Semi-Slant Warped Product Submanifolds of a Kenmotsu Manifold

Abstract

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

In [1] Tanno classified the connected almost contact metric manifold whose automorphism group has maximum dimension, there are three classes:(a)homogeneous normal contact Riemannian manifolds with constant 𝜙 holomorphic sectional curvature if the sectional curvature of the plane section contains 𝜉, say 𝐶(𝑋,𝜉)>0;(b)global Riemannian product of a line or a circle and Kaehlerian manifold with constant holomorphic sectional curvature, 𝐶(𝑋,𝜉)=0;(c)a warped product space 𝑅×_𝑓𝐶^𝑛, if 𝐶(𝑋,𝜉)<0.

Manifolds of class (a) are 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 [2] obtained some tensorial equations to Characterize manifolds of class (c), these manifolds are called Kenmotsu manifolds.

The notion of semi-slant submanifolds of almost Hermitian manifolds was introduced by Papaghiuc [3] after that cabrerizo et al. [4] defined and studied semi-slant submanifolds in the setting of almost contact manifolds.

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 literature (c.f., [3, 8, 9]).

Recently, Atçeken [10] proved that the warped product submanifolds of type 𝑁_𝜃×_𝑓𝑁_𝑇 and 𝑁_𝜃×_𝑓𝑁_⟂ of a Kenmotsu manifold 𝑀 do not exist where the manifolds 𝑁_𝜃 and 𝑁_𝑇 (resp. 𝑁_⟂) are proper slant and invariant (resp., anti-invariant) submanifolds of Kenmotsu manifold 𝑀, respectively. After that Siraj-Uddin et al. [11] investigated warped product of the types 𝑁_𝑇×_𝑓𝑁_𝜃 and 𝑁_⟂×_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 2𝑛+1 dimensional 𝐶^∞ manifold 𝑀 is said to have an almost contact structure if there exist on 𝑀 a tensor field 𝜙 of type (1,1), a vector field 𝜉, and 1-form 𝜂 satisfying the following properties: 𝜙²=−𝐼+𝜂⊗𝜉,𝜙(𝜉)=0,𝜂∘𝜙=0,𝜂(𝜉)=1. (2.1) There always exists a Riemannian metric 𝑔 on an almost contact manifold 𝑀 satisfying the following conditions:𝑔𝜙𝑋,𝜙𝑌=𝑔(𝑋,𝑌)−𝜂(𝑋)𝜂(𝑌),𝜂(𝑋)=𝑔(𝑋,𝜉), (2.2) where 𝑋,𝑌 are vector fields on 𝑀.

An almost contact metric structure (𝜙,𝜉,𝜂,𝑔) is said to be Kenmotsu manifold, if it satisfies the following tensorial equation [2]:∇_𝑋𝜙𝑌=𝑔𝜙𝑋,𝑌𝜉−𝜂(𝑌)𝜙𝑋, (2.3) for any 𝑋,𝑌∈𝑇𝑀, where 𝑇𝑀 is the tangent bundle of 𝑀 and ∇ denotes the Riemannian connection of the metric 𝑔. Moreover, for a Kenmotsu manifold∇_𝑋𝜉=𝑋−𝜂(𝑋)𝜉. (2.4)

Let 𝑀 be a submanifold of an almost contact metric manifold 𝑀 with induced metric 𝑔 and if ∇ and ∇^⟂ are the induced connection on the tangent bundle 𝑇𝑀 and the normal bundle 𝑇^⟂𝑀 of 𝑀, respectively, then Gauss and Weingarten formulae are given by ∇_𝑋𝑌=∇_𝑋𝑌+ℎ(𝑋,𝑌),∇_𝑋𝑁=−𝐴_𝑁𝑋+∇^⟂_𝑋𝑁, (2.5) for each 𝑋,𝑌∈𝑇𝑀 and 𝑁∈𝑇^⟂𝑀, where ℎ and 𝐴_𝑁 are the second fundamental form and the shape operator, respectively, for the immersion of 𝑀 into 𝑀 and they are related as 𝑔(ℎ(𝑋,𝑌),𝑁)=𝑔𝐴_𝑁𝑋,𝑌, (2.6) where 𝑔 denotes the Riemannian metric on 𝑀 as well as on 𝑀.

For any 𝑋∈𝑇𝑀, we write 𝜙𝑋=𝑃𝑋+𝐹𝑋, (2.7) where 𝑃𝑋 is the tangential component and 𝐹𝑋 is the normal component of 𝜙𝑋.

Similarly, for any 𝑁∈𝑇^⟂𝑀, we write 𝜙𝑁=𝑡𝑁+𝑓𝑁, (2.8) where 𝑡𝑁 is the tangential component and 𝑓𝑁 is the normal component of 𝜙𝑁. The covariant derivatives of the tensor field 𝑃 and 𝐹 are defined as ∇_𝑋𝑃𝑌=∇_𝑋𝑃𝑌−𝑃∇_𝑋𝑌,∇_𝑋𝐹𝑌=∇^⟂_𝑋F𝑌−𝐹∇_𝑋𝑌. (2.9) From (2.3), (2.5), (2.7) and (2.8) we have ∇_𝑋𝑃𝑌=𝐴_𝐹𝑌𝑋+𝑡ℎ(𝑋,𝑌)−𝑔(𝑋,𝑃𝑌)𝜉−𝜂(𝑌)𝑃𝑋,∇_𝑋𝐹𝑌=𝑓ℎ(𝑋,𝑌)−ℎ(𝑋,𝑃𝑌)−𝜂(𝑌)𝐹𝑋. (2.10)(2.11)

Definition 2.1 (see [12]). A submanifold 𝑀 of an almost contact metric manifold 𝑀 is said to be slant submanifold if for any 𝑥∈𝑀 and 𝑋∈𝑇_𝑥𝑀−⟨𝜉⟩ is constant. The constant angle 𝜃∈[0,𝜋/2] is then called slant angle of 𝑀 in 𝑀. If 𝜃=0 the submanifold is invariant submanifold, if 𝜃=𝜋/2 then it is anti-invariant submanifold, if 𝜃≠0, 𝜋/2 then it is proper slant submanifold.

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

Lemma 2.2. Let 𝑀 be a submanifold of an almost contact manifold 𝑀, such that 𝜉∈𝑇𝑀 then 𝑀 is slant submanifold if and only if there exists a constant 𝜆∈[0,1] such that 𝑃²=𝜆𝐼−𝜂⊗𝜉, (2.12) where 𝜆=−cos²𝜃.
Thus, one has the following consequences of above formulae: 𝑔(𝑃𝑋,𝑃𝑌)=cos²𝜃𝑔()(𝑋)𝜂(𝑌,𝑔()𝑋,𝑌−𝜂)𝐹𝑋,𝐹𝑌=sin²𝜃𝑔(𝑋,𝑌)−𝜂(𝑋)𝜂(𝑌). (2.13)(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(i)𝑇𝑀=𝐷_𝑇⊕𝐷_𝜃⊕⟨𝜉⟩, (ii)the distribution 𝐷_𝑇 is invariant that is, 𝜙𝐷_𝑇⊆𝐷_𝑇,(iii)the distribution 𝐷_𝜃 is slant with slant angle 𝜃≠0.

It is straight forward to see that semi-invariant submanifolds and slant submanifolds are semi-slant submanifolds with 𝜃=𝜋/2 and 𝐷_𝑇={0}, respectively.

If 𝜇 is invariant subspace under 𝜙 of the normal bundle 𝑇^⟂𝑀, then in the case of semi-slant submanifold, the normal bundle 𝑇^⟂𝑀 can be decomposed as 𝑇^⟂𝑀=𝜇⊕𝐹𝐷_𝜃. (2.15)

A semi-slant submanifold 𝑀 is called a semi-slant product if the distributions 𝐷_𝑇 and 𝐷_𝜃 are parallel on 𝑀. In this case 𝑀 is 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. Let (𝐵,𝑔_𝐵) and (𝐹,𝑔_𝐹) be two Riemannian manifolds with Riemannian metric 𝑔_𝐵 and 𝑔_𝐹, respectively, and 𝑓 be a positive differentiable function on 𝐵. The warped product of 𝐵 and 𝐹 is the Riemannian manifold (𝐵×𝐹,𝑔), where 𝑔=𝑔_𝐵+𝑓²𝑔_𝐹. (2.16)

For a warped product manifold 𝑁₁×_𝑓𝑁₂, we denote by 𝐷₁ and 𝐷₂ the distributions defined by the vectors tangent to the leaves and fibers, respectively. In other words, 𝐷₁ is obtained by the tangent vectors of 𝑁₁ via the horizontal lift, and 𝐷₂ is obtained by the tangent vectors of 𝑁₂ via vertical lift. In case of semi-slant warped product submanifolds 𝐷₁ and 𝐷₂ are replaced by 𝐷_𝑇 and 𝐷_𝜃, respectively.

The warped product manifold (𝐵×𝐹,𝑔) is denoted by 𝐵×_𝑓𝐹. If 𝑋 is the tangent vector field to 𝑀=𝐵×_𝑓𝐹 at (𝑝,𝑞) then‖𝑋‖²=𝑑𝜋₁𝑋²+𝑓²𝑝𝑑𝜋₂𝑋². (2.17)

Bishop and O’Neill [5] proved the following.

Theorem 2.4. Let 𝑀=𝐵×_𝑓𝐹 be warped product manifolds. If 𝑋, 𝑌∈𝑇𝐵 and 𝑉, 𝑊∈𝑇𝐹 then(i)∇_𝑋𝑌∈𝑇𝐵, (ii)∇_𝑋𝑉=∇_𝑉𝑋=(𝑋𝑓/𝑓)𝑉, (iii)∇_𝑉𝑊=(−𝑔(𝑉,𝑊)/𝑓)∇𝑓.
∇𝑓 is the gradient of 𝑓 and is defined as 𝑔∇𝑓,𝑋=𝑋𝑓, (2.18) for all 𝑋∈𝑇𝑀.

Corollary 2.5. On a warped product manifold 𝑀=𝑁₁×_𝑓𝑁₂, the following statements hold:(i)𝑁₁ is totally geodesic in 𝑀,(ii)𝑁₂ is totally umbilical in 𝑀.

Throughout, one denotes by 𝑁_𝑇 and 𝑁_𝜃 an invariant and a slant submanifold, respectively, of an almost contact metric manifold 𝑀.

Khan et al. [14] proved the following corollary.

Corollary 2.6. Let 𝑀 be a Kenmotsu manifold and 𝑁₁ and 𝑁₂ be any Riemannian submanifolds of 𝑀, then there do not exist a warped product submanifold 𝑀=𝑁₁×_𝑓𝑁₂ of 𝑀 such that 𝜉 is tangential to 𝑁₂.
Thus, one assumes that the structure vector field 𝜉 is tangential to 𝑁₁ of a warped product submanifold 𝑁₁×_𝑓𝑁₂ of 𝑀.

In this paper we will consider the warped product of the type 𝑁_𝜃×_𝑓𝑁_𝑇 and 𝑁_𝑇×_𝑓𝑁_𝜃. The warped product of the type 𝑁_𝜃×_𝑓𝑁_𝑇 is called warped product semi-slant submanifolds; this type of warped product is studied by Atçeken [10], 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. [11] and they proved the following Lemma

Lemma 2.7. Let 𝑀=𝑁_𝑇×_𝑓𝑁_𝜃 be warped product semi-slant submanifold of a Kenmotsu manifold 𝑀 such that 𝜉 is tangent to 𝑁_𝑇, where 𝑁_𝑇 and 𝑁_𝜃 are invariant and proper slant submanifolds of 𝑀. then(i)𝑔(ℎ(𝑋,𝑍),𝐹𝑃𝑍)=𝑔(ℎ(𝑋,𝑃𝑍),𝐹𝑍)={𝑋ln𝑓−𝜂(𝑋)}cos²𝜃‖𝑍‖², (ii)𝑔(ℎ(𝑋,𝑍),𝐹𝑍)=−𝑃𝑋ln𝑓‖𝑍‖²,
for any 𝑋∈𝑇𝑁_𝑇 and 𝑍∈𝑇𝑁_𝜃.

Replacing 𝑋 by 𝑃𝑋 in part (ii) of above lemma one has𝑔(ℎ(𝑃𝑋,𝑍),𝐹𝑍)=𝑋ln𝑓‖𝑍‖². (2.19)

3. Semi-Slant Warped Product Submanifolds

Throughout this section we will study the warped product of the type 𝑁_𝑇×_𝑓𝑁_𝜃, for these submanifolds by Theorem 2.4 we have ∇_𝑋𝑍=∇_𝑍𝑋=𝑋ln𝑓𝑍, (3.1) for any 𝑋∈𝑇𝑁_𝑇 and 𝑍∈𝑇𝑁_𝜃.

Lemma 3.1. Let 𝑀=𝑁_𝑇×_𝑓𝑁_𝜃 be a semi-slant warped product submanifolds of a Kenmotsu manifold 𝑀, then 𝑔(ℎ(𝑋,𝑌),𝐹𝑍)=0, (3.2) for any 𝑋,𝑌∈𝑇𝑁_𝑇 and 𝑍∈𝑇𝑁_𝜃.

Proof. As 𝑁_𝑇 is totally geodesic in 𝑀 then (∇_𝑋𝑃)𝑌∈𝑇𝑁_𝑇 and therefore by formula (2.10): ∇_𝑋𝑃𝑌=𝑡ℎ(𝑋,𝑌)−𝑔(𝑋,𝑃𝑌)𝜉−𝜂(𝑌)𝑃𝑋, (3.3) taking inner product with 𝑍∈𝑇𝑁_𝜃 we get (3.2).

Now we have the following Characterization.

Theorem 3.2. A semi-slant submanifold 𝑀 of a Kenmotsu manifold 𝑀 with integrable invariant distribution 𝐷_𝑇⊕⟨𝜉⟩ and integrable slant distribution 𝐷_𝜃 is locally a semi-slant warped product if and only if ∇_𝑍𝑃𝑍∈𝐷_𝜃 and there exists a 𝐶^∞- function 𝛼 on 𝑀 with 𝑍𝛼=0, 𝐴_𝐹𝑍𝑋=𝑋𝛼−𝜂(𝑋)𝑃𝑍−𝑃𝑋𝛼𝑍, (3.4) for all 𝑋∈𝐷_𝑇⊕⟨𝜉⟩ and 𝑍∈𝐷_𝜃.

Proof. From (2.10) and (3.1) we have 𝐴_𝐹𝑍𝑋+𝑡ℎ(𝑋,𝑍)=0. (3.5) Similarly, 𝑃𝑋ln𝑓𝑍−𝑋ln𝑓𝑃𝑍=𝑡ℎ(𝑋,𝑍)−𝜂(𝑋)𝑃𝑍, (3.6) from (3.5) and (3.6), we get 𝐴_𝐹𝑍𝑋=𝑋ln𝑓𝑃𝑍−𝑃𝑋ln𝑓𝑍−𝜂(𝑋)𝑃𝑍, (3.7) taking inner product with 𝑊∈𝑇𝑁_𝜃, we have 𝑔𝐴_𝐹𝑍𝑋,𝑊=𝑋ln𝑓−𝜂(𝑋)𝑔(𝑃𝑍,𝑊)−𝑃𝑋ln𝑓𝑔(𝑍,𝑊). (3.8) From Lemma 3.1 and (3.8) we get the desired result.
Conversely, let 𝑀 be a semi-slant submanifold of 𝑀 satisfying the hypothesis of the theorem, then for any 𝑋,𝑌∈𝐷_𝑇⊕⟨𝜉⟩ and 𝑍∈𝐷_𝜃𝑔(ℎ(𝑋,𝑌),𝐹𝑍)=0, (3.9) that means ℎ(𝑋,𝑌)∈𝜇. Then from (2.11) −𝐹∇_𝑋𝑌=𝑓ℎ(𝑋,𝑌)−ℎ(𝑋,𝑃𝑌). (3.10) Since ℎ(𝑋,𝑌)∈𝜇, then we have 𝐹∇_𝑋𝑌=0, that is, ∇_𝑋𝑌∈𝐷_𝑇⊕⟨𝜉⟩. Hence, each leaf of 𝐷_𝑇⊕⟨𝜉⟩ is totally geodesic in 𝑀.
Further, suppose 𝑁_𝜃 be a leaf of 𝐷_𝜃 and ℎ_𝜃 be second fundamental form of the immersion of 𝑁_𝜃 in 𝑀, then for any 𝑋∈𝐷_𝑇⊕⟨𝜉⟩ and 𝑍∈𝐷_𝜃, we have 𝑔ℎ_𝜃(𝑍,𝑍),𝜙𝑋=𝑔∇_𝑍𝑍,𝜙𝑋, (3.11) using (2.7) and (2.5), the above equation yields 𝑔ℎ_𝜃(𝑍,𝑍),𝜙𝑋=𝑔∇_𝑍𝑃𝑍,𝑋+𝑔𝐴_𝐹𝑍𝑍,𝑋, (3.12) applying (3.4), we get 𝑔ℎ_𝜃(𝑍,𝑍),𝜙𝑋=−𝑃𝑋ln𝑓𝑔(𝑍,𝑍). (3.13) Replacing 𝑋 by 𝑃𝑋, the above equation gives ℎ_𝜃(𝑍,𝑍)=∇𝛼𝑔(𝑍,𝑍). (3.14) From above equation it is easy to derive ℎ_𝜃(𝑍,𝑊)=∇𝛼𝑔(𝑍,𝑊), (3.15) that is, 𝑁_𝜃 is totally umbilical and as 𝑍𝛼=0, for all 𝑍∈𝐷_𝜃,∇𝜇 is defined on 𝑁_𝑇, this mean that mean curvature vector of 𝑁_𝜃 is parallel, that is, the leaves of 𝐷_𝜃 are extrinsic spheres in 𝑀. Hence by virtue of result of [15] which says that if the tangent bundle of a Riemannian manifold 𝑀 splits into an orthogonal sum 𝑇𝑀=𝐸₀⊕𝐸₁ of nontrivial vector subbundles such that 𝐸₁ is spherical and its orthogonal complement 𝐸₀ is autoparallel, then the manifold 𝑀 is locally isometric to a warped product 𝑀₀×_𝑓𝑀₁, we can say 𝑀 is locally semi-slant warped product submanifold 𝑁_𝑇×_𝑓𝑁_𝜃, where the warping function 𝑓=𝑒^𝛼.
Let us denote by 𝐷_𝑇 and 𝐷_𝜃 the tangent bundles on 𝑁_𝑇 and 𝑁_𝜃, respectively, and let {𝑋₀=𝜉,𝑋₁,…,𝑋_𝑝,𝑋_𝑝+1=𝜙𝑋₁,…,𝑋_2𝑝=𝜙𝑋_𝑝} and {𝑍₁,…,𝑍_𝑞,𝑍_𝑞+1=𝑃𝑍₁,…,𝑍_2𝑞=𝑃𝑍_𝑞} be local orthonormal frames of vector fields on 𝑁_𝑇 and 𝑁_𝜃, respectively, with 2𝑝 and 2𝑞 being real dimension. Since ℎ(𝑋,𝜉)=0 for all 𝑋∈𝑇𝑀, then the second fundamental form can be written as ‖ℎ‖²=^2𝑝_𝑖,𝑗=1𝑔ℎ𝑋_𝑖,𝑋_𝑗,ℎ𝑋_𝑖,𝑋_𝑗+^2𝑝_𝑖=1^2𝑞_𝑟=1𝑔ℎ𝑋_𝑖,𝑍_𝑟,ℎ𝑋_𝑖,𝑍_𝑟+^2𝑞_𝑟,𝑠=1𝑔ℎ𝑍_𝑟,𝑍_𝑠,ℎ𝑍_𝑟,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=𝑁_𝑇×_𝑓𝑁_𝜃 be a semi-slant warped product submanifold of a Kenmotsu manifold 𝑀 with 𝑁_𝑇 and 𝑁_𝜃 invariant and slant submanifolds, respectively, of 𝑀. If 𝜂(𝑋)≥2𝑋ln𝑓 for all 𝑋∈𝑇𝑁_𝑇, then the squared norm of the second fundamental form ℎ satisfies ‖ℎ‖²≥4𝑞csc²𝜃1+cos⁴𝜃∇ln𝑓², (3.17) where ∇ln𝑓 is the gradient of ln𝑓 and 2𝑞 is the dimension 𝑁_𝜃.

Proof. In view of the decomposition (2.15), we may write ℎ(𝑈,𝑉)=ℎ_𝐹𝐷𝜃(𝑈,𝑉)+ℎ_𝜇(𝑈,𝑉), (3.18) for each 𝑈,𝑉∈𝑇𝑀, where ℎ_𝐹𝐷𝜃(𝑈,𝑉)∈𝐹𝐷_𝜃 and ℎ_𝜇(𝑈,𝑉)∈𝜇 with ℎ_𝐹𝐷𝜃(𝑈,𝑉)=^2𝑞_𝑟=1ℎ^𝑟(𝑈,𝑉)𝐹𝑍_𝑟,ℎ^𝑟(𝑈,𝑉)=csc²𝜃𝑔ℎ(𝑈,𝑉),𝐹𝑍_𝑟, (3.19)(3.20) for each 𝑈,𝑉∈𝑇𝑀. In view of above formulae we have 𝑔ℎ_𝐹𝐷𝜃𝑃𝑋_𝑖,𝑍_𝑟,ℎ_𝐹𝐷𝜃𝑃𝑋_𝑖,𝑍_𝑟=𝑔ℎ^𝑟𝑃𝑋_𝑖,𝑍_𝑟𝐹𝑍_𝑟,ℎ^𝑟𝑃𝑋_𝑖,𝑍_𝑟𝐹𝑍_𝑟+𝑔ℎ^𝑠𝑃𝑋_𝑖,𝑍_𝑟𝐹𝑍_𝑟,ℎ^𝑠𝑃𝑋_𝑖,𝑍_𝑟𝐹𝑍_𝑟. (3.21) Now using (2.14) and (2.19) 𝑔ℎ_𝐹𝐷𝜃𝑃𝑋_𝑖,𝑍_𝑟,ℎ_𝐹𝐷𝜃𝑃𝑋_𝑖,𝑍_𝑟=ℎ^𝑟𝑃𝑋_𝑖,𝑍_𝑟𝑋_𝑖ln𝑓+sin²𝜃_𝑠≠𝑟ℎ^𝑠𝑃𝑋_𝑖,𝑍_𝑟². (3.22) In view of (3.20) and (2.19), we get 𝑔ℎ_𝐹𝐷𝜃𝑃𝑋_𝑖,𝑍_𝑟,ℎ_𝐹𝐷𝜃𝑃𝑋_𝑖,𝑍_𝑟=csc²𝜃𝑋_𝑖ln𝑓²+sin²𝜃_𝑠≠𝑟ℎ^𝑠𝑃𝑋_𝑖,𝑍_𝑟². (3.23) Summing over 𝑖=1,…,2𝑝 and 𝑟=1,…,2𝑞 the above equation yields ^2𝑝_𝑖=1^2𝑞_𝑟=1𝑔ℎ_𝐹𝐷𝜃𝑃𝑋_𝑖,𝑍_𝑟,ℎ_𝐹𝐷𝜃𝑃𝑋_𝑖,𝑍_𝑟=4𝑞csc²𝜃∇ln𝑓²+sin²𝜃^2𝑝_𝑖=1^2𝑞_{𝑟,𝑠=1,𝑟≠𝑠}ℎ^𝑠𝑃𝑋_𝑖,𝑍_𝑟². (3.24) Since we have choose the orthonormal frame of vector fields on 𝐷_𝜃 as {𝑍₁,…,𝑍_𝑞,𝑍_𝑞+1=𝑃𝑍₁,…,𝑍_2𝑞=𝑃𝑍_𝑞}, then the second term in the right-hand side of (3.24) is written as csc²𝜃^2𝑝_𝑖=1^𝑞_𝑟=1𝑔ℎ𝑃𝑋_𝑖,𝑍_𝑟,𝐹𝑃𝑍_𝑟²+𝑔ℎ𝑃𝑋_𝑖,𝑃𝑍_𝑟,𝐹𝑍_𝑟²+^𝑞𝑞_{𝑟=1𝑠=1,𝑠≠𝑟}𝑔ℎ𝑃𝑋_𝑖,𝑍_𝑟,𝐹𝑃𝑍_𝑠²+𝑔ℎ𝑃𝑋_𝑖,𝑃𝑍_𝑟,𝐹𝑍_𝑠². (3.25) From part (i) of Lemma 2.7, the first two terms of above equation can be written as csc²𝜃^2𝑝_𝑖=12𝑞𝑋_𝑖ln𝑓−𝜂𝑋_𝑖²cos⁴𝜃. (3.26) In account of to hypothesis 𝜂(𝑋_𝑖)≥2𝑋_𝑖ln𝑓 the above expression is greater than equal to the following term: 4𝑞csc²𝜃∇ln𝑓²cos⁴𝜃. (3.27) Using above inequality into (3.24), we have 𝑔ℎ_𝐹𝐷𝜃𝑃𝑋_𝑖,𝑍_𝑟,ℎ_𝐹𝐷𝜃𝑃𝑋_𝑖,𝑍_𝑟≥4𝑞csc²𝜃1+cos⁴𝜃∇ln𝑓². (3.28) The inequality (3.17) follows from (3.16) and (3.28).
The equality holds if ℎ(𝐷_𝑇,𝐷_𝑇)=0, ℎ(𝐷_𝜃,𝐷_𝜃)=0, ℎ(𝑃𝑋,𝑍) is orthogonal to 𝐹𝑍 and 𝐹𝑃𝑍 for all 𝑋∈𝐷_𝑇 and 𝑍∈𝐷_𝜃 and 𝜂(𝑋)=2𝑋ln𝑓.

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 equations (c.f., [8]) and 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.