ (µ).

˜ with Corollary 4.3 ([6]). There are not any compact orientable contact CR product submanifold in a Kenmotsu space form M(c) c < 3. Moreover, we are giving some applications based on the minimum principal properties. n n ˜ 2m+1 (c) is an isometric immersion of a warped product semi-slant Theorem 4.2. Assume that χ : M n = NT 1 ×f Nθ 2 −→ M

˜ 2m+1 (c). Let λT be a non-zero eigenvalue of the Laplacian on the compact invariant submanifold into a Kenmotsu space form M n submanifold NT 1 . Then ∫

2

n

NT 1

∥h∥ dVT ≥

(

∫

c+3 2

n

NT 1

+ 2n2 λT

n1 n2 −

∫ n

NT 1

c−1 2

) n2 dVT

(ln f )2 dVT ,

(4.8)

n

where dVT is the volume element of NT 1 . Proof. Thus, using the minimum principle property, we have

∫ n

NT 1

∥∇ ln f ∥2 dVT ≥ λT

∫ n

NT 1

(ln f )2 dVT .

From (1.2) and (4.9), we get the required result (4.8) and this completes the proof of theorem.

(4.9)

A. Ali, P. Laurian-Ioan / Journal of Geometry and Physics 114 (2017) 276–290 n

287

n

˜ 2m+1 (c) be a warped product semi-slant into a Kenmotsu space form M ˜ 2m+1 (c) Theorem 4.3. Let χ : M n = NT 1 ×f Nθ 2 −→ M n1 n1 such that NT is a compact submanifold and λT be a non-zero eigenvalue of the Laplacian on NT . Then (

∫

n1 n2 ∑ ∑

n

NT 1

) ∥hµ (ei , ej )∥

2

(

∫ dVT ≥

c−3

n

NT 1

i=1 j=1

4

c+1

n1 n2 −

4

∫

− 2n2 cot2 θλT

n

NT 1

) n2 + 2n2 cot θ + 1 dVT 2

(ln f )2 dVT .

(4.10)

Proof. The proof follows from (4.5) and (4.9). 5. Applications to kinetic energy and Hamiltonian In this section, we consider connected, compact Riemannian manifolds with boundary, i.e., ∂ M ̸ = ∅. We will apply connectedness and compactness to the warped product submanifolds. Then, we obtain some necessary and sufficient conditions in terms of kinetic energy and the Hamiltonian of the warping functions which provide that a non-trivial warped product manifold is to be a trivial warped product submanifold. Proof of Theorem 1.4. Combining Eqs. (4.2), and (4.4), we obtain c−3 4

n1 n2 −

c+1 4

n2 + 2n2 cot2 θ + n2 = n2 ∆(ln f ) +

n1 n2 ∑ ∑

∥hµ (ei , ej )∥2 + 2n2 cot2 θ∥∇ ln f ∥2 .

(5.1)

i=1 j=1

Taking integration of the volume element dV over M n with nonempty boundary in the above equation, we find that

∫ (

c−3 4

Mn

n1 n2 −

c+1 4

) n2 + 2n2 cot2 θ + n2 dV = n2

+

∫

( Mn

) ∆(ln f ) dV

∫ (∑ n1 n2 ∑ Mn

) ∥hµ (ei , ej )∥2 dV

i=1 j=1

+ 2n2 cot2 θ

∫

( Mn

) ∥∇ ln f ∥2 dV.

(5.2)

From (2.14) and (5.2), it follows that

∫ ( Mn

c−3 4

n2 −

c+1 4

) + 2cot θ + 1 dV = 2

∫ Mn

+

∆(ln f )dV

∫ (∑ n1 n2 ∑

1 n1

Mn

) ∥hµ (ei , ej )∥

2

dV

i=1 j=1

+ 4cot2 θ E(ln f ).

(5.3)

Equality (1.5) holds if and only if the above condition from (5.3) take place such that

∫ Mn

∆(ln f )dV = 0 on M n ,

which implies that

∆(ln f ) = 0.

(5.4)

n

Since M is a connected, compact warped product semi-slant submanifold, then from (5.4) and Theorem 2.1, imply that ln f = 0 H⇒ f = 1, which means that f is constant on M n . Hence, the theorem is proved completely. In similar way, we derive a characterization in terms of Hamiltonian operator by taking idea from [42] as follows. n

n

˜ 2m+1 (c) be an isometric immersion from a warped product semi-slant into a Theorem 5.1. Let χ : M = NT 1 ×f Nθ 2 −→ M 2m+1 n ˜ Kenmotsu space form M (c). If M is connected compact, then M n is a trivial warped product semi-slant if and only if the Hamiltonian of warping function satisfies the following equality H d(ln f ), x =

(

)

1 4

( tan θ 2

c+3 4

n1 −

c−1 4

+ 2cot θ + 1 − 2

n1 n2 1 ∑∑

n2

i=1 j=1

) ∥hµ (ei , ej )∥

2

.

(5.5)

288

A. Ali, P. Laurian-Ioan / Journal of Geometry and Physics 114 (2017) 276–290

Proof. Using (2.18) in (5.1), we derive 4cot2 θ H d(ln f ), x + ∆(ln f ) =

(

)

c+3 4

n1 −

c−1 4

+ 2cot2 θ + 1 −

n1 n2 1 ∑∑

n2

∥hµ (ei , ej )∥2 .

(5.6)

i=1 j=1

Since, Eq. (5.5) implies that ∆(ln f ) = 0 on M n , by Theorem 2.1, then M n is a trivial warped product submanifold. This completes the proof of theorem. 6. Applications to Hessian of warping functions In this section, we will give some applications in terms of Hessian of positive differentiable functions. We derive some triviality conditions to prove that a warped product semi-slant submanifold in a Kenmotsu space form turn into a Riemannian product manifold without using compactness. Proof of Theorem 1.3. From (2.11), we can express as:

∆(ln f ) = −

n ∑

g(∇ei grad ln f , ei )

i=1

=−

2α ∑

g(∇ei grad ln f , ei ) +

2β ∑

g(∇ej grad ln f , ej )

j=1

i=1

+ g(∇ξ grad ln f , ξ ). In a warped product submanifold of a Kenmotsu manifold, we conclude from (3.3) that g(∇ξ grad ln f , ξ ) = 1. Thus, we simplify as:

∆(ln f ) = −

α ∑ (

g ∇ei grad ln f , ei −

)

i=1

i=1

−

α ∑ ) ( g ∇ϕ ei grad ln f , ϕ ei

β ∑ (

g ∇ej grad ln f , ej

)

j=1

− sec2 θ

β ∑ ) ( g ∇Pej grad ln f , Pej + 1. i=1

Taking account that ∇ is a Levi-Civita connection on M n , by (2.11), we derive

∆(ln f ) = −

( α ∑

) H

ln f

(ei , ei ) + H

ln f

(ϕ e i , ϕ e i )

+1

i=1

−

( β ∑

) ej g(grad ln f , ej ) − g(∇ej ej , grad ln f ) .

j=1

− sec θ 2

β ∑

(

) Pej g(grad ln f , Pej ) − g(∇Pej Pej , grad ln f ) .

j=1

Using the gradient property of functions, from (2.9), we arrive at

∆(ln f ) = −

( α ∑

) H

ln f

(ei , ei ) + H

ln f

(ϕ e i , ϕ e i )

+1

i=1

−

( β ∑

) ej (ej ln f ) − (∇ej ej ln f )

j=1

− sec θ 2

β ∑ j=1

(

) Pej (Pej ln f ) − (∇Pej Pej ln f ) .

A. Ali, P. Laurian-Ioan / Journal of Geometry and Physics 114 (2017) 276–290

289

More simplifying, we get

∆(ln f ) = −

α ∑

(

) H

ln f

(ei , ei ) + H

ln f

(ϕ e i , ϕ e i )

+1

i=1

−

β ∑

( ej

( g(grad f , e ) ) j

f

j=1

( β ∑

− sec θ 2

Pej

)

1

− g(∇ej ej , grad f ) f

( g(grad f , Pe ) ) j

f

j=1

)

1

− g(∇Pej Pej , grad f ) . f

n

But from the hypothesis of warped product we know that NT 1 is defined as totally geodesic in M n . It implies that grad f ∈ Γ (TNT ), and, from Lemma 1.1(ii), we obtain

∆(ln f ) = −

α ∑

(

) H

ln f

(ei , ei ) + H

ln f

(ϕ e i , ϕ e i )

+1

i=1

−

β ∑

(

)

g(ej , ej )∥∇ ln f ∥ + sec θ g(Pej , Pej )∥∇ ln f ∥ 2

2

2

.

j=1

From (2.17), we obtain

∆(ln f ) = −

α ∑

(

)

H

ln f

(ei , ei ) + H

ln f

(ϕ e i , ϕ e i )

− 2n2 ∥∇ ln f ∥2 + 1.

(6.1)

i=1

Thus, from (1.2) and (6.1), it follows that

( 2

∥h∥ ≥ 2n2

c−3 4

+

n1 −

α ∑ [

c+1 4

+ (n2 + 2)∥∇ ln f ∥2

)

H

ln f

(ei , ei ) + H

ln f

(ϕ e i , ϕ e i ) + 1 .

]

(6.2)

i=1

If the inequality (1.4) holds, then, (6.2) implies that (n2 + 2)∥∇ ln f ∥2 ≤ 0, But n2 ≤ −2 is not true and hence we have ∥∇ ln f ∥2 ≤ 0 which is impossible because the norm of a function. Thus it can be concluded that grad ln f = 0, and this means that f is a constant function on M n . Hence, M n becomes a trivial warped product semi-slant or Riemannian product and this completes the proof of the theorem. n

n

Theorem 6.1. Let M n = NT 1 ×f Nθ 2 be a non-trivial warped product semi-slant submanifold into a Kenmotsu space form ˜ 2m+1 (c) such that a slant angle θ ̸= arc cot √n2 . Then necessary and sufficient condition of M n to be a trivial warped product M is given by n1 n2 ∑ ∑

∥hµ (ei , ej )∥ = 2

i=1 j=1

c−3 4

+ n2

n1 n2 −

( α ∑

c+1 4

(

n2 + 1 + n2 1 + 2cot θ 2

)

) H

ln f

(ei , ei ) + H

ln f

(ϕ ei , ϕ ei ) .

(6.3)

i=1

Proof. From the relations (4.2), (4.4) and (6.1), we derive n2 n2 1 ∑∑

n1

i=1 j=1

∥hµ (ei , ej )∥ = 2

( α ∑

) H

ln f

(ei , ei ) + H

ln f

(ϕ e i , ϕ e i )

i=1

1 n2

+

) ( + 2 n2 − cot2 θ ∥∇ ln f ∥2 c−3 4

n1 −

c+1 4

+ 2cot2 θ + 1.

(6.4)

If (6.3) holds, then (6.4) indicates that ∥∇ ln f ∥2 = 0, which implies that f is a constant function on M n , i.e., M n is simply a n n Riemannian product of NT 1 and Nθ 2 (M n is trivial). Similarly, the converse part can be proved easily by using (6.4) and (6.3). This completes the proof of the theorem.

290

A. Ali, P. Laurian-Ioan / Journal of Geometry and Physics 114 (2017) 276–290 n

n

˜ 2m+1 (c) Theorem 6.2. There does not exist a warped product semi-slant submanifold NT 1 ×f Nθ 2 into a Kenmotsu space form M n1 n2 2m+1 ˜ with c ≤ 3 such that NT is a compact invariant submanifold tangent to ξ and Nθ is a slant submanifold of M (c). n

n

˜ 2m+1 (c) with Proof. Assume there exists a warped product semi-slant submanifold NT 1 ×f Nθ 2 in a Kenmotsu space form M n1 n1 c ≤ 3 such that NT is compact. Then the function ln f has an absolute maximum at some point p ∈ NT . At this critical point, the Hessian H ln f is non-positive definite. Thus (6.2) leads to a contradiction. Acknowledgments Authors would like to express their appreciation to the referees for their comments and valuable suggestions to improve the quality of paper. The work is supported to University of Malaya research grant FP074-2015A. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42]

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