Warped Product Semi-Invariant Submanifolds of Nearly Cosymplectic Manifolds

Abstract

We study warped product semi-invariant submanifolds of nearly cosymplectic manifolds. We prove that the warped product of the type 𝑀_⟂×_𝑓𝑀_𝑇 is a usual Riemannian product of 𝑀_⟂ and 𝑀_𝑇, where 𝑀_⟂ and 𝑀_𝑇 are anti-invariant and invariant submanifolds of a nearly cosymplectic manifold 𝑀, respectively. Thus we consider the warped product of the type 𝑀_𝑇×_𝑓𝑀_⟂ and obtain a characterization for such type of warped product.

1. Introduction

The notion of warped product manifolds was introduced by Bishop and O'Neill in 1969 as a natural generalization of the Riemannian product manifolds. Later on, the geometrical aspect of these manifolds has been studied by many researchers (cf., [1–3]). Recently, Chen [1] (see also [4]) studied warped product CR-submanifolds and showed that there exists no warped product CR-submanifolds of the form 𝑀=𝑀_⟂×_𝑓𝑀_𝑇 such that 𝑀_⟂ is a totally real submanifold and 𝑀_𝑇 is a holomorphic submanifold of a Kaehler manifold 𝑀. Therefore he considered warped product CR-submanifold in the form 𝑀=𝑀_𝑇×_𝑓𝑀_⟂ which is called CR-warped product, where 𝑀_𝑇 and 𝑀_⟂ are holomorphic and totally real submanifolds of a Kaehler manifold 𝑀. Motivated by Chen's papers, many geometers studied CR-warped product submanifolds in almost complex as well as contact setting (see [3, 5, 6]).

Almost contact manifolds with Killing structure tensors were defined in [7] as nearly cosymplectic manifolds, and it was shown that normal nearly cosymplectic manifolds are cosymplectic (see also [8]). Later on, Blair and Showers [9] studied nearly cosymplectic structure (𝜙,𝜉,𝜂,𝑔) on a manifold 𝑀 with 𝜂 closed from the topological viewpoint.

In this paper, we have generalized the results of Chen' [1] in this more general setting of nearly cosymplectic manifolds and have shown that the warped product in the form 𝑀=𝑀_⟂×_𝑓𝑀_𝑇 is simply Riemannian product of 𝑀_⟂ and 𝑀_𝑇 where 𝑀_⟂ is an anti-invariant submanifold and 𝑀_𝑇 is an invariant submanifold of a nearly cosymplectic manifold 𝑀. Thus we consider the warped product submanifold of the type 𝑀=𝑀_𝑇×_𝑓𝑀_⟂ by reversing the two factors 𝑀_⟂ and 𝑀_𝑇 and simply will be called warped product semi-invariant submanifold. Thus, we derive the integrability of the involved distributions in the warped product and obtain a characterization result.

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 a 1-form 𝜂 satisfying [9] 𝜙²=−𝐼+𝜂⊗𝜉,𝜙𝜉=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 compatibility condition: 𝜂(𝑋)=𝑔(𝑋,𝜉),𝑔(𝜙𝑋,𝜙𝑌)=𝑔(𝑋,𝑌)−𝜂(𝑋)𝜂(𝑌),(2.2) where 𝑋 and 𝑌 are vector fields on 𝑀 [9].

An almost contact structure (𝜙,𝜉,𝜂) is said to be normal if the almost complex structure 𝐽 on the product manifold 𝑀×ℝ given by 𝐽𝑋,𝑓𝑑𝑑𝑡=𝜙𝑋−𝑓𝜉,𝜂(𝑋)𝑑𝑑𝑡,(2.3) where 𝑓 is a 𝐶^∞-function on 𝑀×ℝ, has no torsion, that is, 𝐽 is integrable, and the condition for normality in terms of 𝜙,𝜉 and 𝜂 is [𝜙,𝜙]+2𝑑𝜂⊗𝜉=0 on 𝑀, where [𝜙,𝜙] is the Nijenhuis tensor of 𝜙. Finally the fundamental 2-form Φ is defined by Φ(𝑋,𝑌)=𝑔(𝑋,𝜙𝑌).

An almost contact metric structure (𝜙,𝜉,𝜂,𝑔) is said to be cosymplectic, if it is normal and both Φ and 𝜂 are closed [9]. The structure is said to be nearly cosymplectic if 𝜙 is Killing, that is, if ∇_𝑋𝜙𝑌+∇_𝑌𝜙𝑋=0,(2.4) for any 𝑋,𝑌∈𝑇𝑀, where 𝑇𝑀 is the tangent bundle of 𝑀 and ∇ denotes the Riemannian connection of the metric 𝑔. Equation (2.4) is equivalent to (∇_𝑋𝜙)𝑋=0, for each 𝑋∈𝑇𝑀. The structure is said to be closely cosymplectic if 𝜙 is Killing and 𝜂 is closed. It is well known that an almost contact metric manifold is cosymplectic if and only if ∇𝜙 vanishes identically, that is, (∇_𝑋𝜙)𝑌=0 and ∇_𝑋𝜉=0.

Proposition 2.1 (see [9]). On a nearly cosymplectic manifold, the vector field 𝜉 is Killing.

From the above proposition we have ∇_𝑋𝜉=0, for any vector field 𝑋 tangent to 𝑀, where 𝑀 is a nearly cosymplectic manifold.

Let 𝑀 be submanifold of an almost contact metric manifold 𝑀 with induced metric 𝑔, and if ∇ and ∇^⟂ are the induced connections 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),(2.5)∇_𝑋𝑁=−𝐴_𝑁𝑋+∇^⟂𝑋𝑁,(2.6) for each 𝑋,𝑌∈𝑇𝑀 and 𝑁∈𝑇^⟂𝑀, where ℎ and 𝐴_𝑁 are the second fundamental form and the shape operator (corresponding to the normal vector field 𝑁), respectively, for the immersion of 𝑀 into 𝑀. They are related as 𝑔(ℎ(𝑋,𝑌),𝑁)=𝑔𝐴_𝑁𝑋,𝑌,(2.7) where 𝑔 denotes the Riemannian metric on 𝑀 as well as being induced on 𝑀.

For any 𝑋∈𝑇𝑀, we write 𝜙𝑋=𝑇𝑋+𝐹𝑋,(2.8) where 𝑇𝑋 is the tangential component and 𝐹𝑋 is the normal component of 𝜙𝑋.

Similarly for any 𝑁∈𝑇^⟂𝑀, we write 𝜙𝑁=𝐵𝑁+𝐶𝑁,(2.9) where 𝐵𝑁 is the tangential component and 𝐶𝑁 is the normal component of 𝜙𝑁. The covariant derivatives of the tensor fields 𝑃 and 𝐹 are defined as ∇_𝑋𝑇𝑌=∇_𝑋𝑇𝑌−𝑇∇_𝑋𝑌,(2.10)∇_𝑋𝐹𝑌=∇^⟂𝑋𝐹𝑌−𝐹∇_𝑋𝑌(2.11) for all 𝑋,𝑌∈𝑇𝑀.

Let 𝑀 be a Riemannian manifold isometrically immersed in an almost contact metric manifold 𝑀. then for every 𝑥∈𝑀 there exists a maximal invariant subspace denoted by 𝒟_𝑥 of the tangent space 𝑇_𝑥𝑀 of 𝑀. If the dimension of 𝒟_𝑥 is the same for all values of 𝑥∈𝑀, then 𝒟_𝑥 gives an invariant distribution 𝒟 on 𝑀.

A submanifold 𝑀 of an almost contact metric manifold 𝑀 is called semi-invariant submanifold if there exists on 𝑀 a differentiable invariant distribution 𝒟 whose orthogonal complementary distribution 𝒟^⟂ is anti-invariant, that is, (i)𝑇𝑀=𝒟⊕𝒟^⟂⊕⟨𝜉⟩, (ii)𝜙(𝒟_𝑥)⊆𝐷_𝑥, (iii)𝜙(𝒟^⟂𝑥)⊂𝑇^⟂𝑥𝑀

for any 𝑥∈𝑀, where 𝑇^⟂𝑥𝑀 denotes the orthogonal space of 𝑇_𝑥𝑀 in 𝑇_𝑥𝑀. A semi-invariant submanifold is called anti-invariant if 𝒟_𝑥={0} and invariant if 𝒟^⟂𝑥={0}, respectively, for any 𝑥∈𝑀. It is called the proper semi-invariant submanifold if neither 𝒟_𝑥={0} nor 𝒟^⟂𝑥={0}, for every 𝑥∈𝑀.

Let 𝑀 be a semi-invariant submanifold of an almost contact metric manifold 𝑀. Then, 𝐹(𝑇_𝑥𝑀) is a subspace of 𝑇^⟂𝑥𝑀. Then for every 𝑥∈𝑀, there exists an invariant subspace 𝜈_𝑥 of 𝑇_𝑥𝑀 such that 𝑇^⟂𝑥𝑀=𝐹𝑇_𝑥𝑀⊕𝜈_𝑥.(2.12)

A semi-invariant submanifold 𝑀 of an almost contact metric manifold 𝑀 is called Riemannian product if the invariant distribution 𝒟 and anti-invariant distribution 𝒟^⟂ are totally geodesic distributions in 𝑀.

Let (𝑀₁,𝑔₁) and (𝑀₂,𝑔₂) be two Riemannian manifolds, and let 𝑓 be a positive differentiable function on 𝑀₁. The warped product of 𝑀₁ and 𝑀₂ is the product manifold 𝑀₁×_𝑓𝑀₂=(𝑀₁×𝑀₂,𝑔), where 𝑔=𝑔₁+𝑓²𝑔₂,(2.13) where 𝑓 is called the warping function of the warped product. The warped product 𝑁₁×_𝑓𝑁₂ is said to be trivial or simply Riemannian product if the warping function 𝑓 is constant. This means that the Riemannian product is a special case of warped product.

We recall the following general results obtained by Bishop and O'Neill [10] for warped product manifolds.

Lemma 2.2. Let 𝑀=𝑀₁×_𝑓𝑀₂ be a warped product manifold with the warping function 𝑓. Then (i)∇_𝑋𝑌∈𝑇𝑀₁,  for each 𝑋,𝑌∈𝑇𝑀₁, (ii)∇_𝑋𝑍=∇_𝑍𝑋=(𝑋ln𝑓)𝑍, for each 𝑋∈𝑇𝑀₁ and 𝑍∈𝑇𝑀₂, (iii)∇_𝑍𝑊=∇^𝑀2_𝑍𝑊−(𝑔(𝑍,𝑊)/𝑓)grad𝑓, where ∇ and ∇^𝑀2 denote the Levi-Civita connections on 𝑀 and 𝑀₂, respectively.

In the above lemma grad𝑓 is the gradient of the function 𝑓 defined by 𝑔(grad𝑓,𝑈)=𝑈𝑓, for each 𝑈∈𝑇𝑀. From the Lemma 2.2, we have that on a warped product manifold 𝑀=𝑀₁×_𝑓𝑀₂(i)𝑀₁ is totally geodesic in 𝑀; (ii)𝑀₂ is totally umbilical in 𝑀.

Now, we denote by 𝒫_𝑋𝑌 and 𝒬_𝑋𝑌 the tangential and normal parts of (∇_𝑋𝜙)𝑌, that is, ∇_𝑋𝜙𝑌=𝒫_𝑋𝑌+𝒬_𝑋𝑌(2.14) for all 𝑋,𝑌∈𝑇𝑀. Making use of (2.5), (2.6), and (2.8)–(2.11), the following relations may easily be obtained 𝒬𝒫_𝑋𝑌=∇_𝑋𝑇𝑌−𝐴_𝐹𝑌𝑋−𝐵ℎ(𝑋,𝑌),(2.15)_𝑋𝑌=∇_𝑋𝐹𝑌+ℎ(𝑋,𝑇𝑌)−𝐶ℎ(𝑋,𝑌).(2.16)

It is straightforward to verify the following properties of 𝒫 and 𝒬, which we enlist here for later use: (𝑝₁)(i) 𝒫_𝑋+𝑌𝑊=𝒫_𝑋𝑊+𝒫_𝑌𝑊, (ii) 𝒬_𝑋+𝑌𝑊=𝒬_𝑋𝑊+𝒬_𝑌𝑊,(𝑝₂)(i) 𝒫_𝑋(𝑌+𝑊)=𝒫_𝑋𝑌+𝒫_𝑋𝑊, (ii) 𝒬_𝑋(𝑌+𝑊)=𝒬_𝑋𝑌+𝒬_𝑋𝑊,(𝑝₃)𝑔(𝒫_𝑋𝑌,𝑊)=−𝑔(𝑌,𝒫_𝑋𝑊)

for all 𝑋,𝑌,𝑊∈𝑇𝑀.

On a submanifold 𝑀 of a nearly cosymplectic manifold 𝑀, we obtain from (2.4) and (2.14) that (i)𝒫_𝑋𝑌+𝒫_𝑌𝑋=0,(ii)𝒬_𝑋𝑌+𝒬_𝑌𝑋=0(2.17) for any 𝑋,𝑌∈𝑇𝑀.

3. Warped Product Semi-Invariant Submanifolds

Throughout the section we consider the submanifold 𝑀 of a nearly cosymplectic manifold 𝑀 such that the structure vector field 𝜉 is tangent to 𝑀. First, we prove that the warped product 𝑀=𝑀₁×_𝑓𝑀₂ is trivial when 𝜉 is tangent to 𝑀₂, where 𝑀₁ and 𝑀₂ are Riemannian submanifolds of a nearly cosymplectic manifold 𝑀. Thus, we consider the warped product 𝑀=𝑀₁×_𝑓𝑀₂, when 𝜉 is tangent to the submanifold 𝑀₁. We have the following nonexistence theorem.

Theorem 3.1. A warped product submanifold 𝑀=𝑀₁×_𝑓𝑀₂ of a nearly cosymplectic manifold 𝑀 is a usual Riemannian product if the structure vector field 𝜉 is tangent to 𝑀₂, where 𝑀₁ and 𝑀₂ are the Riemannian submanifolds of 𝑀.

Proof. For any 𝑋∈𝑇𝑀₁ and 𝜉 tangent to 𝑀₂, we have ∇_𝑋𝜉=∇_𝑋𝜉+ℎ(𝑋,𝜉).(3.1) Using the fact that 𝜉 is Killing on a nearly cosymplectic manifold (see Proposition 2.1) and Lemma 2.2(ii), we get 0=(𝑋ln𝑓)𝜉+ℎ(𝑋,𝜉).(3.2) Equating the tangential component of (3.2), we obtain 𝑋ln𝑓=0, for all 𝑋∈𝑇𝑀₁, that is, 𝑓 is constant function on 𝑀₁. Thus, 𝑀 is Riemannian product. This proves the theorem.

Now, the other case of warped product 𝑀=𝑀₁×_𝑓𝑀₂ when 𝜉∈𝑇𝑀₁, where 𝑀₁ and 𝑀₂ are the Riemannian submanifolds of 𝑀. For any 𝑋∈𝑇𝑀₂, we have ∇_𝑋𝜉=∇_𝑋𝜉+ℎ(𝑋,𝜉).(3.3) By Proposition 2.1, and Lemma 2.2(ii), we obtain (i)𝜉ln𝑓=0,(ii)ℎ(𝑋,𝜉)=0.(3.4) Thus, we consider the warped product semi-invariant submanifolds of a nearly cosymplectic manifold 𝑀 of the types: (i)𝑀=𝑀_⟂×_𝑓𝑀_𝑇, (ii)𝑀=𝑀_𝑇×_𝑓𝑀_⟂,

where 𝑀_𝑇 and 𝑀_⟂ are invariant and anti-invariant submanifolds of 𝑀, respectively. In the following theorem we prove that the warped product semi-invariant submanifold of the type (i) is CR-product.

Theorem 3.2. The warped product semi-invariant submanifold 𝑀=𝑀_⟂×_𝑓𝑀_𝑇 of a nearly cosymplectic manifold 𝑀 is a usual Riemannian product of 𝑀_⟂ and 𝑀_𝑇, where 𝑀_⟂ and 𝑀_𝑇 are anti-invariant and invariant submanifolds of 𝑀, respectively.

Proof. When 𝜉∈𝑇𝑀_𝑇, then by Theorem 3.1, 𝑀 is a Riemannian product. Thus, we consider 𝜉∈𝑇𝑀_⟂. For any 𝑋∈𝑇𝑀_𝑇 and 𝑍∈𝑇𝑀_⟂, we have 𝑔(ℎ(𝑋,𝜙𝑋),𝐹𝑍)==𝑔𝑔(ℎ𝜙(𝑋,𝜙𝑋),𝜙𝑍)=𝑔∇_𝑋𝜙𝑋,𝜙𝑍∇_𝑋𝑋,𝜙𝑍+𝑔∇_𝑋𝜙𝑋,𝜙𝑍.(3.5) From the structure equation of nearly cosymplectic, the second term of right hand side vanishes identically. Thus from (2.2), we derive 𝑔(ℎ(𝑋,𝜙𝑋),𝐹𝑍)==𝑔−𝑔∇𝑋_𝑋𝑋,,𝑍−𝜂(𝑍)𝑔∇_𝑋𝑋,𝜉∇_𝑋𝑍+𝜂(𝑍)𝑔𝑋,∇_𝑋𝜉.(3.6) Then from (2.5), Lemma 2.2(ii), and Proposition 2.1, we obtain 𝑔(ℎ(𝑋,𝜙𝑋),𝐹𝑍)=−(𝑍ln𝑓)‖𝑋‖².(3.7) Interchanging 𝑋 by 𝜙𝑋 in (3.7) and using the fact that 𝜉∈𝑇𝑀_⟂, we obtain 𝑔(ℎ(𝑋,𝜙𝑋),𝐹𝑍)=(𝑍ln𝑓)‖𝑋‖².(3.8) It follows from (3.7) and (3.8) that 𝑍ln𝑓=0, for all 𝑍∈𝑇𝑀_⟂. Also, from (3.4) we have 𝜉ln𝑓=0. Thus, the warping function 𝑓 is constant. This completes the proof of the theorem.

From the above theorem we have seen that the warped product of the type 𝑀=𝑀_⟂×_𝑓𝑀_𝑇 is a usual Riemannian product of an anti-invariant submanifold 𝑀_⟂ and an invariant submanifold 𝑀_𝑇 of a nearly cosymplectic manifold 𝑀. Since both 𝑀_⟂ and 𝑀_𝑇 are totally geodesic in 𝑀, then 𝑀 is CR-product. Now, we study the warped product semi-invariant submanifold 𝑀=𝑀_𝑇×_𝑓𝑀_⟂ of a nearly cosymplectic manifold 𝑀.

Theorem 3.3. Let 𝑀=𝑀_𝑇×_𝑓𝑀_⟂ be a warped product semi-invariant submanifold of a nearly cosymplectic manifold 𝑀. Then the invariant distribution 𝒟 and the anti-invariant distribution 𝒟^⟂ are always integrable.

Proof. For any 𝑋,𝑌∈𝒟, we have 𝐹[𝑋,𝑌]=𝐹∇_𝑋𝑌−𝐹∇_𝑌𝑋.(3.9) Using (2.11), we obtain 𝐹[𝑋,𝑌]=∇_𝑋𝐹𝑌−∇_𝑌𝐹𝑋.(3.10) Then by (2.16), we derive 𝐹[𝑋,𝑌]=𝒬_𝑋𝑌−ℎ(𝑋,𝑇𝑌)+𝐶ℎ(𝑋,𝑌)−𝒬_𝑌𝑋+ℎ(𝑌,𝑇𝑋)−𝐶ℎ(𝑋,𝑌).(3.11) Thus from (2.17)(ii), we get 𝐹[𝑋,𝑌]=2𝒬_𝑋𝑌+ℎ(𝑌,𝑇𝑋)−ℎ(𝑋,𝑇𝑌).(3.12) Now, for any 𝑋,𝑌∈𝐷, we have ℎ(𝑋,𝑇𝑌)+∇_𝑋𝑇𝑌=∇_𝑋𝑇𝑌=∇_𝑋𝜙𝑌.(3.13) Using the covariant derivative property of ∇𝜙, we obtain ℎ(𝑋,𝑇𝑌)+∇_𝑋𝑇𝑌=∇_𝑋𝜙𝑌+𝜙∇_𝑋𝑌.(3.14) Then by (2.5) and (2.14), we get ℎ(𝑋,𝑇𝑌)+∇_𝑋𝑇𝑌=𝑃_𝑋𝑌+𝒬_𝑋𝑌+𝜙∇_𝑋𝑌+ℎ(𝑋,𝑌).(3.15) Since 𝑀_𝑇 is totally geodesic in 𝑀 (see Lemma 2.2(i)), then using (2.8) and (2.9), we obtain ℎ(𝑋,𝑇𝑌)+∇_𝑋𝑇𝑌=𝒫_𝑋𝑌+𝒬_𝑋𝑌+𝑇∇_𝑋𝑌+𝐵ℎ(𝑋,𝑌)+𝐶ℎ(𝑋,𝑌).(3.16) Equating the normal components of (3.16), we get ℎ(𝑋,𝑇𝑌)=𝒬_𝑋𝑌+𝐶ℎ(𝑋,𝑌).(3.17) Similarly, we obtain ℎ(𝑌,𝑇𝑋)=𝒬_𝑌𝑋+𝐶ℎ(𝑋,𝑌).(3.18) Then from (3.17) and (3.18), we arrive at ℎ(𝑌,𝑇𝑋)−ℎ(𝑋,𝑇𝑌)=𝒬_𝑌𝑋−𝒬_𝑋𝑌.(3.19) Hence, using (2.17)(ii), we get ℎ(𝑌,𝑇𝑋)−ℎ(𝑋,𝑇𝑌)=−2𝒬_𝑋𝑌.(3.20) Thus, it follows from (3.12) and (3.20) that 𝐹[𝑋,𝑌]=0, for all 𝑋,𝑌∈𝐷. This proves the integrability of 𝐷. Now, for the integrability of 𝐷^⟂, we consider any 𝑋∈𝐷 and 𝑍,𝑊∈𝐷^⟂, and we have 𝑔([𝑍,𝑊],𝑋)==−𝑔𝑔∇∇_𝑍𝑍𝑊−∇_𝑊𝑍,𝑋.𝑋,𝑊+𝑔∇_𝑊𝑋,𝑍.(3.21) Using Lemma 2.2(ii), we obtain 𝑔([𝑍,𝑊],𝑋)=−(𝑋ln𝑓)𝑔(𝑍,𝑊)+(𝑋ln𝑓)𝑔(𝑍,𝑊)=0.(3.22) Thus from (3.22), we conclude that [𝑍,𝑊]∈𝒟^⟂, for each 𝑍,𝑊∈𝒟^⟂. Hence, the theorem is proved completely.

Lemma 3.4. Let 𝑀=𝑀_𝑇×_𝑓𝑀_⟂ be a warped product submanifold of a nearly cosymplectic manifold 𝑀. If 𝑋,𝑌∈𝑇𝑀_𝑇 and 𝑍,𝑊∈𝑇𝑀_⟂, then (i)𝑔(𝒫_𝑋𝑌,𝑍)=𝑔(ℎ(𝑋,𝑌),𝐹𝑍)=0, (ii)𝑔(𝒫_𝑋𝑍,𝑊)=𝑔(ℎ(𝑋,𝑍),𝐹𝑊)−𝑔(ℎ(𝑋,𝑊),𝐹𝑍)=−(𝜙𝑋ln𝑓)𝑔(𝑍,𝑊)−𝑔(ℎ(𝑋,𝑍),𝐹𝑊), (iii)𝑔(ℎ(𝜙𝑋,𝑍),𝐹𝑍)=(𝑋ln𝑓)‖𝑍‖².

Proof. For a warped product manifold 𝑀=𝑀_𝑇×_𝑓𝑀_⟂, we have that 𝑀_𝑇 is totally geodesic in 𝑀; then by (2.10), (∇_𝑋𝑇)𝑌∈𝑇𝑀_𝑇, for any 𝑋,𝑌∈𝑇𝑀_𝑇, and therefore from (2.15), we get 𝑔𝒫_𝑋𝑌,𝑍=−𝑔(𝐵ℎ(𝑋,𝑌),𝑍)=𝑔(ℎ(𝑋,𝑌),𝐹𝑍).(3.23) The left-hand side of (3.23) is skew symmetric in 𝑋 and 𝑌 whereas the right hand side is symmetric in 𝑋 and 𝑌, which proves (i). Now, from (2.10) and (2.15), we have 𝒫_𝑋𝑍=−𝑇∇_𝑋𝑍−𝐴_𝐹𝑍𝑋−𝐵ℎ(𝑋,𝑍)(3.24) for any 𝑋∈𝑇𝑀_𝑇 and 𝑍∈𝑇𝑀_⟂. Using Lemma 2.2(ii), the first term of right-hand side is zero. Thus, taking the product with 𝑊∈𝑇𝑀_⟂, we obtain 𝑔𝒫_𝑋𝑍,𝑊=−𝑔𝐴_𝐹𝑍𝑋,𝑊−𝑔(𝐵ℎ(𝑋,𝑍),𝑊),(3.25) Then by (2.2) and (2.7), we get 𝑔𝒫_𝑋𝑍,𝑊=−𝑔(ℎ(𝑋,𝑊),𝐹𝑍)+𝑔(ℎ(𝑋,𝑍),𝐹𝑊).(3.26) which proves the first equality of (ii). Again, from (2.10) and (2.15), we have 𝒫_𝑍𝑋=∇_𝑍𝑇𝑋−𝑇∇_𝑍𝑋−𝐵ℎ(𝑋,𝑍).(3.27) Thus using Lemma 2.2(ii), we derive 𝒫_𝑍𝑋=(𝑇𝑋ln𝑓)𝑍−𝐵ℎ(𝑋,𝑍).(3.28) Taking inner product with 𝑊∈𝑇𝑀_⟂ and using (2.2), we obtain 𝑔𝒫_𝑍𝑋,𝑊=(𝜙𝑋ln𝑓)𝑔(𝑍,𝑊)+𝑔(ℎ(𝑋,𝑍),𝐹𝑊).(3.29) Then from (2.17)(i), we get 𝑔𝒫_𝑋𝑍,𝑊=−(𝜙𝑋ln𝑓)𝑔(𝑍,𝑊)−𝑔(ℎ(𝑋,𝑍),𝐹𝑊).(3.30) This is the second equality of (ii). Now, from (3.24) and (3.28), we have 𝒫_𝑋𝑍+𝒫_𝑍𝑋=−𝑇∇_𝑋𝑍−𝐴_𝐹𝑍𝑋+(𝑇𝑋ln𝑓)𝑍−2𝐵ℎ(𝑋,𝑍).(3.31) Left-hand side and the first term of right-hand side are zero on using (2.17)(i) and Lemma 2.2(i), respectively. Thus the above equation takes the form (𝑇𝑋ln𝑓)𝑍=𝐴_𝐹𝑍𝑋+2𝐵ℎ(𝑋,𝑍).(3.32) Taking the product with 𝑍 and on using (2.2) and (2.7), we get (𝜙𝑋ln𝑓)‖𝑍‖²=𝑔(ℎ(𝑋,𝑍),𝐹𝑍)−2𝑔(ℎ(𝑋,𝑍),𝐹𝑍)=−𝑔(ℎ(𝑋,𝑍),𝐹𝑍).(3.33) Interchanging 𝑋 by 𝜙𝑋 and using (2.1), we obtain {−𝑋+𝜂(𝑋)𝜉}ln𝑓‖𝑍‖²=−𝑔(ℎ(𝜙𝑋,𝑍),𝐹𝑍).(3.34) Thus by (3.4)(i), the above equation reduces to (𝑋ln𝑓)‖𝑍‖²=𝑔(ℎ(𝜙𝑋,𝑍),𝐹𝑍).(3.35) This proves the lemma completely.

Theorem 3.5. A proper semi-invariant submanifold 𝑀 of a nearly cosymplectic manifold 𝑀 is locally a semi-invariant warped product if and only if the shape operator of 𝑀 satisfies 𝐴_𝜙𝑍𝑋=−(𝜙𝑋𝜇)𝑍,𝑋∈𝒟⊕⟨𝜉⟩,𝑍∈𝒟^⟂(3.36) for some function 𝜇 on 𝑀 satisfying 𝑉(𝜇)=0 for each 𝑉∈𝒟^⟂.

Proof. If 𝑀=𝑀_𝑇×_𝑓𝑀_⟂ is a warped product semi-invariant submanifold, then by Lemma 3.4 (iii), we obtain (3.36). In this case 𝜇=ln𝑓.
Conversely, suppose 𝑀 is a semi-invariant submanifold of a nearly cosymplectic manifold 𝑀 satisfying (3.36). Then 𝑔(ℎ(𝑋,𝑌),𝜙𝑍)=𝑔𝐴_𝜙𝑍𝑋,𝑌=−(𝜙𝑋𝜇)𝑔(𝑌,𝑍)=0.(3.37) Now, from (2.5) and the property of covariant derivative of ∇, we have 𝑔(ℎ(𝑋,𝑌),𝜙𝑍)==−𝑔𝑔∇_𝑋𝑌,𝜙𝑍=−𝑔𝜙∇_𝑋𝑌,𝑍∇_𝑋𝜙𝑌,𝑍+𝑔∇_𝑋𝜙𝑌,𝑍.(3.38) Then from (2.5), (2.14), and (3.37), the above equation takes the form 𝑔∇_𝑋𝑇𝑌,𝑍=𝑔𝑃_𝑋𝑌,𝑍.(3.39) Using (2.10) and (2.15), we obtain 𝑔∇_𝑋𝑇𝑌,𝑍=𝑔∇_𝑋𝑇𝑌,𝑍−𝑔𝑇∇_𝑋𝑌,𝑍−𝑔(𝐵ℎ(𝑋,𝑌),𝑍).(3.40) Thus by (2.2), the above equation reduces to 𝑔𝑇∇_𝑋𝑌,𝑍=𝑔(ℎ(𝑋,𝑌),𝜙𝑍).(3.41) Hence using (2.7) and (3.36), we get 𝑔𝑇∇_𝑋𝑌,𝑍=𝑔𝐴_𝜙𝑍𝑋,𝑌=0,(3.42) which implies ∇_𝑋𝑌∈𝒟⊕⟨𝜉⟩, that is, 𝒟⊕⟨𝜉⟩ is integrable and its leaves are totally geodesic in 𝑀. Now, for any 𝑍,𝑊∈𝒟^⟂ and 𝑋∈𝒟⊕⟨𝜉⟩, we have 𝑔∇_𝑍𝑊,𝜙𝑋==𝑔𝑔∇_𝑍𝑊,𝜙𝑋=−𝑔𝜙∇_𝑍𝑊,𝑋∇_𝑍𝜙𝑊,𝑋−𝑔∇_𝑍𝜙𝑊,𝑋.(3.43) Then, using (2.6) and (2.14), we obtain 𝑔∇_𝑍𝑊,𝜙𝑋=𝑔𝒫_𝑍𝑊,𝑋+𝑔𝐴_𝜙𝑊𝑍,𝑋.(3.44) Thus from (2.7) and the property (𝑝₃), we arrive at 𝑔∇_𝑍𝑊,𝜙𝑋=−𝑔𝑊,𝒫_𝑍𝑋+𝑔(ℎ(𝑍,𝑋),𝜙𝑊).(3.45) Again using (2.7) and (2.17)(i), we get 𝑔∇_𝑍𝑊,𝜙𝑋=𝑔𝒫_𝑋𝑍,𝑊+𝑔𝐴_𝜙𝑊𝑋,𝑍.(3.46) On the other hand, from (2.10) and (2.15), we have 𝑃_𝑋𝑍=−𝑇∇_𝑋𝑍−𝐴_𝐹𝑍𝑋−𝐵ℎ(𝑋,𝑍).(3.47) Taking the product with 𝑊∈𝐷^⟂ and using (3.36), we obtain 𝑔𝒫_𝑋𝑍,𝑊=−𝑔𝑇∇_𝑋𝑍,𝑊+(𝜙𝑋𝜇)𝑔(𝑍,𝑊)+𝑔(ℎ(𝑋,𝑍),𝐹𝑊).(3.48) The first term of right-hand side of above equation is zero using the fact that 𝑇𝑊=0, for any 𝑊∈𝒟^⟂. Again using (2.7), we get 𝑔𝒫_𝑋𝑍,𝑊=(𝜙𝑋𝜇)𝑔(𝑍,𝑊)+𝑔𝐴_𝜙𝑊𝑋,𝑍.(3.49) Thus from (3.36), we derive 𝑔𝒫_𝑋𝑍,𝑊=(𝜙𝑋𝜇)𝑔(𝑍,𝑊)−(𝜙𝑋𝜇)𝑔(𝑍,𝑊)=0.(3.50) Then from (3.36), (3.46), and (3.50), we obtain 𝑔∇_𝑍𝑊,𝜙𝑋=−(𝜙𝑋𝜇)𝑔(𝑍,𝑊).(3.51) Let 𝑀_⟂ be a leaf of 𝒟^⟂, and let ℎ^⟂ be the second fundamental form of the immersion of 𝑀_⟂ into 𝑀. Then for any 𝑍,𝑊∈𝒟^⟂, we have 𝑔ℎ^⟂(𝑍,𝑊),𝜙𝑋=𝑔∇_𝑍𝑊,𝜙𝑋.(3.52) Hence, from (3.51) and (3.52), we conclude that 𝑔ℎ^⟂(𝑍,𝑊),𝜙𝑋=−(𝜙𝑋𝜇)𝑔(𝑍,𝑊).(3.53) This means that integral manifold 𝑀_⟂ of 𝒟^⟂ is totally umbilical in 𝑀. Since the anti-invariant distribution 𝒟^⟂ of a semi-invariant submanifold 𝑀 is always integrable (Theorem 3.3) and 𝑉(𝜇)=0 for each 𝑉∈𝒟^⟂, which implies that the integral manifold of 𝒟^⟂ is an extrinsic sphere in 𝑀; that is, it is totally umbilical and its mean curvature vector field is nonzero and parallel along 𝑀_⟂. Hence by virtue of results obtained in [11], 𝑀 is locally a warped product 𝑀_𝑇×_𝑓𝑀_⟂, where 𝑀_𝑇 and 𝑀_⟂ denote the integral manifolds of the distributions 𝒟⊕⟨𝜉⟩ and 𝒟^⟂, respectively and 𝑓 is the warping function. Thus the theorem is proved.