Warped Product Semi-Invariant Submanifolds in Almost Paracontact Riemannian Manifolds

Abstract

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 form 𝑁=𝑁_⟂×_𝑓𝑁_𝑇 by reversing two factor manifolds 𝑁_𝑇 and 𝑁_⟂. 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.

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'Neill [1]. Many geometers studied different objects/structures on manifolds endowed with an warped product metric (see [2–6]).

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 form 𝑁=𝑁_⟂×_𝑓𝑁_𝑇 in Kaehlerian manifolds. Therefore, he considered warped product CR-submanifolds in the form 𝑁=𝑁_𝑇×_𝑓𝑁_⟂ which is called CR-warped product, where 𝑁_𝑇 is an invariant submanifold, and 𝑁_⟂ 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 in [3] and Munteanu in [9].

Almost paracontact manifolds and almost paracontact Riemannian manifolds were defined and studied by Şato [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 submanifold [11].

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=𝑁_𝑇×_𝑓𝑁_⟂ in contrast to Kaehlerian manifolds (see Theorem 3.1). So, from now on we consider warped product semi-invariant submanifolds in the form 𝑁=𝑁_⟂×_𝑓𝑁_𝑇, where 𝑁_⟂ is an anti-invariant submanifold, and 𝑁_𝑇 is an invariant submanifold of an almost paracontact Riemannian manifold 𝑀 by reversing the two factor manifolds 𝑁_𝑇 and 𝑁_⟂, and it simply will be called warped product semi-invariant submanifold in the rest of this paper (see Example 3.3 and Theorem 3.4).

2. Preliminaries

Although there are many papers concerning the geometry of semi-invariant submanifolds of almost paracontact Riemannian manifolds (see [11–13]), 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.

Let 𝑀 be an (𝑚+1)-dimensional differentiable manifold. If there exist on 𝑀 a (1,1) type tensor field 𝐹, a vector field 𝜉, and 1-form 𝜂 satisfying

𝐹²=𝐼−𝜂⊗𝜉,𝜂(𝜉)=1, (2.1) then 𝑀 is said to be an almost paracontact manifold, where ⊗, the symbol, denotes the tensor product. In the almost paracontact manifold, the following relations hold good:

𝐹𝜉=0,𝜂∘𝐹=0,rank(𝐹)=𝑚. (2.2)

An almost paracontactmanifold 𝑀 is said to be an almost paracontact metric manifold if Riemannian metric 𝑔 on 𝑀 satisfies

𝑔(𝐹𝑋,𝐹𝑌)=𝑔(𝑋,𝑌)−𝜂(𝑋)𝜂(𝑌),𝜂(𝑋)=𝑔(𝑋,𝜉) (2.3) for all 𝑋,𝑌∈Γ(𝑇𝑀) [14], where Γ(𝑇𝑀) denotes the differentiable vector field set on 𝑀. From (2.2) and (2.3), we can easily derive the relation

𝑔(𝐹𝑋,𝑌)=𝑔(𝑋,𝐹𝑌). (2.4)

An almost paracontact metric manifold is said to be an almost paracontact Riemannian manifold with (𝐹,𝑔,𝜉,𝜂)-connection if ∇𝐹=0 and ∇𝜂=0, where ∇ denotes the connection on 𝑀. Since 𝐹²=𝐼−𝜂⊗𝜉, the vector field 𝜉 is also parallel with respect to ∇ [11, 13].

In the rest of this paper, let us suppose that 𝑀 is an almost paracontact Riemannian manifold with structure (𝐹,𝑔,𝜉,𝜂)-connection.

Let 𝑀 be an almost paracontact Riemannian manifold, and let 𝑁 be a Riemannian manifold isometrically immersed in 𝑀. For each 𝑥∈𝑁, we denote by 𝐷_𝑥 the maximal invariant subspace of the tangent space 𝑇_𝑥𝑁 of 𝑁. If the dimension of 𝐷_𝑥 is the same for all 𝑥 in 𝑁, then 𝐷_𝑥 gives an invariant distribution 𝐷 on 𝑁.

A submanifold 𝑁 in an almost paracontact Riemannian manifold is called semi-invariant submanifold if there exists on 𝑁 a differentiable invariant distribution 𝐷 whose orthogonal complementary 𝐷^⟂ is an anti-invariant distribution, that is, 𝐹(𝐷^⟂)⊂𝑇𝑁^⟂, where 𝑇𝑁^⟂ denotes the orthogonal vector bundle of 𝑇𝑁 in 𝑇𝑀. A semi-invariant submanifold is called anti-invariant (resp., invariant) submanifold if dim(𝐷_𝑥)=0 (resp., dim(𝐷^⟂_𝑥)=0). It is called proper semi-invariant submanifold if it is neither invariant nor anti-invariant submanifold.

A semi-invariant submanifold 𝑁 of an almost paracontact Riemannian manifold 𝑀 is called a Riemannian product if the invariant distribution 𝐷 and 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 submanifolds (see references).

Let 𝑁₁ and 𝑁₂ be two Riemannian manifolds with Riemannian metrics 𝑔₁ and 𝑔₂, respectively, and 𝑓 a differentiable function on 𝑁₁. Consider the product manifold 𝑁₁×𝑁₂ with its projection 𝜋₁∶𝑁₁×𝑁₂→𝑁₁ and 𝜋₂∶𝑁₁×𝑁₂→𝑁₂. The warped product manifold 𝑁=𝑁₁×_𝑓𝑁₂ is the manifold 𝑁₁×𝑁₂ equipped with the Riemannian metric tensor such that

𝑔(𝑋,𝑌)=𝑔₁(𝜋_1∗𝑋,𝜋_1∗𝑌)+𝑓²(𝜋₁(𝑥))𝑔₂(𝜋_2∗𝑋,𝜋_2∗𝑌) (2.5) for any 𝑋,𝑌∈Γ(𝑇𝑁), where ∗ is the symbol for the tangent map. Thus we have 𝑔=𝑔₁+𝑓²𝑔₂, where 𝑓 is called the warping function of the warped product. The warped product manifold 𝑁=𝑁₁×_𝑓𝑁₂ is characterized by the fact that 𝑁₁ and 𝑁₂ are totally geodesic and totally umbilical submanifolds of 𝑁, respectively. Hence Riemannian products are special classes of the warped products [4].

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 manifold 𝑀. We denote by ∇ and ∇ the Levi-Civita connections on 𝑁 and 𝑀, respectively. Then the Gauss and Weingarten formulas are, respectively, defined by

∇_𝑋𝑌=∇_𝑋𝑌+ℎ(𝑋,𝑌),∇_𝑋𝑉=−𝐴_𝑉𝑋+∇^⟂_𝑋𝑉 (2.6) for any 𝑋,𝑌∈Γ(𝑇𝑁), 𝑉∈Γ(𝑇𝑁^⟂), where ∇^⟂ is the connection in the normal bundle 𝑇𝑁^⟂,ℎ is the second fundamental form of 𝑁, and 𝐴_𝑉 is the shape operator. The second fundamental form ℎ and the shape operator 𝐴 are related by

𝑔(𝐴_𝑉𝑋,𝑌)=𝑔(ℎ(𝑋,𝑌),𝑉). (2.7)

Now, let 𝑁 be a differentiable manifold, and we suppose that 𝑁 is an isometrically immersed submanifold in almost paracontact Riemannian manifold 𝑀. We denote by 𝑔 the metric tensor of 𝑀 as well as that induced on 𝑁. For any vector field 𝑋 tangent to 𝑁, we put

𝐹𝑋=𝑡𝑋+𝑛𝑋, (2.8) where 𝑡𝑋 and 𝑛𝑋 denote the tangential and normal components of 𝐹𝑋, respectively. For any vector field 𝑉 normal to 𝑁, we also put

𝐹𝑉=𝐵𝑉+𝐶𝑉, (2.9) where 𝐵𝑉 and 𝐶𝑉 denote the tangential and normal components of 𝐹𝑉, respectively. The submanifold 𝑁 is said to be invariant if 𝑛 is identically zero, that is, 𝐹(𝑇𝑁)=𝑇𝑁. On the other hand, 𝑁 is said to be anti-invariant submanifold if 𝑡 is identically zero, that is, 𝐹(𝑇𝑁)⊂(𝑇𝑁^⟂).

We note that for any invariant submanifold 𝑁 of an almost paracontact Riemannian manifold 𝑀, if 𝜉 is normal to 𝑁, then the induced structure from the almost paracontact structure on 𝑁 is an almost product Riemannian structure whenever 𝑡 is nontrivial. If 𝜉 is tangent to 𝑁, then the induced structure on 𝑁 is an almost paracontact Riemannian structure.

Furthermore, we say that 𝑁 is a semi-invariant submanifold if there exist two orthogonal distributions 𝐷₁ and 𝐷₂ such that

(1)𝑇𝑁 splits into the orthogonal direct sum 𝑇𝑁=𝐷₁⊕𝐷₂;(2)the distribution 𝐷₁ is invariant, that is, 𝐹(𝐷₁)⊆𝐷₁;(3)the distribution 𝐷₂ is anti-invariant, that is, 𝐹(𝐷₂)⊆𝑇𝑁^⟂.

Given any submanifold 𝑁 of an almost paracontact Riemannian manifold 𝑀, from (2.4) and (2.8) we have

𝑔(𝑡𝑋,𝑌)=𝑔(𝑋,𝑡𝑌),𝑔(𝑛𝑋,𝑉)=𝑔(𝑋,𝐵𝑉) (2.10) for any 𝑋,𝑌∈Γ(𝑇𝑁), 𝑉∈Γ(𝑇𝑁^⟂).

From now on we suppose that the vector field 𝜉 is tangent to 𝑁.

We recall the following general lemma from [1] for later use.

Lemma 2.1. Let 𝑁=𝑁₁×_𝑓𝑁₂ be a warped product manifold with warping function 𝑓, then one has (1)∇_𝑋𝑌∈Γ(𝑇𝑁₁)foreach𝑋,𝑌∈Γ(𝑇𝑁₁),(2)∇_𝑋𝑍=∇_𝑍𝑋=𝑋(ln𝑓)𝑍,foreach𝑋∈Γ(𝑇𝑁₁),𝑍∈Γ(𝑇𝑁₂),(3)∇_𝑍𝑊=∇^𝑁2_𝑍𝑊−𝑔(𝑍,𝑊)grad𝑓/𝑓,foreach𝑍,𝑊∈Γ(𝑇𝑁₂), where ∇ and ∇^𝑁2 denote the Levi-Civita connections on 𝑁 and 𝑁₂, respectively.

Let 𝑁 be a semi-invariant submaniold of an almost paracontact Riemannian manifold 𝑀. We denote by the invariant distribution 𝐷 and anti-invariant distribution 𝐷^⟂. We also denote the orthogonal complementary of 𝐹(𝐷^⟂) in 𝑇𝑁^⟂ by 𝜈, then we have direct sum

𝑇𝑁^⟂=𝐹(𝐷^⟂)⊕𝜈. (2.11) We can easily see that 𝜈 is an invariant subbundle with respect to 𝐹.

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

Useful characterizations of warped product semi-invariant submanifolds in almost paracontact 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 manifold 𝑀 such that 𝑁_𝑇 is an invariant submanifold and 𝑁_⟂ is an anti-invariant submanifold of 𝑀, then 𝑁 is a usual Riemannian product.

Proof. Let 𝜉 be normal to 𝑁. Taking into account that ℎ is symmetric and using (2.3), (2.6), (2.7), and considering Lemma 2.1(2), for 𝑋∈Γ(𝑇𝑁_𝑇) and 𝑍,𝑊∈Γ(𝑇𝑁_⟂), we have 𝑔(∇_𝑋𝑍,𝑊)=𝑔(∇_𝑍𝑋,𝑊)=𝑔(∇_𝑍𝑋,𝑊)=𝑔(𝐹∇_𝑍𝑋,𝐹𝑊)+𝜂(∇_𝑍𝑋)𝜂(𝑊),𝑋ln(𝑓)𝑔(𝑍,𝑊)=𝑔(∇_𝑍𝐹𝑋,𝐹𝑊)=𝑔(ℎ(𝑍,𝐹𝑋),𝐹𝑊)=𝑔(∇_𝐹𝑋𝑍,𝐹𝑊)=𝑔(∇_𝐹𝑋𝐹𝑍,𝑊)=−𝑔(𝐴_𝐹𝑍𝐹𝑋,𝑊)=−𝑔(ℎ(𝐹𝑋,𝑊),𝐹𝑍)=−𝑔(∇_𝑊𝐹𝑋,𝐹𝑍)=−𝑔(∇_𝑊𝑋,𝑍)=−𝑋ln(𝑓)𝑔(𝑊,𝑍), (3.1) which implies that 𝑋ln(𝑓)=0.
If 𝜉 is tangent to 𝑁, then 𝜉 can be written as follows: 𝜉=𝜉₁+𝜉₂, (3.2) where 𝜉₁∈Γ(𝑇𝑁_𝑇) and 𝜉₂∈Γ(𝑇𝑁_⟂). Since ∇_𝑋𝜉=0, from the Gauss formulae, we have ℎ(𝑋,𝜉)=0,∇_𝑋𝜉=0 (3.3) for any 𝑋∈Γ(𝑇𝑁). Considering Lemma 2.1(2), we get ∇_𝑍𝜉₁=𝜉₁(ln𝑓)𝑍=0,∇_𝑋𝜉₂=𝑋(ln𝑓)𝜉₂∇=0,_𝜉2𝜉₁=∇_𝜉1𝜉₂=𝜉₁(ln𝑓)𝜉₂=0 (3.4) for any 𝑋∈Γ(𝑇𝑁_𝑇) and 𝑍∈Γ(𝑇𝑁_⟂). If 𝜉₂ is identically zero, then from Lemma 2.1 we have ∇_𝑍𝜉₁=∇_𝜉1𝑍=𝜉₁(ln𝑓)𝑍=0,∇_𝑋𝜉₁∈Γ(𝑇𝑁_𝑇). (3.5) It follows that the warping function 𝑓 is a constant and 𝑁 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 on ℝ^2𝑛+1 (see Example 3.2) and after give an example which is concerning its submanifold (see Example 3.3).

Example 3.2. Let ℝ^2𝑛+1={(𝑥₁,𝑥₂,…,𝑥_𝑛,𝑦₁,𝑦₂,…,𝑦_𝑛,𝑡)∣𝑥_𝑖,𝑦_𝑖,𝑡∈ℝ,𝑖=1,2,…,𝑛}. (3.6) The almost paracontact Riemannian structure (𝐹,𝑔,𝜉,𝜂) is defined on ℝ^2𝑛+1 in the following way: 𝜕𝐹(𝜕𝑥_𝑖𝜕)=𝜕𝑦_𝑖𝜕,𝐹(𝜕𝑦_𝑖𝜕)=𝜕𝑥_𝑖𝜕,𝐹(𝜕𝜕𝑡)=0,𝜉=𝜕𝑡,𝜂=𝑑𝑡. (3.7) If 𝑍=𝜆_𝑖(𝜕/𝜕𝑥_𝑖)+𝜇_𝑖(𝜕/𝜕𝑦_𝑖)+𝜈(𝜕/𝜕𝑡)∈𝑇(ℝ^2𝑛+1), then we have 𝑔(𝑍,𝑍)=^𝑛_𝑖=1𝜆²_𝑖+^𝑛_𝑖=1𝜇²_𝑖+𝜈². (3.8) From this definition, it follows that 𝑔(𝑍,𝜉)=𝜂(𝑍)=𝜈,𝑔(𝐹𝑍,𝐹𝑍)=𝑔(𝑍,𝑍)−𝜂²(𝑍),𝐹𝜉=0,𝜂(𝜉)=1 (3.9) for an arbitrary vector field 𝑍. Thus (ℝ^2𝑛+1,𝐹,𝑔,𝜉,𝜂) becomes an almost paracontact Riemannian manifold, where 𝑔 and {𝜕/𝜕𝑥_𝑖,𝜕/𝜕𝑦_𝑖,𝜕/𝜕𝑡} denote usual inner product and standard basis of 𝑇(ℝ^2𝑛+1), respectively.

Example 3.3. Let 𝑁 be a submanifold in ℝ⁵ with coordinates (𝑥₁,𝑥₂,𝑦₁,𝑦₂,𝑡) given by 𝑥₁=𝑣cos𝜃,𝑥₂=𝑣sin𝜃,𝑦₁=𝑣cos𝛽,𝑦₂=𝑣sin𝛽,𝑡=√2𝑢. (3.10) It is easy to check that the tangent bundle of 𝑁 is spanned by the vectors 𝑍₁𝜕=cos𝜃𝜕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 of ℝ⁵ by 𝜕𝐹(𝜕𝑥_𝑖𝜕)=−𝜕𝑥_𝑖𝜕,𝐹(𝜕𝑦_𝑖𝜕)=𝜕𝑦_𝑖𝜕,𝐹(1𝜕𝑡)=0,𝜂=√2𝑑𝑡. (3.12) Then with respect to the almost paracontact Riemannian structure 𝐹 of ℝ⁵, the space 𝐹(𝑇𝑁) becomes 𝐹𝑍₁𝜕=−cos𝜃𝜕𝑥₁𝜕−sin𝜃𝜕𝑥₂𝜕+cos𝛽𝜕𝑦₁𝜕+sin𝛽𝜕𝑦₂,𝐹𝑍₂𝜕=𝑣sin𝜃𝜕𝑥₁𝜕−𝑣cos𝜃𝜕𝑥₂,𝐹𝑍₃𝜕=−𝑣sin𝛽𝜕𝑦₁𝜕+𝑣cos𝛽𝜕𝑦₂,𝐹𝑍₄=0. (3.13) Since 𝐹𝑍₁ and 𝐹𝑍₄ are orthogonal to 𝑁 and 𝐹𝑍₂, 𝐹𝑍₃ are tangent to 𝑁, 𝐷 and 𝐷^⟂ can be taken subspace sp{𝑍₁,𝑍₄} and subspace sp{𝑍₂,𝑍₃}, respectively, where 𝜉 can be taken as 𝑍₄ for 𝐹𝑍₄=0 and 𝜂(𝑍₄)=1. Furthermore, the metric tensor of 𝑁 is given by 𝑔=2(𝑑𝑢²+𝑑𝑣²)+𝑣²(𝑑𝜃²+𝑑𝛽²)=2𝑔_𝑁⟂+𝑣²𝑔_𝑁𝑇. (3.14) Thus 𝑁 is a warped product semi-invariant submanifold with dimensional 5 of almost paracontact manifold ℝ⁵ with warping function 𝑓=𝑣².

Now, let 𝑁=𝑁_⟂×_𝑓𝑁_𝑇 be a warped product semi-invariant submanifold of an almost paracontact Riemannian manifold 𝑀, where 𝑁_⟂ is an anti-invariant submanifold, and 𝑁_𝑇 is an invariant submanifold of 𝑀. If we denote the Levi-Civita connections on 𝑀 and 𝑁 by ∇ and ∇, respectively, by using (2.6) and (2.8), we have

∇_𝑋𝐹𝑌=𝐹∇_𝑋∇𝑌,_𝑋𝑡𝑌+ℎ(𝑋,𝑡𝑌)−𝐴_𝑛𝑌𝑋+∇^⟂_𝑋𝑛𝑌=𝑡(∇_𝑋𝑌)+𝑛(∇_𝑋𝑌)+𝐵ℎ(𝑋,𝑌)+𝐶ℎ(𝑋,𝑌) (3.15) for any 𝑋,𝑌∈Γ(𝑇𝑁). Taking into account the tangential and normal components of (3.15), respectively, we have

(∇_𝑋𝑡)𝑌=𝐴_𝑛𝑌𝑋+𝐵ℎ(𝑋,𝑌),(∇_𝑋𝑛)𝑌=𝐶ℎ(𝑋,𝑌)−ℎ(𝑋,𝑡𝑌), (3.16)(3.17) where the derivatives of 𝑡 and 𝑛 are, respectively, defined by

(∇_𝑋𝑡)𝑌=∇_𝑋𝑡𝑌−𝑡(∇_𝑋𝑌),(∇_𝑋𝑛)𝑌=∇^⟂_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 manifold 𝑀. Then the invariant distribution 𝐷 and the anti invariant distribution 𝐷^⟂ are always integrable.

Proof. From (3.16) and considering lemma 2.1(1), we have ∇_𝑋𝐹𝑈=𝐹∇_𝑋𝑈,𝑋ln(𝑓)𝑡𝑈+ℎ(𝑋,𝑡𝑈)=𝑋ln(𝑓)𝑡𝑈+𝐵ℎ(𝑋,𝑈)+𝐶ℎ(𝑋,𝑈) (3.19) for any 𝑋∈Γ(𝐷^⟂) and 𝑈∈Γ(𝐷). From the tangential and normal components of (3.19), respectively, we arrive at 𝐵ℎ(𝑋,𝑈)=0,ℎ(𝑋,𝑡𝑈)=𝐶ℎ(𝑋,𝑈). (3.20)(3.21) By using (3.16) and (3.20) we get 𝐴_𝑛𝑋𝑈=−𝑋(ln𝑓)𝑡𝑈. (3.22) Furthermore, by using the Gauss-Weingarten formulas and taking into account that 𝐷^⟂ is totally geodesic in 𝑁 and it is anti-invariant in 𝑀, by direct calculations, it is easily to see that 𝐴_𝑛𝑌𝑋=−𝐵ℎ(𝑋,𝑌), (3.23) which is also equivalent to 𝐴_𝑛𝑌𝑋=𝐴_𝑛𝑋𝑌 (3.24) for any 𝑋,𝑌∈Γ(𝐷^⟂). Moreover, using (2.4) and (2.7) and making use of 𝐴 being self-adjoint, we obtain 𝑔(𝐴_𝑛𝑋𝑌,𝑍)=𝑔(ℎ(𝑌,𝑍),𝑛𝑋)=𝑔(∇_𝑍𝑌,𝐹𝑋)=𝑔(∇_𝑍𝐹𝑌,𝑋)=−𝑔(𝐴_𝑛𝑌𝑍,𝑋)=−𝑔(𝐴_𝑛𝑌𝑋,𝑍), (3.25) which gives us 𝐴_𝑛𝑋𝑌=−𝐴_𝑛𝑌𝑋 (3.26) for any 𝑋,𝑌∈Γ(𝐷^⟂) and 𝑍∈Γ(𝑇𝑁). Thus from (3.24) and (3.26), we arrive at 𝐴_𝑛𝑋𝑌=0,𝐵ℎ(𝑋,𝑌)=0 (3.27) for any 𝑋,𝑌∈Γ(𝐷^⟂). Furthermore, by using (2.6), (2.8), and (2.9) and considering Lemma 2.1(3), we have ℎ(𝑈,𝑡𝑉)+∇_𝑈𝑡𝑉=𝐹(∇_𝑈𝑉)+𝐹ℎ(𝑉,𝑈)=𝐹(∇^′_𝑈𝑉−𝑔(𝑉,𝑈)grad𝑓𝑓)+𝐵ℎ(𝑉,𝑈)+𝐶ℎ(𝑉,𝑈)=𝑡(∇^′_𝑈𝑉)−𝑔(𝑉,𝑈)𝑛(grad𝑓𝑓)+𝐵ℎ(𝑉,𝑈)+𝐶ℎ(𝑉,𝑈) (3.28) for any 𝑉,𝑈∈Γ(𝐷), where ∇^′ denote the Levi-Civita connection on 𝐷. Taking into account the normal and tangential components of (3.28), respectively, we have ℎ(𝑈,𝑡𝑉)=−𝑔(𝑈,𝑉)𝑛(grad𝑓𝑓∇)+𝐶ℎ(𝑈,𝑉),_′𝑈𝑡𝑉−𝑔(𝑡𝑉,𝑈)grad𝑓𝑓=𝑡(∇^′_𝑈𝑉)+𝐵ℎ(𝑉,𝑈). (3.29)(3.30) From (3.29), we can easily see that ℎ(𝑈,𝑡𝑉)=ℎ(𝑉,𝑡𝑈) (3.31) for any 𝑈,𝑉∈Γ(𝐷). Finally, by using (3.17) and (3.31), we have 𝑛([𝑉,𝑈])=𝑛(∇_𝑉𝑈−∇_𝑈𝑉)=∇^⟂_𝑉𝑛𝑈−(∇_𝑉𝑛𝑈)−∇^⟂_𝑈𝑛𝑉+(∇_𝑈𝑛)𝑉=(∇_𝑈𝑛)𝑉−(∇_𝑉𝑛)𝑈=𝐶ℎ(𝑈,𝑉)−ℎ(𝑈,𝑡𝑉)−𝐶ℎ(𝑉,𝑈)+ℎ(𝑉,𝑡𝑈)=0 (3.32) for any 𝑉,𝑈∈Γ(𝐷), that is, [𝑉,𝑈]∈Γ(𝐷).
In the same way, making use of (3.16) and (3.27) for any 𝑋,𝑌∈Γ(𝐷^⟂), we conclude that 𝑡([𝑋,𝑌])=𝑡(∇_𝑋𝑌−∇_𝑌𝑋)=∇_𝑋𝑡𝑌−(∇_𝑋𝑡)𝑌−∇_𝑌𝑡𝑋+(∇_𝑌𝑡)𝑋=(∇_𝑌𝑡)𝑋−(∇_𝑋𝑡)𝑌=𝐴_𝑛𝑋𝑌−𝐴_𝑛𝑌𝑋=0 (3.33) that is, [𝑋,𝑌]∈Γ(𝐷^⟂). So we obtain the desired result.

Since the distributions 𝐷 and 𝐷^⟂ are integrable, we denote the integral manifolds of 𝐷 and 𝐷^⟂ by 𝑁_𝑇 and 𝑁_⟂, respectively.

Now, the following theorem characterizes (warped product or Riemannian product) semi-invariant submanifolds in almost paracontact manifolds.

Theorem 3.5. Let 𝑁 be a submanifold of an almost paracontact Riemannian manifold 𝑀. Then 𝑁 is a semi-invariant submanifold if and only 𝑛𝑡=0.

Proof. Let us assume that 𝑁 is a semi-invariant submanifold of an almost paracontact Riemannian manifold 𝑀 and by 𝑄 and 𝑃; we denote the projection operators on subspaces Γ(𝐷^⟂) and Γ(𝐷), respectively, then we have 𝑃+𝑄=𝐼,𝑃²=𝑃,𝑄²=𝑄,𝑃𝑄=𝑄𝑃=0. (3.34) Moreover, by using (2.1), (2.8), and (2.9), if 𝜉 is tangent to 𝑁, then we get 𝑋−𝜂(𝑋)𝜉=𝑡²𝑋+𝐵𝑛𝑋,𝑛𝑡𝑋+𝐶𝑛𝑋=0,𝑡𝐵𝑉+𝐵𝐶𝑉=0,𝑛𝐵𝑉+𝐶²𝑉=𝑉 (3.35)(3.36) for any 𝑋∈Γ(𝑇𝑁) and 𝑉∈Γ(𝑇𝑁^⟂). On the other hand, if 𝜉 is normal to 𝑁, then (3.35) and (3.36) become, respectively, 𝑋=𝑡²𝑋+𝐵𝑛𝑋,𝑛𝑡𝑋+𝐶𝑛𝑋=0,𝑉−𝜂(𝑉)𝜉=𝑛𝐵𝑉+𝐶²𝑉,𝑡𝐵𝑉+𝐵𝐶𝑉=0. (3.37) From (2.8), we have 𝐹𝑋=𝐹𝑃𝑋+𝐹𝑄𝑋,𝑡𝑋+𝑛𝑋=𝑡𝑃𝑋+𝑡𝑄𝑋+𝑛𝑃𝑋+𝑛𝑄𝑋 (3.38) for any 𝑋∈Γ(𝑇𝑁). From the tangential and normal components, we have 𝑡𝑋=𝑡𝑃𝑋+𝑡𝑄𝑋,𝑛𝑋=𝑛𝑃𝑋+𝑛𝑄𝑋. (3.39) Since 𝐷 is invariant and 𝐷^⟂ is anti-invariant, we get 𝑛𝑃=0,𝑄𝑡=0. (3.40) We have 𝑡𝑃=𝑡 (3.41) by virtue of 𝑄=𝐼−𝑃. Now by using the right-hand side to the second equation of (3.35) and using (3.40) and (3.41), we conclude that 𝑛𝑡=0, (3.42) which is also equivalent to 𝐶𝑛=0. (3.43)
Conversely, for a submanifold 𝑁 of an almost paracontact Riemannian manifold 𝑀, we suppose that 𝑛𝑡=0. For any vector fields tangent 𝑋 to 𝑁 and 𝑉 normal to 𝑁, by using (2.4) and (3.43), we have 𝑔(𝑋,𝐹𝑉)=𝑔(𝐹𝑋,𝑉),𝑔(𝑋,𝐵𝑉)=𝑔(𝑛𝑋,𝑉),𝑔(𝑋,𝐹𝐵𝑉)=𝑔(𝐹𝑛𝑋,𝑉),𝑔(𝑋,𝑡𝐵𝑉)=𝑔(𝐶𝑛𝑋,𝑉)=0 (3.44) for all 𝑋∈Γ(𝑇𝑁). So we have 𝑔(𝑡𝐵𝑉,𝑋)=0. Since 𝑋,𝑡𝐵𝑉∈Γ(𝑇𝑁), it implies 𝑡𝐵=0 which is also equivalent to 𝐵𝐶=0 from (3.36). Since 𝐹𝜉=0, we get 𝑡𝜉=𝑛𝜉=0. So, from (3.35) and (3.36), we conclude 𝑡³=𝑡,𝐶³=𝐶. (3.45) Now, if we put 𝑃=𝑡²,𝑄=𝐼−𝑃, (3.46) then we can derive that 𝑃+𝑄=𝐼,𝑃²=𝑃,𝑄²=𝑄,and𝑃𝑄=𝑄𝑃=0 which show that 𝑄 and 𝑃 are orthogonal complementary projection operators and define complementary distributions 𝐷^⟂ and 𝐷, respectively, where 𝐷 and 𝐷^⟂ denote the distributions which are belong to subspaces 𝑇𝑁_𝑇 and 𝑇𝑁_⟂, respectively. From (3.42), (3.45), and (3.46) we can derive 𝑡𝑃=𝑡,𝑡𝑄=0,𝑄𝑡𝑃=0,𝑛𝑃=0. (3.47) These equations show that the distribution 𝐷 is an invariant and the distribution 𝐷^⟂ is an anti-invariant. The proof is complete.

Theorem 3.6. Let 𝑁 be a semi-invariant submanifold of an almost paracontact Riemannian manifold 𝑀. Then 𝑁 is a warped product semi-invariant submanifold if and only if the shape operator of 𝑁 satisfies 𝐴_𝐹𝑋𝑍=−𝑋(𝜇)𝐹𝑍,𝑋∈Γ(𝐷^⟂),𝑍∈Γ(𝐷) (3.48) for some function 𝜇 on 𝑁 satisfying 𝑊(𝜇)=0, 𝑊∈Γ(𝐷).

Proof. We suppose that 𝑁 is a warped product semi-invariant submanifold in an almost paracontact Riemannian manifold 𝑀. Then from (3.22), we have 𝐴_𝐹𝑋𝑍=−𝑋(ln𝑓)𝐹𝑍 (3.49) for any 𝑋∈Γ(𝐷^⟂) and 𝑍∈Γ(𝐷). Since 𝑓 is the only function on 𝑁_⟂, we can easily see that 𝑊(ln𝑓)=0 for all 𝑊∈Γ(𝐷).
Conversely, let us assume that 𝑁 is a semi-invariant submanifold in an almost paracontact Riemannian manifold 𝑀 satisfying 𝐴_𝐹𝑋𝑍=𝑋(𝜇)𝐹𝑍,𝑋∈Γ(𝐷^⟂),𝑍∈Γ(𝐷) (3.50) for some function 𝜇 on 𝑁 satisying 𝑊(𝜇)=0 for all 𝑊∈Γ(𝐷). Since the ambient space 𝑀 is an almost paracontact Riemannian manifold and making use of (2.4) and (3.27), we arrive at 𝑔(∇_𝑋𝑌,𝐹𝑍)=𝑔(∇_𝑋𝑌,𝐹𝑍)=𝑔(∇_𝑋𝐹𝑌,𝑍)=−𝑔(𝐴_𝐹𝑌𝑋,𝑍)=0 (3.51) for any 𝑋,𝑌∈Γ(𝐷^⟂) and 𝑍∈(𝐷). Thus the anti-invariant distribution 𝐷^⟂ is totally geodesic in 𝑁. In the same way, making use of ∇ being Levi-Civita connection and (3.22), we have 𝑔(∇_𝑍𝑊,𝑋)=𝑔(∇_𝑍𝑊,𝑋)=−𝑔(∇_𝑍𝑋,𝑊)=−𝑔(∇_𝑍𝐹𝑋,𝐹𝑊)=𝑔(𝐴_𝐹𝑋𝑍,𝐹𝑊)=𝑋(𝜇)𝑔(𝑍,𝑊) (3.52) for any 𝑍,𝑊∈Γ(𝐷) and 𝑋∈Γ(𝐷^⟂), where 𝜇=ln(1/𝑓). Since the invariant distribution 𝐷 of semi-invariant submanifold 𝑁 is always integrable (Theorem 3.4) and 𝑊(𝜇)=0, for each 𝑊∈Γ(𝑇𝑁_𝑇), which implies that the integral manifold of 𝐷 is an extrinsic sphere in 𝑁, that is, it is a totally umbilical submanifold and its mean curvature vector field is non-zero and parallel, thus we know that 𝑁 is a Riemannian warped product 𝑁_⟂×_𝑓𝑁_𝑇, where 𝑁_⟂ and 𝑁_𝑇 denote the integral manifolds of the distributions of 𝐷^⟂ and 𝐷, respectively, and 𝑓 is 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 form ℎ of 𝑁 satisfies ℎ(𝑋,𝑍)=0, 𝑋∈Γ(𝐷) and 𝑍∈Γ(𝐷^⟂) (resp., ℎ(𝑋,𝑌)=0, 𝑋,𝑌∈Γ(𝐷) and ℎ(𝑍,𝑊)=0, 𝑍,𝑊∈Γ(𝐷^⟂)).

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 manifold 𝑀. Then one has(1)𝑔(ℎ(𝐷^⟂,𝐷^⟂),𝐹𝐷^⟂)=0,(2)𝑔(ℎ(𝑍,𝑊),𝐹𝑋)=−𝑋(ln𝑓)𝑔(𝑡𝑍,𝑊), 𝑍,𝑊∈Γ(𝐷), 𝑋∈Γ(𝐷^⟂),(3)𝑔(ℎ(𝑋,𝑍),𝐹𝑌)=0, for any 𝑋,𝑌∈Γ(𝐷^⟂) and 𝑍∈Γ(𝐷),(4)𝑔(ℎ(𝐷,𝐹𝐷),𝐹𝐷^⟂)=0 if and only if 𝑁=𝑁_⟂×_𝑓𝑁_𝑇 is a usual Riemannian product, where 𝐷 and 𝐷^⟂ denote the leaves of 𝑁_𝑇 and 𝑁_⟂, respectively.

Proof. (1) For any 𝑋,𝑌,𝑍∈Γ(𝐷^⟂), by using (2.4) and (3.27) and considering that the ambient space is an almost paracontact Riemannian manifold, we have 𝑔(ℎ(𝑋,𝑌),𝐹𝑍)=𝑔(∇_𝑋𝑌,𝐹𝑍)=𝑔(∇_𝑋𝐹𝑌,𝑍)=−𝑔(𝐴_𝑛𝑌𝑋,𝑍)=0. (3.53)
(2) Making use of ∇ being Levi-Civita connection and Lemma 2.1(2.2), we get 𝑔(ℎ(𝑍,𝑊),𝐹𝑋)=𝑔(∇_𝑊𝐹𝑍,𝑋)=−𝑔(∇_𝑊𝑋,𝑡𝑍)=−𝑋ln𝑔⋅𝑔(𝑊,𝑡𝑍) (3.54) for any 𝑍,𝑊∈Γ(𝐷), 𝑋∈Γ(𝐷^⟂). 
(3) In the same way, we have 𝑔(ℎ(𝑋,𝑍),𝐹𝑌)=𝑔(∇_𝑋𝑍,𝐹𝑌)=𝑔(∇_𝑋𝑡𝑍,𝑌)=𝑋ln𝑓𝑔(𝑡𝑍,𝑌)=0 (3.55) for any 𝑋,𝑌∈Γ(𝐷^⟂) and 𝑍∈Γ(𝐷).
(4) Considering Lemma 2.1(3) we derive 𝑔(ℎ(𝑊,𝐹𝑍),𝐹𝑋)=𝑔(∇_𝐹𝑍𝑊,𝐹𝑋)=𝑔(∇_𝐹𝑍𝐹𝑊,𝑋)=−𝑔(𝑡𝑍,𝑡𝑊)𝑋(ln𝑓) (3.56) for any 𝑍,𝑊∈Γ(𝐷) and 𝑋∈Γ(𝐷^⟂).

Theorem 3.8. Let 𝑁=𝑁_⟂×_𝑓𝑁_𝑇 be a warped product semi-invariant submanifold of an almost paracontact Riemannian manifold 𝑀. Then one has the following.(1) The squared norm of the second fundamental form of 𝑁 in 𝑀 satisfies ‖ℎ‖²≥1𝑓²‖grad𝑓‖²(Tr(𝑡))², (3.57) where Tr(𝑡) denote the trace of mapping 𝑡.(2) If the equality sign of (3.57) holds identically, then 𝑁_⟂ is a totally geodesic, 𝑁_𝑇 is a totally umbilical submanifolds of 𝑀, and 𝑁 is a mixed geodesic submanifold in 𝑀. Furthermore, 𝑁 is a minimal submanifold 𝑀 if and only if Tr(𝑡)=0 or 𝑁=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 Γ(𝑇𝑁_⟂),{𝑒¹,𝑒²,…,𝑒^𝑞} is tangent to Γ(𝑇𝑁_𝑇), and {𝑁₁,𝑁₂,…,𝑁_𝑠} is tangent to Γ(𝜈). Taking into account Lemma 3.7 and the basic linear algebra rules, by direct calculations, we have ℎ(𝑋,𝑌)=𝑔(ℎ(𝑋,𝑌),𝑁_𝑗)𝑁_𝑗,1≤𝑗≤𝑠,ℎ(𝑍,𝑊)=−𝑒_𝑖ln𝑓(𝑡𝑍,𝑊)𝐹𝑒_𝑖+𝑔(ℎ(𝑍,𝑊),𝑁_𝑗)𝑁_𝑗,1≤𝑖≤𝑝,ℎ(𝑋,𝑍)=𝑔(ℎ(𝑋,𝑍),𝑁_𝑗)𝑁_𝑗 (3.58) for all 𝑋,𝑌∈Γ(𝑇𝑁_⟂) and 𝑍,𝑊∈Γ(𝑇𝑁_𝑇). Since ‖ℎ‖²=^𝑝𝑠_{𝑖,𝑗=1ℓ=1}𝑔(ℎ(𝑒_𝑖,𝑒_𝑗),𝑁_ℓ)²+2^𝑝𝑞_𝑖=1^𝑠_𝑘=1ℓ=1𝑔(ℎ(𝑒_𝑖,𝑒^𝑘),𝑁_ℓ)²+^𝑞𝑝_{𝑟,𝑘=1𝑖=1}(𝑒_𝑖ln𝑓)²𝑔(𝑡𝑒^𝑘,𝑒^𝑟)²+^𝑞𝑠_{𝑟,𝑘=1ℓ=1}𝑔(ℎ(𝑒^𝑘,𝑒^𝑟),𝑁_ℓ)², (3.59) here by direct calculations, we get (𝑒_𝑖(ln𝑓))²=1𝑓²‖grad𝑓‖²,Tr(𝑡)=^𝑞_𝑘=1𝑔(𝑡𝑒^𝑘,𝑒^𝑘). (3.60) So we conclude that ‖ℎ‖²≥1𝑓²‖grad𝑓‖²(Tr(𝑡))², (3.61) which proves our assertion.
Now we assume that the equality case of (3.57) holds identically, then from (3.58), respectively, we obtain ℎ(𝐷^⟂,𝐷^⟂)=0,ℎ(𝐷,𝐷)∈Γ(𝐹(𝐷^⟂)),ℎ(𝐷^⟂,𝐷)=0. (3.62)(3.63) Since 𝑁_⟂ is totally geodesic submanifold in 𝑁, the first condition in (3.62) implies that 𝑁_⟂ is totally geodesic submanifold in 𝑀. Moreover, Lemma 2.1(3) shows that 𝑁_𝑇 is totally umbilical submanifold in 𝑁. Therefore, the second condition in (3.62) implies that 𝑁_𝑇 is also totally umbilical submanifold in 𝑀. On the other hand, (3.20) and (3.63) imply that 𝑁 is mixed geodesic submanifold in 𝑀.

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 form 𝑁=𝑁_⟂×_𝑓𝑁_𝑇 (see [2, 8]) while there exists no proper warped product semi-invariant submanifold in the form 𝑁=𝑁_𝑇×_𝑓𝑁_⟂ in Riemannian manifolds (see Theorem 3.1). The first condition in (3.62) implies 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.