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Mathematical Problems in Engineering
VolumeΒ 2011Β (2011), Article IDΒ 230374, 12 pages
http://dx.doi.org/10.1155/2011/230374
Research Article

Warped Product Semi-Invariant Submanifolds of Nearly Cosymplectic Manifolds

1Institute of Mathematical Sciences, Faculty of Science, University of Malaya, 50603 Kuala Lumpur, Malaysia
2Department of Mathematics, University of Tabuk, Tabuk, Saudi Arabia
3School of Mathematics and Computer Applications, Thapar University, Patiala 147 004, India

Received 20 April 2011; Accepted 16 June 2011

Academic Editor: AlexΒ Elias-Zuniga

Copyright Β© 2011 Siraj Uddin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study warped product semi-invariant submanifolds of nearly cosymplectic manifolds. We prove that the warped product of the type π‘€βŸ‚Γ—π‘“π‘€π‘‡ is a usual Riemannian product of π‘€βŸ‚ and 𝑀𝑇, where π‘€βŸ‚ and 𝑀𝑇 are anti-invariant and invariant submanifolds of a nearly cosymplectic manifold 𝑀, respectively. Thus we consider the warped product of the type π‘€π‘‡Γ—π‘“π‘€βŸ‚ 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=π‘€βŸ‚Γ—π‘“π‘€π‘‡ such that π‘€βŸ‚ 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=π‘€π‘‡Γ—π‘“π‘€βŸ‚ which is called CR-warped product, where 𝑀𝑇 and π‘€βŸ‚ 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 (πœ™,πœ‰,πœ‚,𝑔) on a manifold 𝑀 with πœ‚ 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=π‘€βŸ‚Γ—π‘“π‘€π‘‡ is simply Riemannian product of π‘€βŸ‚ and 𝑀𝑇 where π‘€βŸ‚ 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=π‘€π‘‡Γ—π‘“π‘€βŸ‚ by reversing the two factors π‘€βŸ‚ 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 𝐢∞ manifold 𝑀 is said to have an almost contact structure if there exist on 𝑀 a tensor field πœ™ of type (1,1), a vector field πœ‰, and a 1-form πœ‚ satisfying [9] πœ™2=βˆ’πΌ+πœ‚βŠ—πœ‰,πœ™πœ‰=0,πœ‚βˆ˜πœ™=0,πœ‚(πœ‰)=1.(2.1) There always exists a Riemannian metric 𝑔 on an almost contact manifold 𝑀 satisfying the following compatibility condition: πœ‚(𝑋)=𝑔(𝑋,πœ‰),𝑔(πœ™π‘‹,πœ™π‘Œ)=𝑔(𝑋,π‘Œ)βˆ’πœ‚(𝑋)πœ‚(π‘Œ),(2.2) where 𝑋 and π‘Œ are vector fields on 𝑀 [9].

An almost contact structure (πœ™,πœ‰,πœ‚) is said to be normal if the almost complex structure 𝐽 on the product manifold 𝑀×ℝ given by 𝐽𝑑𝑋,𝑓=ξ‚€π‘‘π‘‘π‘‘πœ™π‘‹βˆ’π‘“πœ‰,πœ‚(𝑋),𝑑𝑑(2.3) where 𝑓 is a 𝐢∞-function on 𝑀×ℝ, has no torsion, that is, 𝐽 is integrable, and the condition for normality in terms of πœ™,πœ‰ and πœ‚ is [πœ™,πœ™]+2π‘‘πœ‚βŠ—πœ‰=0 on 𝑀, where [πœ™,πœ™] is the Nijenhuis tensor of πœ™. Finally the fundamental 2-form Ξ¦ is defined by Ξ¦(𝑋,π‘Œ)=𝑔(𝑋,πœ™π‘Œ).

An almost contact metric structure (πœ™,πœ‰,πœ‚,𝑔) is said to be cosymplectic, if it is normal and both Ξ¦ and πœ‚ are closed [9]. The structure is said to be nearly cosymplectic if πœ™ is Killing, that is, if ξ‚€βˆ‡π‘‹πœ™ξ‚ξ‚€π‘Œ+βˆ‡π‘Œπœ™ξ‚π‘‹=0,(2.4) for any 𝑋,π‘Œβˆˆπ‘‡π‘€, where 𝑇𝑀 is the tangent bundle of 𝑀 and βˆ‡ denotes the Riemannian connection of the metric 𝑔. Equation (2.4) is equivalent to (βˆ‡π‘‹πœ™)𝑋=0, for each π‘‹βˆˆπ‘‡π‘€. The structure is said to be closely cosymplectic if πœ™ is Killing and πœ‚ is closed. It is well known that an almost contact metric manifold is cosymplectic if and only if βˆ‡πœ™ vanishes identically, that is, (βˆ‡π‘‹πœ™)π‘Œ=0 and βˆ‡π‘‹πœ‰=0.

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

From the above proposition we have βˆ‡π‘‹πœ‰=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 βˆ‡ and βˆ‡βŸ‚ are the induced connections on the tangent bundle 𝑇𝑀 and the normal bundle π‘‡βŸ‚π‘€ of 𝑀, respectively, then, Gauss and Weingarten formulae are given by βˆ‡π‘‹π‘Œ=βˆ‡π‘‹π‘Œ+β„Ž(𝑋,π‘Œ),(2.5)βˆ‡π‘‹π‘=βˆ’π΄π‘π‘‹+βˆ‡βŸ‚π‘‹π‘,(2.6) for each 𝑋,π‘Œβˆˆπ‘‡π‘€ and π‘βˆˆπ‘‡βŸ‚π‘€, where β„Ž 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(β„Ž(𝑋,π‘Œ),𝑁)=𝑔𝑁𝑋,π‘Œ,(2.7) where 𝑔 denotes the Riemannian metric on 𝑀 as well as being induced on 𝑀.

For any π‘‹βˆˆπ‘‡π‘€, we write πœ™π‘‹=𝑇𝑋+𝐹𝑋,(2.8) where 𝑇𝑋 is the tangential component and 𝐹𝑋 is the normal component of πœ™π‘‹.

Similarly for any π‘βˆˆπ‘‡βŸ‚π‘€, we write πœ™π‘=𝐡𝑁+𝐢𝑁,(2.9) where 𝐡𝑁 is the tangential component and 𝐢𝑁 is the normal component of πœ™π‘. The covariant derivatives of the tensor fields 𝑃 and 𝐹 are defined as ξ€·βˆ‡π‘‹π‘‡ξ€Έπ‘Œ=βˆ‡π‘‹π‘‡π‘Œβˆ’π‘‡βˆ‡π‘‹ξ‚€π‘Œ,(2.10)βˆ‡π‘‹πΉξ‚π‘Œ=βˆ‡βŸ‚π‘‹πΉπ‘Œβˆ’πΉβˆ‡π‘‹π‘Œ(2.11) for all 𝑋,π‘Œβˆˆπ‘‡π‘€.

Let 𝑀 be a Riemannian manifold isometrically immersed in an almost contact metric manifold 𝑀. then for every π‘₯βˆˆπ‘€ there exists a maximal invariant subspace denoted by π’Ÿπ‘₯ of the tangent space 𝑇π‘₯𝑀 of 𝑀. If the dimension of π’Ÿπ‘₯ is the same for all values of π‘₯βˆˆπ‘€, then π’Ÿπ‘₯ gives an invariant distribution π’Ÿ 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 π’Ÿ whose orthogonal complementary distribution π’ŸβŸ‚ is anti-invariant, that is, (i)𝑇𝑀=π’ŸβŠ•π’ŸβŸ‚βŠ•βŸ¨πœ‰βŸ©, (ii)πœ™(π’Ÿπ‘₯)βŠ†π·π‘₯, (iii)πœ™(π’ŸβŸ‚π‘₯)βŠ‚π‘‡βŸ‚π‘₯𝑀

for any π‘₯βˆˆπ‘€, where π‘‡βŸ‚π‘₯𝑀 denotes the orthogonal space of 𝑇π‘₯𝑀 in 𝑇π‘₯𝑀. A semi-invariant submanifold is called anti-invariant if π’Ÿπ‘₯={0} and invariant if π’ŸβŸ‚π‘₯={0}, respectively, for any π‘₯βˆˆπ‘€. It is called the proper semi-invariant submanifold if neither π’Ÿπ‘₯={0} nor π’ŸβŸ‚π‘₯={0}, for every π‘₯βˆˆπ‘€.

Let 𝑀 be a semi-invariant submanifold of an almost contact metric manifold 𝑀. Then, 𝐹(𝑇π‘₯𝑀) is a subspace of π‘‡βŸ‚π‘₯𝑀. Then for every π‘₯βˆˆπ‘€, there exists an invariant subspace 𝜈π‘₯ of 𝑇π‘₯𝑀 such that π‘‡βŸ‚π‘₯𝑇𝑀=𝐹π‘₯π‘€ξ€ΈβŠ•πœˆπ‘₯.(2.12)

A semi-invariant submanifold 𝑀 of an almost contact metric manifold 𝑀 is called Riemannian product if the invariant distribution π’Ÿ and anti-invariant distribution π’ŸβŸ‚ 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)βˆ‡π‘‹π‘Œβˆˆπ‘‡π‘€1,  for each 𝑋,π‘Œβˆˆπ‘‡π‘€1, (ii)βˆ‡π‘‹π‘=βˆ‡π‘π‘‹=(𝑋ln𝑓)𝑍, for each π‘‹βˆˆπ‘‡π‘€1 and π‘βˆˆπ‘‡π‘€2, (iii)βˆ‡π‘π‘Š=βˆ‡π‘€2π‘π‘Šβˆ’(𝑔(𝑍,π‘Š)/𝑓)grad𝑓, where βˆ‡ and βˆ‡π‘€2 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,π‘ˆ)=π‘ˆπ‘“, for each π‘ˆβˆˆπ‘‡π‘€. 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 π’«π‘‹π‘Œ and π’¬π‘‹π‘Œ the tangential and normal parts of (βˆ‡π‘‹πœ™)π‘Œ, that is, ξ‚€βˆ‡π‘‹πœ™ξ‚π‘Œ=π’«π‘‹π‘Œ+π’¬π‘‹π‘Œ(2.14) for all 𝑋,π‘Œβˆˆπ‘‡π‘€. Making use of (2.5), (2.6), and (2.8)–(2.11), the following relations may easily be obtained π’«π‘‹ξ€·βˆ‡π‘Œ=π‘‹π‘‡ξ€Έπ‘Œβˆ’π΄πΉπ‘Œπ’¬π‘‹βˆ’π΅β„Ž(𝑋,π‘Œ),(2.15)π‘‹ξ‚€π‘Œ=βˆ‡π‘‹πΉξ‚π‘Œ+β„Ž(𝑋,π‘‡π‘Œ)βˆ’πΆβ„Ž(𝑋,π‘Œ).(2.16)

It is straightforward to verify the following properties of 𝒫 and 𝒬, which we enlist here for later use: (𝑝1)(i) 𝒫𝑋+π‘Œπ‘Š=π’«π‘‹π‘Š+π’«π‘Œπ‘Š, (ii) 𝒬𝑋+π‘Œπ‘Š=π’¬π‘‹π‘Š+π’¬π‘Œπ‘Š,(𝑝2)(i) 𝒫𝑋(π‘Œ+π‘Š)=π’«π‘‹π‘Œ+π’«π‘‹π‘Š, (ii) 𝒬𝑋(π‘Œ+π‘Š)=π’¬π‘‹π‘Œ+π’¬π‘‹π‘Š,(𝑝3)𝑔(π’«π‘‹π‘Œ,π‘Š)=βˆ’π‘”(π‘Œ,π’«π‘‹π‘Š)

for all 𝑋,π‘Œ,π‘Šβˆˆπ‘‡π‘€.

On a submanifold 𝑀 of a nearly cosymplectic manifold 𝑀, we obtain from (2.4) and (2.14) that (i)π’«π‘‹π‘Œ+π’«π‘Œπ‘‹=0,(ii)π’¬π‘‹π‘Œ+π’¬π‘Œπ‘‹=0(2.17) for any 𝑋,π‘Œβˆˆπ‘‡π‘€.

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 πœ‰ is tangent to 𝑀. First, we prove that the warped product 𝑀=𝑀1×𝑓𝑀2 is trivial when πœ‰ 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 πœ‰ 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 πœ‰ is tangent to 𝑀2, where 𝑀1 and 𝑀2 are the Riemannian submanifolds of 𝑀.

Proof. For any π‘‹βˆˆπ‘‡π‘€1 and πœ‰ tangent to 𝑀2, we have βˆ‡π‘‹πœ‰=βˆ‡π‘‹πœ‰+β„Ž(𝑋,πœ‰).(3.1) Using the fact that πœ‰ is Killing on a nearly cosymplectic manifold (see Proposition 2.1) and Lemma 2.2(ii), we get 0=(𝑋ln𝑓)πœ‰+β„Ž(𝑋,πœ‰).(3.2) Equating the tangential component of (3.2), we obtain 𝑋ln𝑓=0, for all π‘‹βˆˆπ‘‡π‘€1, 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 πœ‰βˆˆπ‘‡π‘€1, where 𝑀1 and 𝑀2 are the Riemannian submanifolds of 𝑀. For any π‘‹βˆˆπ‘‡π‘€2, we have βˆ‡π‘‹πœ‰=βˆ‡π‘‹πœ‰+β„Ž(𝑋,πœ‰).(3.3) By Proposition 2.1, and Lemma 2.2(ii), we obtain (i)πœ‰ln𝑓=0,(ii)β„Ž(𝑋,πœ‰)=0.(3.4) Thus, we consider the warped product semi-invariant submanifolds of a nearly cosymplectic manifold 𝑀 of the types: (i)𝑀=π‘€βŸ‚Γ—π‘“π‘€π‘‡, (ii)𝑀=π‘€π‘‡Γ—π‘“π‘€βŸ‚,

where 𝑀𝑇 and π‘€βŸ‚ 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=π‘€βŸ‚Γ—π‘“π‘€π‘‡ of a nearly cosymplectic manifold 𝑀 is a usual Riemannian product of π‘€βŸ‚ and 𝑀𝑇, where π‘€βŸ‚ and 𝑀𝑇 are anti-invariant and invariant submanifolds of 𝑀, respectively.

Proof. When πœ‰βˆˆπ‘‡π‘€π‘‡, then by Theorem 3.1, 𝑀 is a Riemannian product. Thus, we consider πœ‰βˆˆπ‘‡π‘€βŸ‚. For any π‘‹βˆˆπ‘‡π‘€π‘‡ and π‘βˆˆπ‘‡π‘€βŸ‚, we have 𝑔(β„Ž(𝑋,πœ™π‘‹),𝐹𝑍)=𝑔(β„Ž(𝑋,πœ™π‘‹),πœ™π‘)=π‘”βˆ‡π‘‹ξ‚ξ‚€πœ™πœ™π‘‹,πœ™π‘=π‘”βˆ‡π‘‹ξ‚π‘‹,πœ™π‘+π‘”ξ‚€ξ‚€βˆ‡π‘‹πœ™ξ‚ξ‚.𝑋,πœ™π‘(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(β„Ž(𝑋,πœ™π‘‹),𝐹𝑍)=π‘”βˆ‡π‘‹ξ‚ξ‚€π‘‹,π‘βˆ’πœ‚(𝑍)π‘”βˆ‡π‘‹ξ‚ξ‚€π‘‹,πœ‰=βˆ’π‘”π‘‹,βˆ‡π‘‹π‘ξ‚ξ‚€+πœ‚(𝑍)𝑔𝑋,βˆ‡π‘‹πœ‰ξ‚.(3.6) Then from (2.5), Lemma 2.2(ii), and Proposition 2.1, we obtain 𝑔(β„Ž(𝑋,πœ™π‘‹),𝐹𝑍)=βˆ’(𝑍ln𝑓)‖𝑋‖2.(3.7) Interchanging 𝑋 by πœ™π‘‹ in (3.7) and using the fact that πœ‰βˆˆπ‘‡π‘€βŸ‚, we obtain 𝑔(β„Ž(𝑋,πœ™π‘‹),𝐹𝑍)=(𝑍ln𝑓)‖𝑋‖2.(3.8) It follows from (3.7) and (3.8) that 𝑍ln𝑓=0, for all π‘βˆˆπ‘‡π‘€βŸ‚. Also, from (3.4) we have πœ‰ln𝑓=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=π‘€βŸ‚Γ—π‘“π‘€π‘‡ is a usual Riemannian product of an anti-invariant submanifold π‘€βŸ‚ and an invariant submanifold 𝑀𝑇 of a nearly cosymplectic manifold 𝑀. Since both π‘€βŸ‚ and 𝑀𝑇 are totally geodesic in 𝑀, then 𝑀 is CR-product. Now, we study the warped product semi-invariant submanifold 𝑀=π‘€π‘‡Γ—π‘“π‘€βŸ‚ of a nearly cosymplectic manifold 𝑀.

Theorem 3.3. Let 𝑀=π‘€π‘‡Γ—π‘“π‘€βŸ‚ be a warped product semi-invariant submanifold of a nearly cosymplectic manifold 𝑀. Then the invariant distribution π’Ÿ and the anti-invariant distribution π’ŸβŸ‚ are always integrable.

Proof. For any 𝑋,π‘Œβˆˆπ’Ÿ, we have 𝐹[]𝑋,π‘Œ=πΉβˆ‡π‘‹π‘Œβˆ’πΉβˆ‡π‘Œπ‘‹.(3.9) Using (2.11), we obtain 𝐹[]=𝑋,π‘Œβˆ‡π‘‹πΉξ‚ξ‚€π‘Œβˆ’βˆ‡π‘ŒπΉξ‚π‘‹.(3.10) Then by (2.16), we derive 𝐹[]𝑋,π‘Œ=π’¬π‘‹π‘Œβˆ’β„Ž(𝑋,π‘‡π‘Œ)+πΆβ„Ž(𝑋,π‘Œ)βˆ’π’¬π‘Œπ‘‹+β„Ž(π‘Œ,𝑇𝑋)βˆ’πΆβ„Ž(𝑋,π‘Œ).(3.11) Thus from (2.17)(ii), we get 𝐹[]𝑋,π‘Œ=2π’¬π‘‹π‘Œ+β„Ž(π‘Œ,𝑇𝑋)βˆ’β„Ž(𝑋,π‘‡π‘Œ).(3.12) Now, for any 𝑋,π‘Œβˆˆπ·, we have β„Ž(𝑋,π‘‡π‘Œ)+βˆ‡π‘‹π‘‡π‘Œ=βˆ‡π‘‹π‘‡π‘Œ=βˆ‡π‘‹πœ™π‘Œ.(3.13) Using the covariant derivative property of βˆ‡πœ™, we obtain β„Ž(𝑋,π‘‡π‘Œ)+βˆ‡π‘‹ξ‚€π‘‡π‘Œ=βˆ‡π‘‹πœ™ξ‚π‘Œ+πœ™βˆ‡π‘‹π‘Œ.(3.14) Then by (2.5) and (2.14), we get β„Ž(𝑋,π‘‡π‘Œ)+βˆ‡π‘‹π‘‡π‘Œ=π‘ƒπ‘‹π‘Œ+π’¬π‘‹ξ€·βˆ‡π‘Œ+πœ™π‘‹ξ€Έπ‘Œ+β„Ž(𝑋,π‘Œ).(3.15) Since 𝑀𝑇 is totally geodesic in 𝑀 (see Lemma 2.2(i)), then using (2.8) and (2.9), we obtain β„Ž(𝑋,π‘‡π‘Œ)+βˆ‡π‘‹π‘‡π‘Œ=π’«π‘‹π‘Œ+π’¬π‘‹π‘Œ+π‘‡βˆ‡π‘‹π‘Œ+π΅β„Ž(𝑋,π‘Œ)+πΆβ„Ž(𝑋,π‘Œ).(3.16) Equating the normal components of (3.16), we get β„Ž(𝑋,π‘‡π‘Œ)=π’¬π‘‹π‘Œ+πΆβ„Ž(𝑋,π‘Œ).(3.17) Similarly, we obtain β„Ž(π‘Œ,𝑇𝑋)=π’¬π‘Œπ‘‹+πΆβ„Ž(𝑋,π‘Œ).(3.18) Then from (3.17) and (3.18), we arrive at β„Ž(π‘Œ,𝑇𝑋)βˆ’β„Ž(𝑋,π‘‡π‘Œ)=π’¬π‘Œπ‘‹βˆ’π’¬π‘‹π‘Œ.(3.19) Hence, using (2.17)(ii), we get β„Ž(π‘Œ,𝑇𝑋)βˆ’β„Ž(𝑋,π‘‡π‘Œ)=βˆ’2π’¬π‘‹π‘Œ.(3.20) Thus, it follows from (3.12) and (3.20) that 𝐹[𝑋,π‘Œ]=0, for all 𝑋,π‘Œβˆˆπ·. This proves the integrability of 𝐷. Now, for the integrability of π·βŸ‚, we consider any π‘‹βˆˆπ· and 𝑍,π‘Šβˆˆπ·βŸ‚, and we have []𝑔(𝑍,π‘Š,𝑋)=π‘”βˆ‡π‘π‘Šβˆ’βˆ‡π‘Šξ‚.ξ€·βˆ‡π‘,𝑋=βˆ’π‘”π‘ξ€Έξ€·βˆ‡π‘‹,π‘Š+π‘”π‘Šξ€Έ.𝑋,𝑍(3.21) Using Lemma 2.2(ii), we obtain []𝑔(𝑍,π‘Š,𝑋)=βˆ’(𝑋ln𝑓)𝑔(𝑍,π‘Š)+(𝑋ln𝑓)𝑔(𝑍,π‘Š)=0.(3.22) Thus from (3.22), we conclude that [𝑍,π‘Š]βˆˆπ’ŸβŸ‚, for each 𝑍,π‘Šβˆˆπ’ŸβŸ‚. Hence, the theorem is proved completely.

Lemma 3.4. Let 𝑀=π‘€π‘‡Γ—π‘“π‘€βŸ‚ be a warped product submanifold of a nearly cosymplectic manifold 𝑀. If 𝑋,π‘Œβˆˆπ‘‡π‘€π‘‡ and 𝑍,π‘Šβˆˆπ‘‡π‘€βŸ‚, then (i)𝑔(π’«π‘‹π‘Œ,𝑍)=𝑔(β„Ž(𝑋,π‘Œ),𝐹𝑍)=0, (ii)𝑔(𝒫𝑋𝑍,π‘Š)=𝑔(β„Ž(𝑋,𝑍),πΉπ‘Š)βˆ’π‘”(β„Ž(𝑋,π‘Š),𝐹𝑍)=βˆ’(πœ™π‘‹ln𝑓)𝑔(𝑍,π‘Š)βˆ’π‘”(β„Ž(𝑋,𝑍),πΉπ‘Š), (iii)𝑔(β„Ž(πœ™π‘‹,𝑍),𝐹𝑍)=(𝑋ln𝑓)‖𝑍‖2.

Proof. For a warped product manifold 𝑀=π‘€π‘‡Γ—π‘“π‘€βŸ‚, we have that 𝑀𝑇 is totally geodesic in 𝑀; then by (2.10), (βˆ‡π‘‹π‘‡)π‘Œβˆˆπ‘‡π‘€π‘‡, for any 𝑋,π‘Œβˆˆπ‘‡π‘€π‘‡, and therefore from (2.15), we get π‘”ξ€·π’«π‘‹ξ€Έπ‘Œ,𝑍=βˆ’π‘”(π΅β„Ž(𝑋,π‘Œ),𝑍)=𝑔(β„Ž(𝑋,π‘Œ),𝐹𝑍).(3.23) The left-hand side of (3.23) is skew symmetric in 𝑋 and π‘Œ whereas the right hand side is symmetric in 𝑋 and π‘Œ, which proves (i). Now, from (2.10) and (2.15), we have 𝒫𝑋𝑍=βˆ’π‘‡βˆ‡π‘‹π‘βˆ’π΄πΉπ‘π‘‹βˆ’π΅β„Ž(𝑋,𝑍)(3.24) for any π‘‹βˆˆπ‘‡π‘€π‘‡ and π‘βˆˆπ‘‡π‘€βŸ‚. Using Lemma 2.2(ii), the first term of right-hand side is zero. Thus, taking the product with π‘Šβˆˆπ‘‡π‘€βŸ‚, we obtain 𝑔𝒫𝑋𝐴𝑍,π‘Š=βˆ’π‘”πΉπ‘ξ€Έπ‘‹,π‘Šβˆ’π‘”(π΅β„Ž(𝑋,𝑍),π‘Š),(3.25) Then by (2.2) and (2.7), we get 𝑔𝒫𝑋𝑍,π‘Š=βˆ’π‘”(β„Ž(𝑋,π‘Š),𝐹𝑍)+𝑔(β„Ž(𝑋,𝑍),πΉπ‘Š).(3.26) which proves the first equality of (ii). Again, from (2.10) and (2.15), we have 𝒫𝑍𝑋=βˆ‡π‘π‘‡π‘‹βˆ’π‘‡βˆ‡π‘π‘‹βˆ’π΅β„Ž(𝑋,𝑍).(3.27) Thus using Lemma 2.2(ii), we derive 𝒫𝑍𝑋=(𝑇𝑋ln𝑓)π‘βˆ’π΅β„Ž(𝑋,𝑍).(3.28) Taking inner product with π‘Šβˆˆπ‘‡π‘€βŸ‚ and using (2.2), we obtain 𝑔𝒫𝑍=𝑋,π‘Š(πœ™π‘‹ln𝑓)𝑔(𝑍,π‘Š)+𝑔(β„Ž(𝑋,𝑍),πΉπ‘Š).(3.29) Then from (2.17)(i), we get 𝑔𝒫𝑋𝑍,π‘Š=βˆ’(πœ™π‘‹ln𝑓)𝑔(𝑍,π‘Š)βˆ’π‘”(β„Ž(𝑋,𝑍),πΉπ‘Š).(3.30) This is the second equality of (ii). Now, from (3.24) and (3.28), we have 𝒫𝑋𝑍+𝒫𝑍𝑋=βˆ’π‘‡βˆ‡π‘‹π‘βˆ’π΄πΉπ‘π‘‹+(𝑇𝑋ln𝑓)π‘βˆ’2π΅β„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+2π΅β„Ž(𝑋,𝑍).(3.32) Taking the product with 𝑍 and on using (2.2) and (2.7), we get (πœ™π‘‹ln𝑓)‖𝑍‖2=𝑔(β„Ž(𝑋,𝑍),𝐹𝑍)βˆ’2𝑔(β„Ž(𝑋,𝑍),𝐹𝑍)=βˆ’π‘”(β„Ž(𝑋,𝑍),𝐹𝑍).(3.33) Interchanging 𝑋 by πœ™π‘‹ and using (2.1), we obtain {βˆ’π‘‹+πœ‚(𝑋)πœ‰}ln𝑓‖𝑍‖2=βˆ’π‘”(β„Ž(πœ™π‘‹,𝑍),𝐹𝑍).(3.34) Thus by (3.4)(i), the above equation reduces to (𝑋ln𝑓)‖𝑍‖2=𝑔(β„Ž(πœ™π‘‹,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 π΄πœ™π‘π‘‹=βˆ’(πœ™π‘‹πœ‡)𝑍,π‘‹βˆˆπ’ŸβŠ•βŸ¨πœ‰βŸ©,π‘βˆˆπ’ŸβŸ‚(3.36) for some function πœ‡ on 𝑀 satisfying 𝑉(πœ‡)=0 for each π‘‰βˆˆπ’ŸβŸ‚.

Proof. If 𝑀=π‘€π‘‡Γ—π‘“π‘€βŸ‚ is a warped product semi-invariant submanifold, then by Lemma 3.4 (iii), we obtain (3.36). In this case πœ‡=ln𝑓.
Conversely, suppose 𝑀 is a semi-invariant submanifold of a nearly cosymplectic manifold 𝑀 satisfying (3.36). Then 𝑔𝐴(β„Ž(𝑋,π‘Œ),πœ™π‘)=π‘”πœ™π‘ξ€Έπ‘‹,π‘Œ=βˆ’(πœ™π‘‹πœ‡)𝑔(π‘Œ,𝑍)=0.(3.37) Now, from (2.5) and the property of covariant derivative of βˆ‡, we have 𝑔(β„Ž(𝑋,π‘Œ),πœ™π‘)=π‘”βˆ‡π‘‹ξ‚ξ‚€πœ™π‘Œ,πœ™π‘=βˆ’π‘”βˆ‡π‘‹ξ‚ξ‚€π‘Œ,𝑍=βˆ’π‘”βˆ‡π‘‹ξ‚πœ™π‘Œ,𝑍+π‘”ξ‚€ξ‚€βˆ‡π‘‹πœ™ξ‚ξ‚.π‘Œ,𝑍(3.38) Then from (2.5), (2.14), and (3.37), the above equation takes the form π‘”ξ€·βˆ‡π‘‹ξ€Έξ€·π‘ƒπ‘‡π‘Œ,𝑍=π‘”π‘‹ξ€Έπ‘Œ,𝑍.(3.39) Using (2.10) and (2.15), we obtain π‘”ξ€·βˆ‡π‘‹ξ€Έξ€·βˆ‡π‘‡π‘Œ,𝑍=π‘”π‘‹ξ€Έξ€·π‘‡π‘Œ,π‘βˆ’π‘”π‘‡βˆ‡π‘‹ξ€Έπ‘Œ,π‘βˆ’π‘”(π΅β„Ž(𝑋,π‘Œ),𝑍).(3.40) Thus by (2.2), the above equation reduces to π‘”ξ€·π‘‡βˆ‡π‘‹ξ€Έπ‘Œ,𝑍=𝑔(β„Ž(𝑋,π‘Œ),πœ™π‘).(3.41) Hence using (2.7) and (3.36), we get π‘”ξ€·π‘‡βˆ‡π‘‹ξ€Έξ€·π΄π‘Œ,𝑍=π‘”πœ™π‘ξ€Έπ‘‹,π‘Œ=0,(3.42) which implies βˆ‡π‘‹π‘Œβˆˆπ’ŸβŠ•βŸ¨πœ‰βŸ©, that is, π’ŸβŠ•βŸ¨πœ‰βŸ© is integrable and its leaves are totally geodesic in 𝑀. Now, for any 𝑍,π‘Šβˆˆπ’ŸβŸ‚ and π‘‹βˆˆπ’ŸβŠ•βŸ¨πœ‰βŸ©, we have π‘”ξ€·βˆ‡π‘ξ€Έξ‚€π‘Š,πœ™π‘‹=π‘”βˆ‡π‘ξ‚ξ‚€πœ™π‘Š,πœ™π‘‹=βˆ’π‘”βˆ‡π‘ξ‚π‘Š,𝑋=π‘”ξ‚€ξ‚€βˆ‡π‘πœ™ξ‚ξ‚ξ‚€π‘Š,π‘‹βˆ’π‘”βˆ‡π‘ξ‚.πœ™π‘Š,𝑋(3.43) Then, using (2.6) and (2.14), we obtain π‘”ξ€·βˆ‡π‘ξ€Έξ€·π’«π‘Š,πœ™π‘‹=π‘”π‘ξ€Έξ€·π΄π‘Š,𝑋+π‘”πœ™π‘Šξ€Έ.𝑍,𝑋(3.44) Thus from (2.7) and the property (𝑝3), we arrive at π‘”ξ€·βˆ‡π‘ξ€Έξ€·π‘Š,πœ™π‘‹=βˆ’π‘”π‘Š,𝒫𝑍𝑋+𝑔(β„Ž(𝑍,𝑋),πœ™π‘Š).(3.45) Again using (2.7) and (2.17)(i), we get π‘”ξ€·βˆ‡π‘ξ€Έξ€·π’«π‘Š,πœ™π‘‹=𝑔𝑋𝐴𝑍,π‘Š+π‘”πœ™π‘Šξ€Έ.𝑋,𝑍(3.46) On the other hand, from (2.10) and (2.15), we have 𝑃𝑋𝑍=βˆ’π‘‡βˆ‡π‘‹π‘βˆ’π΄πΉπ‘π‘‹βˆ’π΅β„Ž(𝑋,𝑍).(3.47) Taking the product with π‘Šβˆˆπ·βŸ‚ and using (3.36), we obtain 𝑔𝒫𝑋𝑍,π‘Š=βˆ’π‘”π‘‡βˆ‡π‘‹ξ€Έ+𝑍,π‘Š(πœ™π‘‹πœ‡)𝑔(𝑍,π‘Š)+𝑔(β„Ž(𝑋,𝑍),πΉπ‘Š).(3.48) The first term of right-hand side of above equation is zero using the fact that π‘‡π‘Š=0, for any π‘Šβˆˆπ’ŸβŸ‚. Again using (2.7), we get 𝑔𝒫𝑋=𝐴𝑍,π‘Š(πœ™π‘‹πœ‡)𝑔(𝑍,π‘Š)+π‘”πœ™π‘Šξ€Έ.𝑋,𝑍(3.49) Thus from (3.36), we derive 𝑔𝒫𝑋=𝑍,π‘Š(πœ™π‘‹πœ‡)𝑔(𝑍,π‘Š)βˆ’(πœ™π‘‹πœ‡)𝑔(𝑍,π‘Š)=0.(3.50) Then from (3.36), (3.46), and (3.50), we obtain π‘”ξ€·βˆ‡π‘ξ€Έπ‘Š,πœ™π‘‹=βˆ’(πœ™π‘‹πœ‡)𝑔(𝑍,π‘Š).(3.51) Let π‘€βŸ‚ be a leaf of π’ŸβŸ‚, and let β„ŽβŸ‚ be the second fundamental form of the immersion of π‘€βŸ‚ into 𝑀. Then for any 𝑍,π‘Šβˆˆπ’ŸβŸ‚, we have π‘”ξ€·β„ŽβŸ‚(ξ€Έξ€·βˆ‡π‘,π‘Š),πœ™π‘‹=𝑔𝑍.π‘Š,πœ™π‘‹(3.52) Hence, from (3.51) and (3.52), we conclude that π‘”ξ€·β„ŽβŸ‚(𝑍,π‘Š),πœ™π‘‹=βˆ’(πœ™π‘‹πœ‡)𝑔(𝑍,π‘Š).(3.53) This means that integral manifold π‘€βŸ‚ of π’ŸβŸ‚ is totally umbilical in 𝑀. Since the anti-invariant distribution π’ŸβŸ‚ of a semi-invariant submanifold 𝑀 is always integrable (Theorem 3.3) and 𝑉(πœ‡)=0 for each π‘‰βˆˆπ’ŸβŸ‚, which implies that the integral manifold of π’ŸβŸ‚ is an extrinsic sphere in 𝑀; that is, it is totally umbilical and its mean curvature vector field is nonzero and parallel along π‘€βŸ‚. Hence by virtue of results obtained in [11], 𝑀 is locally a warped product π‘€π‘‡Γ—π‘“π‘€βŸ‚, where 𝑀𝑇 and π‘€βŸ‚ denote the integral manifolds of the distributions π’ŸβŠ•βŸ¨πœ‰βŸ© and π’ŸβŸ‚, respectively and 𝑓 is the warping function. Thus the theorem is proved.

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