Abstract

We study warped product semi-invariant submanifolds of nearly cosymplectic manifolds. We prove that the warped product of the type 𝑀⟂×𝑓𝑀𝑇 is a usual Riemannian product of 𝑀⟂ and 𝑀𝑇, where 𝑀⟂ and 𝑀𝑇 are anti-invariant and invariant submanifolds of a nearly cosymplectic manifold 𝑀, respectively. Thus we consider the warped product of the type 𝑀𝑇×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⟂×𝑓𝑀𝑇 such that 𝑀⟂ is a totally real submanifold and 𝑀𝑇 is a holomorphic submanifold of a Kaehler manifold 𝑀. Therefore he considered warped product CR-submanifold in the form 𝑀=𝑀𝑇×𝑓𝑀⟂ which is called CR-warped product, where 𝑀𝑇 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⟂×𝑓𝑀𝑇 is simply Riemannian product of 𝑀⟂ and 𝑀𝑇 where 𝑀⟂ is an anti-invariant submanifold and 𝑀𝑇 is an invariant submanifold of a nearly cosymplectic manifold 𝑀. Thus we consider the warped product submanifold of the type 𝑀=𝑀𝑇×𝑓𝑀⟂ by reversing the two factors 𝑀⟂ and 𝑀𝑇 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 ğ¶âˆž 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𝑑𝑋,𝑓=𝑑𝑑𝑡𝜙𝑋−𝑓𝜉,𝜂(𝑋),𝑑𝑡(2.3) where 𝑓 is a ğ¶âˆž-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=−𝐴𝑁𝑋+∇⟂𝑋𝑁,(2.6) for each 𝑋,𝑌∈𝑇𝑀 and 𝑁∈𝑇⟂𝑀, where ℎ and 𝐴𝑁 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𝐴(ℎ(𝑋,𝑌),𝑁)=𝑔𝑁𝑋,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∇𝑋𝑌,(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 𝒟𝑥 of the tangent space 𝑇𝑥𝑀 of 𝑀. If the dimension of 𝒟𝑥 is the same for all values of 𝑥∈𝑀, then 𝒟𝑥 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(𝒟𝑥)⊆𝐷𝑥, (iii)𝜙(𝒟⟂𝑥)⊂𝑇⟂𝑥𝑀

for any 𝑥∈𝑀, where 𝑇⟂𝑥𝑀 denotes the orthogonal space of 𝑇𝑥𝑀 in 𝑇𝑥𝑀. A semi-invariant submanifold is called anti-invariant if 𝒟𝑥={0} and invariant if 𝒟⟂𝑥={0}, respectively, for any 𝑥∈𝑀. It is called the proper semi-invariant submanifold if neither 𝒟𝑥={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(𝑇𝑥𝑀) is a subspace of 𝑇⟂𝑥𝑀. Then for every 𝑥∈𝑀, there exists an invariant subspace 𝜈𝑥 of 𝑇𝑥𝑀 such that 𝑇⟂𝑥𝑇𝑀=𝐹𝑥𝑀⊕𝜈𝑥.(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=∇𝑀2𝑍𝑊−(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 𝒫𝑋𝑌 and 𝒬𝑋𝑌 the tangential and normal parts of (∇𝑋𝜙)𝑌, that is, ∇𝑋𝜙𝑌=𝒫𝑋𝑌+𝒬𝑋𝑌(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 𝒫𝑋∇𝑌=ğ‘‹ğ‘‡î€¸ğ‘Œâˆ’ğ´ğ¹ğ‘Œğ’¬ğ‘‹âˆ’ğµâ„Ž(𝑋,𝑌),(2.15)𝑋𝑌=∇𝑋𝐹𝑌+ℎ(𝑋,𝑇𝑌)âˆ’ğ¶â„Ž(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) 𝒫𝑋+𝑌𝑊=𝒫𝑋𝑊+𝒫𝑌𝑊, (ii) 𝒬𝑋+𝑌𝑊=𝒬𝑋𝑊+𝒬𝑌𝑊,(𝑝2)(i) 𝒫𝑋(𝑌+𝑊)=𝒫𝑋𝑌+𝒫𝑋𝑊, (ii) 𝒬𝑋(𝑌+𝑊)=𝒬𝑋𝑌+𝒬𝑋𝑊,(𝑝3)𝑔(𝒫𝑋𝑌,𝑊)=−𝑔(𝑌,𝒫𝑋𝑊)

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)𝒫𝑋𝑌+𝒫𝑌𝑋=0,(ii)𝒬𝑋𝑌+𝒬𝑌𝑋=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⟂×𝑓𝑀𝑇, (ii)𝑀=𝑀𝑇×𝑓𝑀⟂,

where 𝑀𝑇 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⟂×𝑓𝑀𝑇 of a nearly cosymplectic manifold 𝑀 is a usual Riemannian product of 𝑀⟂ and 𝑀𝑇, where 𝑀⟂ and 𝑀𝑇 are anti-invariant and invariant submanifolds of 𝑀, respectively.

Proof. When 𝜉∈𝑇𝑀𝑇, then by Theorem 3.1, 𝑀 is a Riemannian product. Thus, we consider 𝜉∈𝑇𝑀⟂. For any 𝑋∈𝑇𝑀𝑇 and 𝑍∈𝑇𝑀⟂, we have 𝑔(ℎ(𝑋,𝜙𝑋),𝐹𝑍)=𝑔(ℎ(𝑋,𝜙𝑋),𝜙𝑍)=𝑔∇𝑋𝜙𝜙𝑋,𝜙𝑍=𝑔∇𝑋𝑋,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∇𝑋𝑋,𝑍−𝜂(𝑍)𝑔∇𝑋𝑋,𝜉=−𝑔𝑋,∇𝑋𝑍+𝜂(𝑍)𝑔𝑋,∇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⟂×𝑓𝑀𝑇 is a usual Riemannian product of an anti-invariant submanifold 𝑀⟂ and an invariant submanifold 𝑀𝑇 of a nearly cosymplectic manifold 𝑀. Since both 𝑀⟂ and 𝑀𝑇 are totally geodesic in 𝑀, then 𝑀 is CR-product. Now, we study the warped product semi-invariant submanifold 𝑀=𝑀𝑇×𝑓𝑀⟂ of a nearly cosymplectic manifold 𝑀.

Theorem 3.3. Let 𝑀=𝑀𝑇×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=ğ’¬ğ‘‹ğ‘Œâˆ’â„Ž(𝑋,𝑇𝑌)+ğ¶â„Ž(𝑋,𝑌)−𝒬𝑌𝑋+ℎ(𝑌,𝑇𝑋)âˆ’ğ¶â„Ž(𝑋,𝑌).(3.11) Thus from (2.17)(ii), we get 𝐹[]𝑋,𝑌=2𝒬𝑋𝑌+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)+∇𝑋𝑇𝑌=∇𝑋𝜙𝑌+𝜙∇𝑋𝑌.(3.14) Then by (2.5) and (2.14), we get ℎ(𝑋,𝑇𝑌)+∇𝑋𝑇𝑌=𝑃𝑋𝑌+𝒬𝑋∇𝑌+𝜙𝑋𝑌+ℎ(𝑋,𝑌).(3.15) Since 𝑀𝑇 is totally geodesic in 𝑀 (see Lemma 2.2(i)), then using (2.8) and (2.9), we obtain ℎ(𝑋,𝑇𝑌)+∇𝑋𝑇𝑌=𝒫𝑋𝑌+𝒬𝑋𝑌+𝑇∇𝑋𝑌+ğµâ„Ž(𝑋,𝑌)+ğ¶â„Ž(𝑋,𝑌).(3.16) Equating the normal components of (3.16), we get ℎ(𝑋,𝑇𝑌)=𝒬𝑋𝑌+ğ¶â„Ž(𝑋,𝑌).(3.17) Similarly, we obtain ℎ(𝑌,𝑇𝑋)=𝒬𝑌𝑋+ğ¶â„Ž(𝑋,𝑌).(3.18) Then from (3.17) and (3.18), we arrive at ℎ(𝑌,𝑇𝑋)−ℎ(𝑋,𝑇𝑌)=𝒬𝑌𝑋−𝒬𝑋𝑌.(3.19) Hence, using (2.17)(ii), we get ℎ(𝑌,𝑇𝑋)−ℎ(𝑋,𝑇𝑌)=−2𝒬𝑋𝑌.(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)=𝑔∇𝑍𝑊−∇𝑊.∇𝑍,𝑋=−𝑔𝑍∇𝑋,𝑊+𝑔𝑊.𝑋,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=𝑀𝑇×𝑓𝑀⟂ be a warped product submanifold of a nearly cosymplectic manifold 𝑀. If 𝑋,𝑌∈𝑇𝑀𝑇 and 𝑍,𝑊∈𝑇𝑀⟂, then (i)𝑔(𝒫𝑋𝑌,𝑍)=𝑔(ℎ(𝑋,𝑌),𝐹𝑍)=0, (ii)𝑔(𝒫𝑋𝑍,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=𝑀𝑇×𝑓𝑀⟂, we have that 𝑀𝑇 is totally geodesic in 𝑀; then by (2.10), (∇𝑋𝑇)𝑌∈𝑇𝑀𝑇, for any 𝑋,𝑌∈𝑇𝑀𝑇, and therefore from (2.15), we get 𝑔𝒫𝑋𝑌,𝑍=−𝑔(ğµâ„Ž(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 𝒫𝑋𝑍=âˆ’ğ‘‡âˆ‡ğ‘‹ğ‘âˆ’ğ´ğ¹ğ‘ğ‘‹âˆ’ğµâ„Ž(𝑋,𝑍)(3.24) for any 𝑋∈𝑇𝑀𝑇 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𝒫𝑋𝐴𝑍,𝑊=−𝑔𝐹𝑍𝑋,𝑊−𝑔(ğµâ„Ž(𝑋,𝑍),𝑊),(3.25) Then by (2.2) and (2.7), we get 𝑔𝒫𝑋𝑍,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 𝒫𝑍𝑋=âˆ‡ğ‘ğ‘‡ğ‘‹âˆ’ğ‘‡âˆ‡ğ‘ğ‘‹âˆ’ğµâ„Ž(𝑋,𝑍).(3.27) Thus using Lemma 2.2(ii), we derive 𝒫𝑍𝑋=(𝑇𝑋ln𝑓)ğ‘âˆ’ğµâ„Ž(𝑋,𝑍).(3.28) Taking inner product with 𝑊∈𝑇𝑀⟂ and using (2.2), we obtain 𝑔𝒫𝑍=𝑋,𝑊(𝜙𝑋ln𝑓)𝑔(𝑍,𝑊)+𝑔(ℎ(𝑋,𝑍),𝐹𝑊).(3.29) Then from (2.17)(i), we get 𝑔𝒫𝑋𝑍,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 𝒫𝑋𝑍+𝒫𝑍𝑋=−𝑇∇𝑋𝑍−𝐴𝐹𝑍𝑋+(𝑇𝑋ln𝑓)𝑍−2ğµâ„Ž(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=𝐴𝐹𝑍𝑋+2ğµâ„Ž(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 𝐴𝜙𝑍𝑋=−(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=𝑀𝑇×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𝐴(ℎ(𝑋,𝑌),𝜙𝑍)=𝑔𝜙𝑍𝑋,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)=𝑔∇𝑋𝜙𝑌,𝜙𝑍=−𝑔∇𝑋𝑌,𝑍=−𝑔∇𝑋𝜙𝑌,𝑍+𝑔∇𝑋𝜙.𝑌,𝑍(3.38) Then from (2.5), (2.14), and (3.37), the above equation takes the form 𝑔∇𝑋𝑃𝑇𝑌,𝑍=𝑔𝑋𝑌,𝑍.(3.39) Using (2.10) and (2.15), we obtain 𝑔∇𝑋∇𝑇𝑌,𝑍=𝑔𝑋𝑇𝑌,𝑍−𝑔𝑇∇𝑋𝑌,𝑍−𝑔(ğµâ„Ž(𝑋,𝑌),𝑍).(3.40) Thus by (2.2), the above equation reduces to 𝑔𝑇∇𝑋𝑌,𝑍=𝑔(ℎ(𝑋,𝑌),𝜙𝑍).(3.41) Hence using (2.7) and (3.36), we get 𝑔𝑇∇𝑋𝐴𝑌,𝑍=𝑔𝜙𝑍𝑋,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∇𝑍𝑊,𝜙𝑋=𝑔∇𝑍𝜙𝑊,𝜙𝑋=−𝑔∇𝑍𝑊,𝑋=𝑔∇𝑍𝜙𝑊,𝑋−𝑔∇𝑍.𝜙𝑊,𝑋(3.43) Then, using (2.6) and (2.14), we obtain 𝑔∇𝑍𝒫𝑊,𝜙𝑋=𝑔𝑍𝐴𝑊,𝑋+𝑔𝜙𝑊.𝑍,𝑋(3.44) Thus from (2.7) and the property (𝑝3), we arrive at 𝑔∇𝑍𝑊,𝜙𝑋=−𝑔𝑊,𝒫𝑍𝑋+𝑔(ℎ(𝑍,𝑋),𝜙𝑊).(3.45) Again using (2.7) and (2.17)(i), we get 𝑔∇𝑍𝒫𝑊,𝜙𝑋=𝑔𝑋𝐴𝑍,𝑊+𝑔𝜙𝑊.𝑋,𝑍(3.46) On the other hand, from (2.10) and (2.15), we have 𝑃𝑋𝑍=âˆ’ğ‘‡âˆ‡ğ‘‹ğ‘âˆ’ğ´ğ¹ğ‘ğ‘‹âˆ’ğµâ„Ž(𝑋,𝑍).(3.47) Taking the product with 𝑊∈𝐷⟂ and using (3.36), we obtain 𝑔𝒫𝑋𝑍,𝑊=−𝑔𝑇∇𝑋+𝑍,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𝒫𝑋=𝐴𝑍,𝑊(𝜙𝑋𝜇)𝑔(𝑍,𝑊)+𝑔𝜙𝑊.𝑋,𝑍(3.49) Thus from (3.36), we derive 𝑔𝒫𝑋=𝑍,𝑊(𝜙𝑋𝜇)𝑔(𝑍,𝑊)−(𝜙𝑋𝜇)𝑔(𝑍,𝑊)=0.(3.50) Then from (3.36), (3.46), and (3.50), we obtain 𝑔∇𝑍𝑊,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 ğ‘”î€·â„ŽâŸ‚(∇𝑍,𝑊),𝜙𝑋=𝑔𝑍.𝑊,𝜙𝑋(3.52) Hence, from (3.51) and (3.52), we conclude that ğ‘”î€·â„ŽâŸ‚(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 𝑀𝑇×𝑓𝑀⟂, where 𝑀𝑇 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.