Warped Product Submanifolds of Riemannian Product Manifolds

Abstract

We study warped product of the type 𝑁_𝜃×_𝑓𝑁_𝑇 and 𝑁_𝜃×_𝑓𝑁_⟂, where 𝑁_𝜃, 𝑁_𝑇, and 𝑁_⟂ are proper slant, invariant, and anti-invariant submanifolds, respectively, and we prove some basic results and finally obtain some inequalities for squared norm of second fundamental form.

1. Introduction

Bishop and O’Neil [1] introduced the notion of warped product manifolds that occur naturally; for example, surface of revolution is a warped product manifold. With regard to physical applications of these manifolds, one may realize that space time around a massive star or a black hole can be modeled on warped product manifolds [2]. CR-warped product was introduced by Chen [3]; he studied warped product CR-submanifolds in the setting of Kaehler manifolds and showed that there does not exist warped product of the form 𝑁_⟂×_𝑓𝑁_𝑇; therefore he considered warped product CR-submanifolds of type 𝑁_𝑇×𝑁_⟂ and established a relationship between the warping function 𝑓 and the squared norm of second fundamental form [3]. In [4] Atçeken studied semi-slant warped product of Riemannian product manifolds. In fact they proved that there exists no warped product if spheric submanifold of warped product submanifold is proper slant submanifold. On the other hand they proved the existence of warped product of the type 𝑁_𝜃×_𝑓𝑁_𝑇 and 𝑁_𝜃×_𝑓𝑁_⟂ via some examples. In this continuation we have studied the warped product submanifolds in which proper slant submanifolds are totally geodesic; that is, we study the warped product of the types 𝑁_𝜃×𝑁_𝑇 and 𝑁_𝜃×𝑁_⟂ and called them semi-slant warped product and hemi-slant warped product submanifolds, respectively.

2. Preliminaries

Let (𝑀₁,𝑔₁) and (𝑀₂,𝑔₂) be the Riemannian manifolds with dimensions 𝑚₁ and 𝑚₂, respectively, and let 𝑀₁×𝑀₂ be Riemannian product manifold of 𝑀₁ and 𝑀₂. We denote projection mapping of 𝑇(𝑀₁×𝑀₂) onto 𝑇𝑀₁ and 𝑇𝑀₂ by 𝜎_★ and 𝜋_★, respectively. Then we have 𝜎_★+𝜋_★=𝐼,𝜎²_★=𝜎_★,𝜋²_★=𝜋_★, and 𝜎_★∘𝜋_★=𝜋_★∘𝜎_★=0, where ★ denotes the differential.

Riemannian metric of the Riemannian product manifold 𝑀=𝑀₁×𝑀₂ is defined by 𝑔(𝑋,𝑌)=𝑔₁𝜎_★𝑋,𝜎_★𝑌+𝑔₂𝜋_★𝑋,𝜋_★𝑌, (2.1) for any 𝑋,𝑌∈𝑇𝑀. If we set 𝐹=𝜎_★−𝜋_★, then 𝐹²=𝐼,  𝐹≠𝐼, and 𝑔 satisfies the condition 𝑔(𝐹𝑋,𝑌)=𝑔(𝑋,𝐹𝑌), (2.2) for any 𝑋,𝑌∈𝑇𝑀; thus 𝐹 defines an almost Riemannian product structure on 𝑀. We denote Levi-Civita connection on 𝑀 by ∇; then the covariant derivative of 𝐹 is defined as ∇_𝑋𝐹𝑌=∇_𝑋𝐹𝑌−𝐹∇_𝑋𝑌, (2.3) for any 𝑋,𝑌∈𝑇𝑀. We say that 𝐹 is parallel with respect to the connection ∇ if we have (∇_𝑋𝐹)𝑌=0. Here from [5], we know that 𝐹 is parallel; that is, 𝐹 is Riemannian product structure.

Let 𝑀 be a Riemannian product manifold with Riemannian product structure 𝐹 and 𝑀 an immersed submanifold of 𝑀; we also denote by 𝑔 the induced metric tensor on 𝑀 as well as on 𝑀. If ∇ is the Levi-Civita connection on 𝑀, then the Gauss and Weingarten formulas are given, respectively, as ∇_𝑋𝑌=∇_𝑋𝑌+ℎ(𝑋,𝑌),∇_𝑋𝑉=−𝐴_𝑉𝑋+∇^⟂_𝑋𝑉, (2.4)(2.5) for any 𝑋,𝑌∈𝑇𝑀 and 𝑉∈𝑇^⟂𝑀, where ∇ is the connection on 𝑀 and ∇^⟂ is the connection in the normal bundle, ℎ is the second fundamental form of 𝑀, and 𝐴_𝑉 is the shape operator of 𝑀. The second fundamental form ℎ and the shape operator 𝐴_𝑉 are related by 𝑔𝐴_𝑉𝑋,𝑌=𝑔(ℎ(𝑋,𝑌),𝑉). (2.6) For any 𝑋∈𝑇𝑀, we can write 𝐹𝑋=𝑇𝑋+𝑁𝑋, (2.7) where 𝑇𝑋 and 𝑁𝑋 are the tangential and normal components of 𝐹𝑋, respectively, and for 𝑉∈𝑇^⟂𝑀, 𝐹𝑉=𝑡𝑉+𝑛𝑉, (2.8) where 𝑡𝑉 and 𝑛𝑉 are the tangential and normal components of 𝐹𝑉, respectively, and the submanifold 𝑀 is said to be invariant if 𝑁 is identically zero. On the other hand 𝑀 is said to be an anti-invariant submanifold if 𝑇 is identically zero.

The covariant derivatives of 𝑇, 𝑁, 𝑡, and 𝑛 are defined as ∇_𝑋𝑇𝑌=∇_𝑋𝑇𝑌−𝑇∇_𝑋𝑌,∇_𝑋𝑁𝑌=∇^⟂_𝑋𝑁𝑌−𝑁∇_𝑋𝑌,∇_𝑋𝑡𝑉=∇_𝑋𝑡𝑉−𝑡∇_𝑋𝑉,∇_𝑋𝑛𝑉=∇^⟂_𝑋𝑛𝑉−𝑛∇^⟂_𝑋𝑌. (2.9)(2.10)(2.11)(2.12) Using (2.4)–(2.9) we get ∇_𝑋𝑇𝑌=𝐴_𝑁𝑌𝑋+𝑡ℎ(𝑋,𝑌),∇_𝑋𝑁𝑌=ℎ(𝑋,𝑇𝑌)−𝑛ℎ(𝑋,𝑌). (2.13)(2.14)

Let 𝑀 be an immersed submanifold of a Riemannian product manifold 𝑀, for each nonzero vector 𝑋 tangent to 𝑀 at a point 𝑥, and we denote by 𝜃(𝑥) the angle between 𝐹𝑋 and 𝑇_𝑥𝑀. The angle 𝜃(𝑥) is called the slant angle of immersion.

Let 𝑀 be an immersed submanifold of a Riemannian product manifold 𝑀. 𝑀 is said to be slant submanifold of Riemannian product manifold 𝑀 if the slant angle 𝜃(𝑥) is constant which is independent of choice of 𝑥∈𝑀 and 𝑋∈𝑇𝑀.

Invariant and anti-invariant submanifolds are particular cases of slant submanifolds with angles 𝜃=0 and 𝜃=𝜋/2, respectively. A slant submanifold which is neither invariant nor anti-invariant is called proper slant submanifold. The following characterization of slant submanifolds of Riemannian product manifolds is proved by Atçeken [6].

Theorem 2.1. Let 𝑀 be an immersed submanifold of a Riemannian product manifold 𝑀. Then 𝑀 is a slant submanifold if and only if there exists a constant 𝜆∈[0,1] such that 𝑇²=𝜆𝐼.

Moreover, if 𝜃 is the slant angle of 𝑀, then it satisfies 𝜆=cos²𝜃.

Hence, for a slant submanifold we have the following relations which are consequences of the above theorem: 𝑔(𝑇𝑋,𝑇𝑌)=cos²(),𝑔()𝜃𝑔𝑋,𝑌𝑁𝑋,𝑁𝑌=sin²𝜃𝑔(𝑋,𝑌) (2.15)(2.16) for any 𝑋,𝑌∈𝑇𝑀.

Papaghuic [7] introduced a class of submanifolds in almost Hermitian manifolds called semi-slant submanifolds; this class includes the class of proper CR-submanifolds and slant submanifolds. Cabrerizo et al. [8] initiated the study of contact version of semi-slant submanifolds and also gave the notion of Bi-slant submanifolds. A step forward Carriazo [9] defined and studied Bi-slant submanifolds and simultaneously gave the notion of anti-slant submanifolds; after that V. A. Khan and M. A. Khan [10] have studied anti-slant submanifolds with the name pseudo-slant submanifolds. Recently, Sahin [11] renamed these submanifolds and studied these submanifolds with the name hemi-slant submanifolds for their warped product.

Definition 2.2. A submanifold 𝑀 of a Riemannian product manifold is said to be semi-slant submanifold if there exist two orthogonal complementary distributions 𝐷_𝑇 and 𝐷_𝜃 such that 𝐷_𝑇 is invariant and 𝐷_𝜃 is slant distribution 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 𝐹 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⊕𝑁𝐷_𝜃. (2.17)

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.

Definition 2.3. A submanifold 𝑀 of a Riemannian product manifold is called hemi-slant submanifold if it is endowed with two orthogonal complementary distributions 𝐷_⟂ and 𝐷_𝜃 such that 𝐷_⟂ is totally real and 𝐷_𝜃 is slant distribution with slant angle 𝜃≠𝜋/2.

It is easy to see that semi-invariant submanifolds and slant submanifolds are semi-slant submanifolds with 𝜃=0 and 𝐷_⟂={0}, respectively. The normal bundle 𝑇^⟂𝑀 can be decomposed as follows: 𝑇^⟂𝑀=𝜇⊕𝑁𝐷_⟂⊕𝑁𝐷_𝜃. (2.18)

As 𝐷_⟂ and 𝐷_𝜃 are orthogonal distributions on 𝑀, then it is easy to see that the distributions 𝑁𝐷_⟂ and 𝑁𝐷_𝜃 are mutually perpendicular. In fact, the decomposition (2.18) is an orthogonal direct decomposition. A hemi-slant submanifold 𝑀 is called a hemi-slant product if the distributions 𝐷_⟂ and 𝐷_𝜃 are parallel on 𝑀.

As a generalization of product manifold and in particular of semi-slant product submanifolds (hemi-slant product submanifolds) one can consider warped product of manifolds which are defined as.

Definition 2.4. Let (𝐵,𝑔_𝐵) and (𝐹,𝑔_𝐹) be two Riemannian manifolds with Riemannian metric 𝑔_𝐵 and 𝑔_𝐹, respectively, and 𝑓 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. (2.19)
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. (2.20)
Bishop and O’Neill [1] proved the following.

Theorem 2.5. 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, (2.21) for all 𝑋∈𝑇𝑀.

Corollary 2.6. 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, we denote by 𝑁_𝑇, 𝑁_⟂, and 𝑁_𝜃 invariant, anti-invariant, and slant submanifolds, respectively, of a Riemannian product manifold 𝑀.

3. Semi-Slant Warped Product Submanifolds

In this section we will consider the warped product of the type 𝑁_𝜃×_𝑓𝑁_𝑇.

For the warped product of the type 𝑁_𝜃×_𝑓𝑁_𝑇 by Theorem 2.5 we have ∇_𝑋𝑍=∇_𝑍𝑋=𝑍ln𝑓𝑋, (3.1) for any 𝑍∈𝑇𝑁_𝜃 and 𝑋∈𝑇𝑁_𝑇.

Lemma 3.1. Let 𝑀=𝑁_𝜃×_𝑓𝑁_𝑇 be a semi-slant warped product submanifold of a Riemannian product manifold; then(i)𝑔(ℎ(𝑋,𝑍),𝑁𝑊)=0, (ii)𝑔(ℎ(𝑋,𝑋),𝑁𝑍)=𝑇𝑍ln𝑓‖𝑋‖²,
for any 𝑋∈𝑇𝑁_𝑇 and 𝑍,𝑊∈𝑇𝑁_𝜃.

Proof. For any 𝑍,𝑊∈𝑇𝑁_𝜃, (∇_𝑍𝑇)𝑊∈𝑇𝑁_𝜃; then from (2.13) ∇_𝑍𝑇𝑊=𝐴_𝑁𝑊𝑍+𝑡ℎ(𝑍,𝑊). (3.2) Taking inner product with 𝑋∈𝑇𝑁_𝑇 we have 𝑔(ℎ(𝑋,𝑍),𝑁𝑊)=0. (3.3) This is part (i) of the lemma.
Now for any 𝑋,𝑌∈𝑇𝑁_𝑇, from (2.13) and (2.9), ∇_𝑋𝑇𝑌−𝑇∇_𝑋𝑌=𝑡ℎ(𝑋,𝑌). (3.4) Taking inner product with 𝑍∈𝑇𝑁_𝜃, the above equation yields −𝑔∇_𝑋𝑍,𝑇𝑌−𝑔∇_𝑋𝑇𝑍,𝑌=𝑔(ℎ(𝑋,𝑌),𝑁𝑍). (3.5) Using (3.1), the above equation gives −𝑍ln𝑓𝑔(𝑋,𝑇𝑌)+𝑇𝑍ln𝑓𝑔(𝑋,𝑌)=𝑔(ℎ(𝑋,𝑌),𝑁𝑍). (3.6) In particular 𝑇𝑍ln𝑓𝑔(𝑋,𝑋)=𝑔(ℎ(𝑋,𝑋),𝑁𝑍). (3.7) This proves part (ii) of the lemma. Now we have the following corollary.

Corollary 3.2. For the warped product of the type 𝑁_𝜃×_𝑓𝑁_𝑇 following statements are equivalent: (i)𝐻∈𝜇, (ii)𝜃=𝜋/2 or the warping function 𝑓 is constant; that is, there does not exist warped product.

Proof. Since 𝑁_𝑇 is totally umbilical, then from (3.7) 𝑔(𝐻,𝑁𝑍)=𝑇𝑍ln𝑓. (3.8) Replacing 𝑍 by 𝑇𝑍 and using Theorem 2.1, we get 𝑔(𝐻,𝑁𝑇𝑍)=cos²𝜃𝑍ln𝑓. (3.9) The proof follows from (3.9).

Now we have the following characterization for semi-slant warped product submanifolds.

Theorem 3.3. A semi-slant submanifold 𝑀 of Riemannian product manifolds 𝑀 with integrable invariant distribution 𝐷_𝑇 and the slant distribution 𝐷_𝜃 is locally a semi-slant warped product if and only if ∇_𝑋𝑇𝑍∈𝐷_𝜃 and there exist a 𝐶^∞-function 𝛼 on 𝑀 with 𝑋𝛼=0 for all 𝑋∈𝐷_𝑇 such that 𝐴_𝑁𝑍𝑋=𝑇𝑍ln𝑓𝑋, (3.10) for all 𝑋∈𝐷_𝑇 and 𝑍∈𝐷_𝜃.

Proof. If 𝑀 is a semi-slant warped product of the type 𝑁_𝜃×_𝑓𝑁_𝑇, then for any 𝑋∈𝑇𝑁_𝑇 and 𝑍∈𝑇𝑁_𝜃 from (2.9), (2.13), and (3.1), we have 𝑇𝑍ln𝑓𝑋−𝑍ln𝑓𝑇𝑋=𝐴_𝑁𝑍𝑋+𝑡ℎ(𝑋,𝑍). (3.11) Taking inner product with 𝑋, the above equation gives 𝑔𝐴_𝑁𝑍𝑋,𝑋=𝑇𝑍ln𝑓𝑔(𝑋,𝑋). (3.12) By part (i) of Lemma 3.1, we also have 𝑔𝐴_𝑁𝑍𝑋,𝑊=0. (3.13) From (3.12) and (3.13) we have the following equation: 𝐴_𝑁𝑍𝑋=𝑇𝑍ln𝑓𝑋. (3.14) Conversely, let 𝑀 be a semi-slant submanifold of 𝑀 satisfying the hypothesis of the theorem; then for any 𝑍∈𝑇𝑁_𝜃 and 𝑌∈𝑇𝑁_𝑇 we have 𝑔(ℎ(𝑍,𝑌),𝑁𝑍)=𝑔𝐴_𝑁𝑍𝑌,𝑍=0. (3.15) This mean ℎ(𝑍,𝑌)∈𝜇.
From (2.14), we have −𝑁∇_𝑍𝑌=𝑓ℎ(𝑍,𝑌)−ℎ(𝑍,𝑇𝑌). (3.16) Comparing components of 𝜇 and 𝑁𝐷_𝜃, we get 𝑁∇_𝑍𝑌=0. (3.17) It is evident from the above equation that ∇_𝑍𝑌∈𝐷_𝑇; this means ∇_𝑍𝑊∈𝐷_𝜃 for any 𝑍,𝑊∈𝐷_𝜃 and hence 𝐷_𝜃 is totally geodesic. Further, let 𝑁_𝑇 be a leaf of 𝐷_𝑇 and ℎ^𝑇 a second fundamental form of the immersion 𝑁_𝑇 in 𝑀; then for any 𝑋,𝑌∈𝐷_𝑇 and 𝑍,𝑊∈𝐷_𝜃𝑔ℎ^𝑇(𝑋,𝑌),𝐹𝑊=𝑔∇_𝑋𝑌,𝐹𝑊. (3.18) or 𝑔ℎ^𝑇(𝑋,𝑌),𝐹𝑊=−𝑔∇_𝑋𝑇𝑊−𝐴_𝑁𝑊𝑋,𝑌. (3.19) Using the hypothesis, we get 𝑔ℎ^𝑇(𝑋,𝑌),𝐹𝑊=𝑇𝑊ln𝑓𝑔(𝑋,𝑌), (3.20) Finally, the above equation yields ℎ^𝑇(𝑋,𝑌)=𝑔(𝑋,𝑌)∇𝛼. (3.21) That is, 𝑁_𝑇 is totally umbilical and as 𝑋𝛼=0, for all 𝑋∈𝐷_𝑇, this means that mean curvature vector of 𝑁_𝑇 is parallel; that is, the leaves of 𝐷_𝑇 are extrinsic spheres in 𝑀. Hence by virtue of result of [12] 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 auto parallel, 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_𝜃×_𝑓𝑁_𝑇, where 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 {𝑋₁,…,𝑋_𝑝,𝑋_𝑝+1=𝐹𝑋₁,…,𝑋_2𝑝=𝐹𝑋_𝑝} and {𝑍₁,…,𝑍_𝑞,𝑍_𝑞+1=𝑇𝑍₁,…,𝑍_2𝑞=𝑇𝑍_𝑞} be local orthonormal frames of vector fields on 𝑁_𝑇 and 𝑁_𝜃, respectively, with 2𝑝 and 2𝑞 being real dimensions: ‖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. (3.22) Now, on a semi-slant warped product submanifold of a Riemannian product manifold, we prove the following.

Theorem 3.4. Let 𝑀=𝑁_𝜃×_𝑓𝑁_𝑇 be a semi-slant warped product submanifold of a Riemannian product manifold 𝑀 with 𝑁_𝑇 and 𝑁_𝜃 invariant and slant submanifolds, respectively, of 𝑀. Then the squared norm of the second fundamental form ℎ satisfies ‖ℎ‖²≥4𝑝1+cos²𝜃csc²𝜃∇ln𝑓². (3.23)

Proof. For 𝑁_𝜃×_𝑓𝑁_𝑇, in view of decomposition (2.17), we may write ℎ(𝑋,𝑌)=ℎ_𝑁𝐷𝜃(𝑋,𝑌)+ℎ_𝜇(𝑋,𝑌), (3.24) for each 𝑋,𝑌∈𝑇𝑀, where ℎ_𝑁𝐷𝜃(𝑋,𝑌)∈𝑁𝐷_𝜃 and ℎ_𝜇(𝑋,𝑌)∈𝜇 with ℎ_𝑁𝐷𝜃(𝑋,𝑌)=^2𝑞_𝑟=1ℎ^𝑟(𝑋,𝑌)𝑁𝑍_𝑟, (3.25) where ℎ^𝑟(𝑋,𝑌)=csc²𝜃𝑔ℎ(𝑋,𝑌),𝑁𝑍_𝑟, (3.26) for each 𝑍∈𝑇𝑁_𝜃. For any 𝑋∈𝑇𝑁_𝑇 and 𝑍∈𝑇𝑁_𝜃, by (3.25) we have 𝑔ℎ_𝑁𝐷𝜃𝑋_𝑖,𝑋_𝑖,ℎ_𝑁𝐷𝜃𝑋_𝑖,𝑋_𝑖=𝑔ℎ^𝑟𝑋_𝑖,𝑋_𝑖𝑁𝑍_𝑟,ℎ^𝑟𝑋_𝑖,𝑋_𝑖𝑁𝑍_𝑟+_𝑠≠𝑟𝑔ℎ^𝑠𝑋_𝑖,𝑋_𝑖𝑁𝑍_𝑟,ℎ^𝑠𝑋_𝑖,𝑋_𝑖𝑁𝑍_𝑟. (3.27) Now, using (3.26), (3.7), and (2.16), the above equation takes the form 𝑔ℎ_𝑁𝐷𝜃𝑋_𝑖,𝑋_𝑖,ℎ_𝑁𝐷𝜃𝑋_𝑖,𝑋_𝑖=csc²𝜃𝑇𝑍_𝑟ln𝑓²+sin²𝜃_𝑠≠𝑟ℎ^𝑠𝑋_𝑖,𝑋_𝑖². (3.28) Now summing over 𝑖=1,…,2𝑝, 𝑟,𝑠=1,…,2𝑞 and again using (3.7) and (3.26), we have 𝑔ℎ_𝑁𝐷𝜃𝑋_𝑖,𝑋_𝑖,ℎ_𝑁𝐷𝜃𝑋_𝑖,𝑋_𝑖=2𝑝1+cos²𝜃csc²𝜃∇ln𝑓²1+sin²𝜃csc²𝜃, (3.29) or 𝑔ℎ_𝑁𝐷𝜃𝑋_𝑖,𝑋_𝑖,ℎ_𝑁𝐷𝜃𝑋_𝑖,𝑋_𝑖=4𝑝1+cos²𝜃csc²𝜃∇ln𝑓², (3.30) By similar calculation, from (3.25), (3.26), (3.3), and (2.16) it is easy to see that 𝑔ℎ_𝑁𝐷𝜃𝑋_𝑖,𝑍_𝑟,ℎ_𝑁𝐷𝜃𝑋_𝑖,𝑍_𝑟=0. (3.31) The result follows from (3.22), (3.30), and (3.31).

4. Hemi-Slant Warped Product Submanifolds

In this section we will study the warped product of the type 𝑁_𝜃×_𝑓𝑁_⟂. For warped product of type 𝑁_𝜃×_𝑓𝑁_⟂ from Theorem 2.5 we have ∇_𝑋𝑍=∇_𝑍𝑋=𝑋ln𝑓𝑍, (4.1) for any 𝑋∈𝑇𝑁_𝜃 and 𝑍∈𝑇𝑁_⟂.

Now we have the following lemma.

Lemma 4.1. Let 𝑀=𝑁_𝜃×_𝑓𝑁_⟂ be a hemi-slant warped product submanifold of a Riemannian product manifold; then(i)𝑔(ℎ(𝑋,𝑍),𝑁𝑌)+𝑔(ℎ(𝑋,𝑌),𝑁𝑍)=0, (ii)𝑔(ℎ(𝑋,𝑍),𝑁𝑍)=0, (iii)𝑔(ℎ(𝑍,𝑍),𝑁𝑋)=𝑇𝑋ln𝑓‖𝑍‖², for any 𝑋,𝑌∈𝑇𝑁_𝜃 and 𝑍∈𝑇𝑁_⟂.

Proof. For any 𝑋,𝑌∈𝑇𝑁_𝜃, (∇_𝑋𝑇)𝑌∈𝑇𝑁_𝜃; then from (2.13) we have 𝑔∇_𝑋𝑇𝑌,𝑍=𝑔𝐴_𝑁𝑌𝑋,𝑍+𝑔(𝑡ℎ(𝑋,𝑌),𝑍), (4.2) or equivalently the above equation gives 𝑔(ℎ(𝑋,𝑍),𝑁𝑌)+𝑔(ℎ(𝑋,𝑌),𝑁𝑍)=0, (4.3) which proves part (i).
From (2.9), (2.13), we have 𝐴_𝑁𝑍𝑋+𝑡ℎ(𝑋,𝑍)=0. (4.4) Taking inner product with 𝑍∈𝑇𝑁_⟂ the above equation is reduced to 𝑔(ℎ(𝑋,𝑍),𝑁𝑍)=0. (4.5) Using (2.9), (2.13), and (4.1), we derive 𝑇𝑋ln𝑓𝑍=𝐴_𝑁𝑋𝑍+𝑡ℎ(𝑋,𝑍). (4.6) Taking inner product with 𝑍∈𝑇𝑁_⟂ and using (4.5), 𝑔(ℎ(𝑍,𝑍),𝑁𝑋)=𝑇𝑋ln𝑓‖𝑍‖². (4.7) Now we have the following corollary.

Corollary 4.2. For the warped product of the type 𝑁_𝜃×_𝑓𝑁_⟂ following statements are equivalent: (i)𝐻∈𝜇⊕𝑁𝐷^⟂, (ii)the warping function 𝑓 is constant; that is, there does not exist warped product.

Proof. As 𝑁_⟂ is totally umbilical, then from (4.7) and from Theorem 2.5𝑔(𝐻,𝑁𝑇𝑋)=cos²𝜃𝑋ln𝑓. (4.8) Since 𝜃≠𝜋/2, hence from the previous equation it is easy to see that statements (i) and (ii) are equivalent.
Let us denote by 𝐷_𝜃 and 𝐷_⟂ the tangent bundles on 𝑁_𝜃 and 𝑁_⟂, respectively, and let {𝑋₁,𝑋₂,…,𝑋_𝑞,𝑋_𝑞+1=𝑇𝑋₁,…,𝑋_2𝑞=𝑇𝑋_𝑞} and {𝑍₁,𝑍₂,…,𝑍_𝑝} be local orthonormal frames of vector fields on 𝑁_𝜃 and 𝑁_⟂, respectively, with 2𝑝 and 𝑞 being their real dimensions; then ‖ℎ‖²=^2𝑞_𝑟,𝑠=1𝑔ℎ𝑋_𝑟,𝑋_𝑠,ℎ𝑋_𝑟,𝑋_𝑠+^2𝑞𝑝_𝑟=1𝑖=1𝑔ℎ𝑋_𝑟,𝑍_𝑖,ℎ𝑋_𝑟,𝑍_𝑖+^𝑝_𝑖,𝑗=1𝑔ℎ𝑍_𝑖,𝑍_𝑗,ℎ𝑍_𝑖,𝑍_𝑗. (4.9) Now, on a hemi-slant warped product submanifold of a Riemannian product manifold, we prove the following inequality.

Theorem 4.3. Let 𝑀=𝑁_𝜃×_𝑓𝑁_⟂ be a hemi-slant warped product submanifold of a Riemannian product manifold 𝑀 with 𝑁_⟂ and 𝑁_𝜃 anti-invariant and slant submanifolds, respectively, of 𝑀. Then the squared norm of the second fundamental form ℎ satisfies ‖ℎ‖²≥2𝑝1+cos²𝜃csc²𝜃∇ln𝑓². (4.10)

Proof. In view of decomposition (2.18), the second fundamental form can be decomposed as follows: ℎ(𝑋,𝑌)=ℎ_𝑁𝐷𝜃(𝑋,𝑌)+ℎ_𝑁𝐷⟂(𝑋,𝑌)+ℎ_𝜇(𝑋,𝑌), (4.11) for each 𝑋,𝑌∈𝑇𝑀, where ℎ_𝑁𝐷𝜃(𝑋,𝑌)∈𝑁𝐷_𝜃, ℎ_𝑁𝐷⟂(𝑋,𝑌)∈𝑁𝐷_⟂, and ℎ_𝜇(𝑋,𝑌)∈𝜇 with ℎ_𝑁𝐷𝜃(𝑋,𝑌)=^2𝑞_𝑟=1ℎ^𝑟(𝑋,𝑌)𝑁𝑋^′_𝑟, (4.12) where ℎ^𝑟(𝑋,𝑌)=csc²𝜃𝑔ℎ(𝑋,𝑌),𝑁𝑋^′_𝑟, (4.13) for each 𝑋^′_𝑟∈𝑇𝑁_𝜃.
Now making use of (2.18), (4.12), (4.13), and (4.7) we have 𝑔ℎ_𝑁𝐷𝜃𝑍_𝑖,𝑍_𝑖,ℎ_𝑁𝐷𝜃𝑍_𝑖,𝑍_𝑖=csc²𝜃𝑇𝑋_𝑟ln𝑓²+sin²𝜃_𝑠≠𝑟ℎ^𝑠𝑍_𝑖,𝑍_𝑖², (4.14) for any 𝑋∈𝑇𝑁_𝜃 and 𝑍∈𝑇𝑁_⟂.
Again using (4.12) and (4.13), the previous equation gives 𝑔ℎ_𝑁𝐷𝜃𝑍_𝑖,𝑍_𝑖,ℎ_𝑁𝐷𝜃𝑍_𝑖,𝑍_𝑖=csc²𝜃𝑇𝑋_𝑟ln𝑓²+sin²𝜃csc⁴𝜃𝑇𝑋_𝑠ln𝑓². (4.15) Summing over 𝑖=1,…,𝑝 and 𝑟,𝑠=1,…,2𝑞, we have 𝑔ℎ_𝑁𝐷𝜃𝑍_𝑖,𝑍_𝑖,ℎ_𝑁𝐷𝜃𝑍_𝑖,𝑍_𝑖=2𝑝1+cos²𝜃csc²𝜃∇ln𝑓². (4.16) Similarly, for any 𝑋∈𝑇𝑁_𝜃 and 𝑍∈𝑇𝑁_⟂ by (4.12), (4.13), and (4.5) it is easy to see that 𝑔ℎ_𝑁𝐷𝜃𝑋_𝑟,𝑍_𝑖,ℎ_𝑁𝐷𝜃𝑋_𝑟,𝑍_𝑖=0. (4.17) The result follows from (4.9), (4.16), and (4.17).

Acknowledgment

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