Warped Product Pseudo-Slant Submanifolds of a Nearly Cosymplectic Manifold

Abstract

We study warped product pseudo-slant submanifolds of a nearly cosymplectic manifold. We obtain some characterization results on the existence or nonexistence of warped product pseudo-slant submanifolds of a nearly cosymplectic manifold in terms of the canonical structures 𝑃 and 𝐹.

1. Introduction

To study the manifolds with negative curvature, Bishop and O'Neill [1] introduced the notion of warped product manifolds by homothetically warping the product metric of a product manifold 𝑁₁×𝑁₂ onto the fibers 𝑝×𝑁₂ for each 𝑝∈𝑁₁. Later on, the geometrical aspect of these manifolds has been studied by many researchers (cf. [2–4]). Pseudo-slant submanifolds were introduced by Carriazo [5] as a special case of bislant submanifolds.

Almost contact manifolds with Killing structure tensors were defined in [6] as nearly cosymplectic manifolds, and it was shown that the normal nearly cosymplectic manifolds are cosymplectic (see also [7]). Later on, Blair and Showers [8] studied nearly cosymplectic structure (𝜙,𝜉,𝜂,𝑔) on a Riemannian manifold 𝑀 with 𝜂 closed from the topological viewpoint.

Recently, Sahin [9] studied the warped product hemislant (pseudo-slant) submanifolds of Kaehler manifolds. He proved that the warped product submanifolds of the type 𝑀=𝑁_⟂×_𝑓𝑁_𝜃 of a Kaehler manifold 𝑀 do not exist and obtained some characterization results on the existence of warped product submanifold 𝑀=𝑁_𝜃×_𝑓𝑁_⟂, where 𝑁_⟂ and 𝑁_𝜃 are totally real and proper slant submanifolds of a Kaehler manifold 𝑀, respectively. After that, we have extended this study to the more general setting of nearly Kaehler manifolds [4]. The warped product semi-invariant submanifolds of a nearly cosymplectic manifold had been studied in [10].

In this paper, we study warped product pseudo-slant submanifolds of a nearly cosymplectic manifold. We obtain some characterization results of warped product submanifolds of the types 𝑁_⟂×_𝑓𝑁_𝜃 and 𝑁_𝜃×_𝑓𝑁_⟂ in terms of the canonical structures 𝑃 and 𝐹, where 𝑁_⟂ and 𝑁_𝜃 are anti-invariant and proper slant submanifolds of a nearly cosymplectic manifold 𝑀, respectively.

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 [8] 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),𝑔𝜙𝑋,𝜙𝑌=𝑔(𝑋,𝑌)−𝜂(𝑋)𝜂(𝑌), (2.2) where 𝑋 and 𝑌 are vector fields on 𝑀 [8].

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 𝐽𝑑𝑋,𝑓=(𝑋)𝑑𝑑𝑡𝜙𝑋−𝑓𝜉,𝜂𝑑𝑡, (2.3) where 𝑓 is a 𝐶^∞-function on 𝑀×ℝ has no torsion, that is, 𝐽 is integrable, 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 [8]. The structure is said to be nearly cosymplectic if 𝜙 is Killing, that is, if ∇_𝑋𝜙𝑌+∇_𝑌𝜙𝑋=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 [8]). On a nearly cosymplectic manifold the vector field 𝜉 is Killing.

From the above proposition, one has ∇_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),∇_𝑋𝑁=−𝐴_𝑁𝑋+∇^⟂_𝑋𝑁, (2.5)(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)=𝑔𝐴_𝑁𝑋,𝑌, (2.7) where 𝑔 denotes the Riemannian metric on 𝑀 as well as induced on 𝑀.

For any 𝑋∈𝑇𝑀, one writes 𝜙𝑋=𝑃𝑋+𝐹𝑋, (2.8) where 𝑃𝑋 is the tangential component and 𝐹𝑋 is the normal component of 𝜙𝑋.

Similarly for any 𝑁∈𝑇^⟂𝑀, one writes 𝜙𝑁=𝐵𝑁+𝐶𝑁, (2.9) where 𝐵𝑁 is the tangential component and 𝐶𝑁 is the normal component of 𝜙𝑁.

Now, denote by 𝒫_𝑋𝑌 and 𝒬_𝑋𝑌 the tangential and normal parts of (∇_𝑋𝜙)𝑌, that is, ∇_𝑋𝜙𝑌=𝒫_𝑋𝑌+𝒬_𝑋𝑌 (2.10) for all 𝑋,𝑌∈𝑇𝑀. Making use of (2.8), (2.10), and the Gauss and Weingarten formulae, the following equations may easily be obtained: 𝒫_𝑋𝑌=∇_𝑋𝑃𝑌−𝐴_𝐹𝑌(),𝒬𝑋−𝐵ℎ𝑋,𝑌_𝑋𝑌=∇_𝑋𝐹𝑌+ℎ(𝑋,𝑃𝑌)−𝐶ℎ(𝑋,𝑌). (2.11) Similarly, for any 𝑁∈𝑇^⟂𝑀, denoting tangential and normal parts of (∇_𝑋𝜙)𝑁 by 𝒫_𝑋𝑁 and 𝒬_𝑋𝑁, respectively, one obtains 𝒫_𝑋𝑁=∇_𝑋𝐵𝑁+𝑃𝐴_𝑁𝑋−𝐴_𝐶𝑁𝒬𝑋,_𝑋𝑁=∇_𝑋𝐶𝑁+ℎ(𝐵𝑁,X)+𝐹𝐴_𝑁𝑋, (2.12) where the covariant derivatives of 𝑃,𝐹,𝐵, and 𝐶 are defined by ∇_𝑋𝑃𝑌=∇_𝑋𝑃𝑌−𝑃∇_𝑋𝑌,∇_𝑋𝐹𝑌=∇^⟂_𝑋𝐹𝑌−𝐹∇_𝑋𝑌,∇_𝑋𝐵𝑁=∇_𝑋𝐵𝑁−𝐵∇^⟂_𝑋𝑁,∇_𝑋𝐶𝑁=∇^⟂_𝑋𝐶𝑁−𝐶∇^⟂_𝑋𝑁, (2.13)(2.14)(2.15)(2.16) for all 𝑋,𝑌∈𝑇𝑀 and 𝑁∈𝑇^⟂𝑀.

It is straightforward to verify the following properties of 𝒫 and 𝒬, which one enlists 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)  (i)  𝑔(𝒫_𝑋𝑌,𝑊)=−𝑔(𝑌,𝒫_𝑋𝑊),  (ii)  𝑔(𝒬_𝑋𝑌,𝑁)=−𝑔(𝑌,𝒫_𝑋𝑁),(p₄) 𝒫_𝑋𝜙𝑌+𝒬_𝑋𝜙𝑌=−𝜙(𝒫_𝑋𝑌+𝒬_𝑋𝑌),

for all 𝑋,𝑌,𝑊∈𝑇𝑀 and 𝑁∈𝑇^⟂𝑀.

On a submanifold 𝑀 of a nearly cosymplectic manifold, by (2.4) and (2.10), one has (𝑎)𝒫_𝑋𝑌+𝒫_𝑌𝑋=0,(𝑏)𝒬_𝑋𝑌+𝒬_𝑌𝑋=0, (2.17)

for any 𝑋,𝑌∈𝑇𝑀.

The submanifold 𝑀 is said to be invariant if 𝐹 is identically zero, that is, 𝜙𝑋∈𝑇𝑀 for any 𝑋∈𝑇𝑀. On the other hand, 𝑀 is said to be anti-invariant if 𝑃 is identically zero, that is, 𝜙𝑋∈𝑇^⟂𝑀, for any 𝑋∈𝑇𝑀.

One will always consider 𝜉 to be tangent to the submanifold 𝑀. There is another class of submanifolds that is called the slant submanifold. For each nonzero vector 𝑋 tangent to 𝑀 at any 𝑥∈𝑀, such that 𝑋 is not proportional to 𝜉_𝑥, one denotes by 0≤𝜃(𝑋)≤𝜋/2, the angle between 𝜙𝑋 and 𝑇_𝑥𝑀 is called the slant angle. If the slant angle 𝜃(𝑋) is constant for all 𝑋∈𝑇_𝑥𝑀−⟨𝜉_𝑥⟩ and 𝑥∈𝑀, then M is said to be a slant submanifold [11]. Obviously, if 𝜃=0, then 𝑀 is an invariant submanifold and if 𝜃 = 𝜋/2, then 𝑀 is an anti-invariant submanifold. A slant submanifold is said to be proper slant if it is neither invariant nor anti-invariant.

One recalls the following result for a slant submanifold.

Theorem 2.2 (see [11]). Let M be a submanifold of an almost contact metric manifold 𝑀, such that 𝜉∈𝑇𝑀. Then M is slant if and only if there exists a constant 𝜆∈[0,1] such that 𝑃²=𝜆−𝐼+𝜂⊗𝜉. (2.18)
Furthermore, if 𝜃 is slant angle, then 𝜆=cos²𝜃.

The following relations are straightforward consequence of (2.18): 𝑔(𝑃𝑋,𝑃𝑌)=cos²𝜃𝑔()(𝑌)𝜂(𝑋,𝑔()𝑋,𝑌−𝜂)𝐹𝑋,𝐹𝑌=sin²𝜃𝑔(𝑋,𝑌)−𝜂(𝑌)𝜂(𝑋), (2.19)(2.20) for all 𝑋,𝑌∈𝑇𝑀.

A submanifold 𝑀 of an almost contact manifold 𝑀 is said to be a pseudo-slant submanifold if there exist two orthogonal complementary distributions 𝐷₁ and 𝐷₂ satisfying: (i)𝑇𝑀=𝐷₁⊕𝐷₂⊕⟨𝜉⟩, (ii)𝐷₁ is a slant distribution with slant angle 𝜃≠𝜋/2,(iii)𝐷₂ is anti-invariant that is, 𝜙𝐷₂⊆𝑇^⟂𝑀.

A pseudo-slant submanifold 𝑀 of an almost contact manifold 𝑀 is mixed geodesic if ℎ(𝑋,𝑍)=0, (2.21) for any 𝑋∈𝐷₁ and 𝑍∈𝐷₂.

If 𝜇 is the invariant subspace of the normal bundle 𝑇^⟂𝑀, then in the case of pseudo-slant submanifold, the normal bundle 𝑇^⟂𝑀 can be decomposed as follows: 𝑇^⟂𝑀=𝐹𝐷₁⊕𝐹𝐷₂⊕𝜇. (2.22)

3. Warped Product Pseudo-Slant Submanifolds

Bishop and O'Neill [1] introduced the notion of warped product manifolds. These manifolds are the natural generalizations of Riemannian product manifolds. They defined these manifolds as follows Let (𝑁₁,𝑔₁) and (𝑁₂,𝑔₂) be two Riemannian manifolds and 𝑓, a positive differentiable function on 𝑁₁. The warped product of 𝑁₁ and 𝑁₂ is the Riemannian manifold 𝑁₁×_𝑓𝑁₂=(𝑁₁×𝑁₂,𝑔), where 𝑔=𝑔₁+𝑓²𝑔₂. (3.1) A warped product manifold 𝑁₁×_𝑓𝑁₂ is said to be trivial if the warping function 𝑓 is constant. We recall the following general formula on a warped product manifold [1]: ∇_𝑋𝑍=∇_𝑍𝑋=𝑋ln𝑓𝑍, (3.2) where 𝑋 is tangential to 𝑁₁ and 𝑍 is tangential to 𝑁₂.

Let 𝑀=𝑁₁×_𝑓𝑁₂ be a warped product manifold. This means that 𝑁₁ is totally geodesic and 𝑁₂ is a totally umbilical submanifold of 𝑀, respectively [1].

Throughout this section, we consider the warped product pseudo-slant submanifolds which are either in the form 𝑁_⟂×_𝑓𝑁_𝜃 or 𝑁_𝜃×_𝑓𝑁_⟂ in a nearly cosymplectic manifold 𝑀, where 𝑁_𝜃 and 𝑁_⟂ are proper slant and anti-invariant submanifolds of a nearly cosymplectic manifold 𝑀, respectively. On a warped product submanifold 𝑀=𝑁₁×_𝑓𝑁₂ of a nearly cosymplectic manifold 𝑀, we have the following result.

Theorem 3.1 (see [10]). A warped product submanifold 𝑀=𝑁₁×_𝑓𝑁₂ of a nearly cosymplectic manifold 𝑀 is an usual Riemannian product if the structure vector field 𝜉 is tangential to 𝑀₂, where 𝑀₁ and 𝑀₂ are the Riemannian submanifolds of 𝑀.

Now, one considers the warped product pseudo-slant submanifolds in the form 𝑀=𝑁_⟂×_𝑓𝑁_𝜃 of a nearly cosymplectic manifold 𝑀. If one considers the structure vector field 𝜉∈𝑇𝑁_𝜃 then by Theorem 3.1, the warping function 𝑓 is constant and hence one will considers 𝜉∈𝑇𝑁_⟂.

Proposition 3.2. Let 𝑀=𝑁_⟂×_𝑓𝑁_𝜃 be a warped product pseudo-slant submanifold of a nearly cosymplectic manifold 𝑀. Then, 𝑔∇^⟂_𝑃𝑋𝐹𝑃𝑋−∇^⟂_𝑋𝐹𝑋,𝐹𝑍=𝑍ln𝑓sin²𝜃‖𝑋‖²+1+cos²𝜃𝑔(ℎ(𝑋,𝑃𝑋),𝐹𝑍), (3.3) for any 𝑋∈𝑇𝑁_𝜃 and 𝑍∈𝑇𝑁_⟂, where 𝑁_𝜃 and 𝑁_⟂ are proper slant and anti-invariant submanifolds of 𝑀, respectively.

Proof. For any 𝑋∈𝑇𝑁_𝜃 and 𝑍∈𝑇𝑁_⟂, by (2.8), we have 𝑔∇_𝑋𝜙𝑋,𝐹𝑍=𝑔∇_𝑋𝑃𝑋,𝐹𝑍+𝑔∇_𝑋𝐹𝑋,𝐹𝑍. (3.4) Using (2.5), (2.6), and the covariant derivative property of 𝜙, we obtain 𝑔∇_𝑋𝜙𝑋,𝐹𝑍+𝑔𝜙∇_𝑋𝑋,𝜙𝑍=𝑔(ℎ(𝑋,𝑃𝑋),𝐹𝑍)+𝑔∇^⟂_𝑋𝐹𝑋,𝐹𝑍. (3.5) Then from (2.2), (2.4), and the fact that 𝜉 is a Killing vector field on 𝑀, thus we obtain 𝑔∇_𝑋𝑋,𝑍=𝑔(ℎ(𝑋,𝑃𝑋),𝐹𝑍)+𝑔∇^⟂_𝑋𝐹𝑋,𝐹𝑍. (3.6) Using the property of ∇, we get −𝑔𝑋,∇_𝑋𝑍=𝑔(ℎ(𝑋,𝑃𝑋),𝐹𝑍)+𝑔∇^⟂_𝑋𝐹𝑋,𝐹𝑍. (3.7) Then by (2.5) and (3.2), we derive −𝑍ln𝑓‖𝑋‖²=𝑔(ℎ(𝑃𝑋,𝑋),𝐹𝑍)+𝑔∇^⟂_𝑋𝐹𝑋,𝐹𝑍. (3.8) Interchanging 𝑋 by 𝑃𝑋 in (3.8) and using (2.18), (2.19), and the fact that 𝜉∈𝑇𝑁_⟂, we obtain −𝑍ln𝑓cos²𝜃‖𝑋‖²=−cos²𝜃𝑔(ℎ(𝑋,𝑃𝑋),𝐹𝑍)+𝑔∇^⟂_𝑃𝑋𝐹𝑃𝑋,𝐹𝑍. (3.9) Thus, the result follows from (3.8) and (3.9).

Proposition 3.3. Let 𝑀=𝑁_⟂×_𝑓𝑁_𝜃 be a warped product pseudo-slant submanifold of a nearly cosymplectic manifold 𝑀. Then, 𝑔∇_𝑋𝐹𝑋,𝐹𝑍=sec²𝜃𝑔∇_𝑃𝑋𝐹𝑃𝑋,𝐹𝑍 (3.10) for any 𝑋∈𝑇𝑁_𝜃 and 𝑍∈𝑇𝑁_⟂, where 𝑁_𝜃 and 𝑁_⟂ are proper slant and anti-invariant submanifolds of 𝑀, respectively.

Proof. For any 𝑋∈𝑇𝑁_𝜃 and 𝑍∈𝑇𝑁_⟂ by (2.14), we have 𝑔∇^⟂_𝑋𝐹𝑋,𝐹𝑍=𝑔∇_𝑋𝐹𝑋,𝐹𝑍+𝑔𝐹∇_𝑋𝑋,𝐹𝑍. (3.11) Using (2.20), (2.5), and the fact that 𝜉 is killing vector field, we obtain 𝑔∇^⟂_𝑋𝐹𝑋,𝐹𝑍=𝑔∇_𝑋𝐹𝑋,𝐹𝑍−sin²𝜃𝑔𝑋,∇_𝑋𝑍. (3.12) Then from (3.2), we derive 𝑔∇^⟂_𝑋𝐹𝑋,𝐹𝑍=𝑔∇_𝑋𝐹𝑋,𝐹𝑍−𝑍ln𝑓sin²𝜃‖𝑋‖². (3.13) Now, from (3.8) and (3.13), we obtain 𝑔∇_𝑋𝐹𝑋,𝐹𝑍=−𝑍ln𝑓cos²𝜃‖𝑋‖²−𝑔(ℎ(𝑋,𝑃𝑋),𝐹𝑍). (3.14) Interchanging 𝑋 by 𝑃𝑋 in (3.14) and then using (2.18), (2.19), and the fact that 𝜉∈𝑇𝑁_⟂, we get 𝑔∇_𝑃𝑋𝐹𝑃𝑋,𝐹𝑍=−𝑍ln𝑓cos⁴𝜃‖𝑋‖²−cos²𝜃𝑔(ℎ(𝑋,𝑃𝑋),𝐹𝑍). (3.15) From (3.14) and (3.15), we arrive at 𝑔∇_𝑋𝐹𝑋,𝐹𝑍=sec²𝜃𝑔∇_𝑃𝑋𝐹𝑃𝑋,𝐹𝑍. (3.16) Hence, the result is proved.

Lemma 3.4. Let 𝑀=𝑁_⟂×_𝑓𝑁_𝜃 be a warped product pseudo-slant submanifold of a nearly cosymplectic manifold 𝑀. Then, 𝑔𝒫_𝑋𝑃𝑋,𝑍=𝑔(ℎ(𝑋,𝑍),𝐹𝑃𝑋)−𝑔(ℎ(𝑃𝑋,𝑍),𝐹𝑋) (3.17) for any 𝑋∈𝑇𝑁_𝜃 and 𝑍∈𝑇𝑁_⟂, where 𝑁_𝜃 and 𝑁_⟂ are proper slant and anti-invariant submanifolds of 𝑀, respectively.

Proof. For any 𝑋∈𝑇𝑁_𝜃 and 𝑍∈𝑇𝑁_⟂ by (2.5), we have 𝑔(ℎ(𝑃𝑋,𝑍),𝐹𝑋)=𝑔∇_𝑍𝑃𝑋,𝐹𝑋=−𝑔𝑃𝑋,∇_𝑍𝐹𝑋. (3.18) Then from (2.8), we derive 𝑔(ℎ(𝑃𝑋,𝑍),𝐹𝑋)=𝑔𝑃𝑋,∇_𝑍𝑃𝑋−𝑔𝑃𝑋,∇_𝑍𝜙𝑋. (3.19) From the covariant derivative property of 𝜙 and (2.5), we obtain 𝑔(ℎ(𝑃𝑋,𝑍),𝐹𝑋)=𝑔𝑃𝑋,∇_𝑍𝑃𝑋−𝑔𝑃𝑋,∇_𝑍𝜙𝑋−𝑔𝑃𝑋,𝜙∇_𝑍𝑋. (3.20) By (2.2), (2.10), and (3.2), we derive 𝑔(ℎ(𝑃𝑋,𝑍),𝐹𝑋)=𝑍ln𝑓𝑔(𝑃𝑋,𝑃𝑋)−𝑔𝑃𝑋,𝒫_𝑍𝑋+𝑔𝜙𝑃𝑋,∇_𝑍𝑋. (3.21) Using (2.5), (2.8), (2.17)(a), (2.19) and the fact that 𝜉∈𝑇𝑁_⟂, we get 𝑔(ℎ(𝑃𝑋,𝑍),𝐹𝑋)=𝑍ln𝑓cos²𝜃‖𝑋‖²+𝑔𝑃𝑋,𝒫_𝑋𝑍+𝑔𝑃²𝑋,∇_𝑍𝑋+𝑔(ℎ(𝑋,𝑍),𝐹𝑃𝑋). (3.22) Thus, by property (𝑝₃)(i), (2.18), and (3.2) and the fact that 𝜉∈𝑇𝑁_⟂, we obtain 𝑔(ℎ(𝑃𝑋,𝑍),𝐹𝑋)=𝑍ln𝑓cos²𝜃‖𝑋‖²−𝑔𝒫_𝑋−𝑃𝑋,𝑍𝑍ln𝑓cos²𝜃‖𝑋‖²+𝑔(ℎ(𝑋,𝑍),𝐹𝑃𝑋). (3.23) Hence, the above equation takes the form 𝑔𝒫_𝑋𝑃𝑋,𝑍=𝑔(ℎ(𝑋,𝑍),𝐹𝑃𝑋)−𝑔(ℎ(𝑃𝑋,𝑍),𝐹𝑋), (3.24) which proves our assertion.

Theorem 3.5. Let 𝑀=𝑁_⟂×_𝑓𝑁_𝜃 be a warped product submanifold of a nearly cosymplectic manifold 𝑀. Then 𝑀 is Riemannian product of 𝑁_⟂ and 𝑁_𝜃 if and only if 𝒫_𝑋𝑇𝑋∈𝑇𝑁_𝜃, for any 𝑋∈𝑇𝑁_𝜃, where 𝑁_𝜃 and 𝑁_⟂ are proper slant and anti-invariant submanifolds of 𝑀, respectively.

Proof. If the structure vector field 𝜉∈𝑇𝑁_𝜃, then, by Theorem 3.1, 𝑀 is Riemannian product of 𝑁_⟂ and 𝑁_𝜃. Now, we consider 𝜉∈𝑇𝑁_⟂, then for any 𝑋∈𝑇𝑁_𝜃 and 𝑍∈𝑇𝑁_⟂ from (2.5), we have 𝑔(ℎ(𝑋,𝑃𝑋),𝐹𝑍)=𝑔∇_𝑃𝑋𝑋,𝜙𝑍. (3.25) Then by (2.2), we get 𝑔(ℎ(𝑋,𝑃𝑋),𝐹𝑍)=−𝑔𝜙∇_𝑃𝑋𝑋,𝑍. (3.26) Using the covariant derivative formula of 𝜙, we derive 𝑔(ℎ(𝑋,𝑃𝑋),𝐹𝑍)=𝑔∇_𝑃𝑋𝜙𝑋,𝑍−𝑔∇_𝑃𝑋𝜙𝑋,𝑍. (3.27) Then from (2.10) and the property of ∇, we obtain 𝑔(ℎ(𝑋,𝑃𝑋),𝐹𝑍)=𝑔𝒫_𝑃𝑋𝑋,𝑍+𝑔𝜙𝑋,∇_𝑃𝑋𝑍. (3.28) Thus by (2.5), (2.8), and (2.17)(a), we arrive at 𝑔(ℎ(𝑋,𝑃𝑋),𝐹𝑍)=−𝑔𝒫_𝑋𝑃𝑋,𝑍+𝑔𝑃𝑋,∇_𝑃𝑋𝑍+𝑔(ℎ(𝑃𝑋,𝑍),𝐹𝑋). (3.29) Using (3.2) and then (2.19) and the fact that 𝜉∈𝑇𝑁_⟂, we get 𝑔(ℎ(𝑋,𝑃𝑋),𝐹𝑍)=−𝑔𝒫_𝑋𝑃𝑋,𝑍+𝑍ln𝑓cos²𝜃‖𝑋‖²+𝑔(ℎ(𝑃𝑋,𝑍),𝐹𝑋). (3.30) By property (𝑝₃)(𝑖), we derive 𝑔(ℎ(𝑋,𝑃𝑋),𝐹𝑍)=𝑔𝑃𝑋,𝒫_𝑋𝑍+𝑍ln𝑓cos²𝜃‖𝑋‖²+𝑔(ℎ(𝑃𝑋,𝑍),𝐹𝑋). (3.31) Interchanging 𝑋 by 𝑃𝑋 in (3.30) and then using (2.18), (2.19), and the fact that 𝜉∈𝑇𝑁_⟂, we obtain −cos²𝜃𝑔(ℎ(𝑋,𝑃𝑋),𝐹𝑍)=−cos²𝜃𝑔𝑋,𝒫_𝑃𝑋𝑍+𝑍ln𝑓cos⁴𝜃‖𝑋‖²−cos²𝜃𝑔(ℎ(𝑋,𝑍),𝐹𝑃𝑋). (3.32) Using the property (𝑝₃)(i) and then (2.17)(a), we arrive at −𝑔(ℎ(𝑋,𝑃𝑋),𝐹𝑍)=−𝑔𝒫_𝑋𝑃𝑋,𝑍+𝑍ln𝑓cos²𝜃‖𝑋‖²−𝑔(ℎ(𝑋,𝑍),𝐹𝑃𝑋). (3.33)
Then from (3.30) and (3.33), we obtain 2𝑍ln𝑓cos²𝜃‖𝑋‖²=2𝑔𝒫_𝑋(ℎ())(ℎ()).𝑃𝑋,𝑍+𝑔𝑋,𝑍,𝐹𝑃𝑋−𝑔𝑃𝑋,𝑍,𝐹𝑋 (3.34)
Thus, by Lemma 3.4, we conclude that 𝑍ln𝑓cos²𝜃‖𝑋‖²=32𝑔𝒫_𝑋𝑃𝑋,𝑍. (3.35) Since 𝑁_𝜃 is proper slant, thus we get (𝑍ln𝑓)=0, if and only if 𝒫_𝑋𝑃𝑋 lies in 𝑇𝑁_𝜃 for all 𝑋∈𝑇𝑁_𝜃 and 𝑍∈𝑇𝑁_⟂. This proves the theorem completely.

Now, we discuss the other case, that is, the warped product submanifold 𝑀=𝑁_𝜃×_𝑓𝑁_⟂ of a nearly cosymplectic manifold 𝑀. In this case also, if the structure vector filed 𝜉∈𝑇𝑁_⟂ then the warping function 𝑓 is constant (by Theorem 3.1), thus we consider 𝜉∈𝑇𝑁_𝜃.

Proposition 3.6. Let 𝑀=𝑁_𝜃×_𝑓𝑁_⟂ be a warped product pseudo-slant submanifold of a nearly cosymplectic manifold 𝑀. Then, 𝑔∇_𝑋𝐹𝑍,𝐹𝑋+𝑔∇_𝑃𝑋𝐹𝑍,𝐹𝑃𝑋=sin²(ℎ())+𝜃𝑔𝑋,𝑃𝑋,𝐹𝑍1+cos²𝜃𝑔𝒫_𝑋𝑍,𝑃𝑋−cos²𝜃𝜂(𝑋)𝑔𝒫_𝜉𝒬𝑍,𝑃𝑋−𝑔_𝑍𝑋,𝐹𝑋−𝑔𝒬_𝑍𝑃𝑋,𝐹𝑃𝑋 (3.36) for any 𝑋∈𝑇𝑁_𝜃 and 𝑍∈𝑇𝑁_⟂, where 𝑁_𝜃 and 𝑁_⟂ are proper slant and anti-invariant submanifolds of 𝑀, respectively.

Proof. For any 𝑋∈𝑇𝑁_𝜃 and 𝑍∈𝑇𝑁_⟂, by (2.2) we have 𝑔𝜙∇_𝑋𝑍,𝜙𝑋=𝑔∇_𝑋𝑍,𝑋−𝜂(𝑋)𝑔∇_𝑋𝑍,𝜉. (3.37) Using the property of the connection ∇ and the fact that 𝜉 is a Killing vector field, then, from (2.5), we obtain 𝑔𝜙∇_𝑋𝑍,𝜙𝑋=𝑔∇_𝑋𝑍,𝑋. (3.38) Thus by (3.2) and the covariant derivative formula of 𝜙, we derive 𝑔∇_𝑋𝜙𝑍,𝜙𝑋−𝑔∇_𝑋𝜙𝑍,𝜙𝑋=𝑋ln𝑓𝑔(𝑍,𝑋). (3.39) Then form (2.6), (2.8), (2.10), and by the orthogonality of two distributions, we get −𝑔𝐴_𝐹𝑍𝑋,𝑃𝑋+𝑔∇^⟂_𝑋𝐹𝑍,𝐹𝑋−𝑔𝒫_𝑋𝑍,𝑃𝑋−𝑔𝒬_𝑋𝑍,𝐹𝑋=0. (3.40) Thus, on using (2.7) and (2.17)(b), the above equation takes the form 𝑔∇^⟂_𝑋𝐹𝑍,𝐹𝑋=𝑔(ℎ(𝑋,𝑃𝑋),𝐹𝑍)+𝑔𝒫_𝑋𝑍,𝑃𝑋−𝑔𝒬_𝑍𝑋,𝐹𝑋. (3.41) Now, for any 𝑋∈𝑇𝑁_𝜃 and 𝑍∈𝑇𝑁_⟂ from (2.14), we have 𝑔∇^⟂_𝑋𝐹𝑍,𝐹𝑋=𝑔∇_𝑋𝐹𝑍,𝐹𝑋+𝑔𝐹∇_𝑋𝑍,𝐹𝑋. (3.42) Using (3.2), we obtain 𝑔∇^⟂_𝑋𝐹𝑍,𝐹𝑋=𝑔∇_𝑋𝐹𝑍,𝐹𝑋+𝑋ln𝑓𝑔(𝐹𝑍,𝐹𝑋). (3.43) By orthogonality of two normal distributions, we get 𝑔∇^⟂_𝑋𝐹𝑍,𝐹𝑋=𝑔∇_𝑋𝐹𝑍,𝐹𝑋. (3.44) Then, from (3.41) and (3.44), we obtain 𝑔∇_𝑋𝐹𝑍,𝐹𝑋=𝑔(ℎ(𝑋,𝑃𝑋),𝐹𝑍)+𝑔𝒫_𝑋𝑍,𝑃𝑋−𝑔𝒬_𝑍𝑋,𝐹𝑋. (3.45) Interchanging 𝑋 by 𝑃𝑋 in (3.45) and using (2.18) and the fact that ℎ(𝑋,𝜉)=0, for any 𝑋 on a nearly cosymplectic manifold 𝑀, hence we get 𝑔∇_𝑃𝑋𝐹𝑍,𝐹𝑃𝑋=−cos²𝜃𝑔(ℎ(𝑋,𝑃𝑋),𝐹𝑍)−cos²𝜃𝑔𝒫_𝑃𝑋𝑍,𝑋+cos²𝜃𝜂(𝑋)𝑔𝒫_𝑃𝑋𝑍,𝜉−𝑔𝒬_𝑍𝑃𝑋,𝐹𝑃𝑋. (3.46) Using property (𝑝₃)(i) and (2.17), we derive 𝑔∇_𝑃𝑋𝐹𝑍,𝐹𝑃𝑋=−cos²𝜃𝑔(ℎ(𝑋,𝑃𝑋),𝐹𝑍)−cos²𝜃𝑔𝒫_𝑋𝑃𝑋,𝑍+cos²𝜃𝜂(𝑋)𝑔𝒫_𝜉𝑃𝑋,𝑍−𝑔𝒬_𝑍𝑃𝑋,𝐹𝑃𝑋. (3.47) Again, by property (𝑝₃)(i), we obtain 𝑔∇_𝑃𝑋𝐹𝑍,𝐹𝑃𝑋=−cos²𝜃𝑔(ℎ(𝑋,𝑃𝑋),𝐹𝑍)+cos²𝜃𝑔𝒫_𝑋𝑍,𝑃𝑋−cos²𝜃𝜂(𝑋)𝑔𝒫_𝜉𝑍,𝑃𝑋−𝑔𝒬_𝑍𝑃𝑋,𝐹𝑃𝑋. (3.48) Thus, the result follows from (3.45) and (3.48).

Theorem 3.7. Let 𝑀=𝑁_𝜃×_𝑓𝑁_⟂ be a warped product submanifold of a nearly cosymplectic manifold 𝑀. Then 𝑀 is Riemannian product of 𝑁_𝜃 and 𝑁_⟂ if and only if 𝑔(ℎ(𝑋,𝑍),𝐹𝑍)=𝑔(ℎ(𝑍,𝑍),𝐹𝑋), (3.49) for any 𝑋∈𝑇𝑁_𝜃 and 𝑍∈𝑇𝑁_⟂, where 𝑁_𝜃 and 𝑁_⟂ are proper slant and anti-invariant submanifolds of 𝑀, respectively.

Proof. If 𝜉∈𝑇𝑁_⟂, then by Theorem 3.1, 𝑓 is constant on 𝑀. Now, we consider 𝜉∈𝑇𝑁_𝜃. In this case, for any 𝑋∈𝑇𝑁_𝜃 and 𝑍∈𝑇𝑁_⟂ by (2.5), we have 𝑔(ℎ(𝑃𝑋,𝑍),𝐹𝑍)=𝑔∇_𝑍𝑃𝑋,𝜙𝑍. (3.50) Using (2.2), we get 𝑔(ℎ(𝑃𝑋,𝑍),𝐹𝑍)=−𝑔𝜙∇_𝑍𝑃𝑋,𝑍. (3.51) Thus, on using the covariant derivative property of 𝜙, we obtain 𝑔(ℎ(𝑃𝑋,𝑍),𝐹𝑍)=𝑔∇_𝑍𝜙𝑃𝑋,𝑍−𝑔∇_𝑍𝜙𝑃𝑋,𝑍. (3.52) Then from (2.8) and (2.10), we get 𝑔(ℎ(𝑃𝑋,𝑍),𝐹𝑍)=𝑔𝒫_𝑍𝑃𝑋,𝑍−𝑔∇_𝑍𝑃²𝑋,𝑍−𝑔∇_𝑍𝐹𝑃𝑋,𝑍. (3.53) Using property (𝑝₃)(i) and the property of the connection ∇, we derive 𝑔(ℎ(𝑃𝑋,𝑍),𝐹𝑍)=−𝑔𝒫_𝑍𝑍,𝑃𝑋+𝑔𝑃²𝑋,∇_𝑍𝑍+𝑔𝐹𝑃𝑋∇_𝑍𝑍. (3.54) As we have 𝒫_𝑍𝑍=0 from (2.4) and (2.10), then by (2.18) the above equation reduced to 𝑔(ℎ(𝑃𝑋,𝑍),𝐹𝑍)=−cos²𝜃𝑔𝑋,∇_𝑍𝑍+cos²𝜃𝜂(𝑋)𝑔𝜉,∇_𝑍𝑍+g(ℎ(𝑍,𝑍),𝐹𝑃𝑋). (3.55) Since 𝜉 is a Killing vector field on 𝑀, then by (2.5), (3.2), and the property of the connection ∇, the above equation takes the form 𝑔(ℎ(𝑃𝑋,𝑍),𝐹𝑍)=𝑋ln𝑓cos²𝜃‖𝑍‖²+𝑔(ℎ(𝑍,𝑍),𝐹𝑃𝑋). (3.56) Interchanging 𝑋 by 𝑃𝑋 in (3.56) and using (2.18), we obtain cos²𝜃𝑔(ℎ(𝑋,𝑍),𝐹𝑍)+cos²𝜃𝜂(𝑋)𝑔(ℎ(𝑍,𝜉),𝐹𝑍)=−𝑃𝑋ln𝑓cos²𝜃‖𝑍‖²+cos²𝜃𝑔(ℎ(𝑍,𝑍),𝐹𝑋). (3.57) Since ℎ(𝑍,𝜉)=0, for nearly cosymplectic, then the above equation reduces to 𝑃𝑋ln𝑓‖𝑍‖²=𝑔(ℎ(𝑍,𝑍),𝐹𝑋)−𝑔(ℎ(𝑋,𝑍),𝐹𝑍). (3.58) Thus, from (3.58), we obtain (𝑃𝑋ln𝑓)=0 if and only if 𝑔(ℎ(𝑍,𝑍),𝐹𝑋)=𝑔(ℎ(𝑋,𝑍),𝐹𝑍). This proves the theorem completely.