Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 230374 | https://doi.org/10.1155/2011/230374

Siraj Uddin, S. H. Kon, M. A. Khan, Khushwant Singh, "Warped Product Semi-Invariant Submanifolds of Nearly Cosymplectic Manifolds", Mathematical Problems in Engineering, vol. 2011, Article ID 230374, 12 pages, 2011. https://doi.org/10.1155/2011/230374

Warped Product Semi-Invariant Submanifolds of Nearly Cosymplectic Manifolds

Academic Editor: Alex Elias-Zuniga
Received20 Apr 2011
Accepted16 Jun 2011
Published04 Sep 2011

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.

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Copyright © 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.


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