Some Results on Warped Product Submanifolds of a Sasakian Manifold

Abstract

We study warped product Pseudo-slant submanifolds of Sasakian manifolds. We prove a theorem for the existence of warped product submanifolds of a Sasakian manifold in terms of the canonical structure 𝐹.

1. Introduction

The notion of slant submanifold of almost contact metric manifold was introduced by Lotta [1]. Latter, Cabrerizo et al. investigated slant and semislant submanifolds of a Sasakian manifold and obtained many interesting results [2, 3].

The notion of warped product manifolds was introduced by Bishop and O'Neill in [4]. Latter on, many research articles appeared exploring the existence or nonexistence of warped product submanifolds in different spaces (cf. [5–7]). The study of warped product semislant submanifolds of Kaehler manifolds was introduced by Sahin [8]. Recently, Hasegawa and Mihai proved that warped product of the type 𝑁_⟂×_𝜆𝑁_𝑇 in Sasakian manifolds is trivial where 𝑁_𝑇 and 𝑁_⟂ are 𝜙−invariant and anti-invariant submanifolds of a Sasakian manifold, respectively [9].

In this paper we study warped product submanifolds of a Sasakian manifold. We will see in this paper that for a warped product of the type 𝑀=𝑁₁×_𝜆𝑁₂, if 𝑁₁ is any Riemannian submanifold tangent to the structure vector field 𝜉 of a Sasakian manifold 𝑀 then 𝑁₂ is an anti-invariant submanifold and if 𝜉 is tangent to 𝑁₂ then there is no warped product. Also, we will show that the warped product of the type 𝑀=𝑁_⟂×_𝜆𝑁_𝜃 of a Sasakian manifold 𝑀 is trivial and that the warped product of the type 𝑁_𝑇×_𝜆𝑁_⟂ exists and obtains a result in terms of canonical structure.

2. Preliminaries

Let 𝑀 be a (2𝑚+1)-dimensional manifold with almost contact structure (𝜙,𝜉,𝜂) defined by a (1,1) tensor field 𝜙, a vector field 𝜉, and the dual 1−form 𝜂 of 𝜉, satisfying the following properties [10]:

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:

𝑔(𝜙𝑋,𝜙𝑌)=𝑔(𝑋,𝑌)−𝜂(𝑋)𝜂(𝑌). (2.2) An almost contact metric manifold 𝑀 is called Sasakian if

(∇_𝑋𝜙)𝑌=𝑔(𝑋,𝑌)𝜉−𝜂(𝑌)𝑋 (2.3) for all 𝑋,𝑌 in 𝑇𝑀, where ∇ is the Levi-Civita connection of 𝑔 on 𝑀. From (2.3), it follows that

∇_𝑋𝜉=−𝜙𝑋. (2.4)

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)=𝑔(𝐴_𝑁𝑋,𝑌), (2.7) where 𝑔 denotes the Riemannian metric on 𝑀 as well as the one induced on 𝑀.

For any 𝑋∈𝑇𝑀, we write

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, we write

𝜙𝑁=𝑡𝑁+𝑓𝑁, (2.9) where 𝑡𝑁 is the tangential component and 𝑓𝑁 is the normal component of 𝜙𝑁. We shall always consider 𝜉 to be tangent to 𝑀. 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.

For each nonzero vector 𝑋 tangent to 𝑀 at 𝑥, such that 𝑋 is not proportional to 𝜉, we denote by 𝜃(𝑋) the angle between 𝜙𝑋 and 𝑃𝑋.

𝑀 is said to be slant [3] if the angle 𝜃(𝑋) is constant for all 𝑋∈𝑇𝑀−{𝜉} and 𝑥∈𝑀. The angle 𝜃 is called slant angle or Wirtinger angle. Obviously, if 𝜃=0,𝑀 is invariant and if 𝜃=𝜋/2,𝑀 is an anti-invariant submanifold. If the slant angle of 𝑀 is different from 0 and 𝜋/2 then it is called proper slant.

A characterization of slant submanifolds is given by the following.

Theorem 2.1 (see [3]). 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 𝛿∈[0,1] such that 𝑃²=𝛿(−𝐼+𝜂⊗𝜉). (2.10) Furthermore, in such case, if 𝜃 is slant angle, then 𝛿=cos²𝜃.

Following relations are straightforward consequences of (2.10)

𝑔(𝑃𝑋,𝑃𝑌)=cos²𝜃[𝑔(𝑋,𝑌)−𝜂(𝑋)𝜂(𝑌)],𝑔(𝐹𝑋,𝐹𝑌)=sin²𝜃[𝑔(𝑋,𝑌)−𝜂(𝑋)𝜂(𝑌)] (2.11)(2.12) for any 𝑋,𝑌 tangent to 𝑀.

3. Warped and Doubly Warped Product Manifolds

Let (𝑁₁,𝑔₁) and (𝑁₂,𝑔₂) be two Riemannian manifolds and 𝜆 a positive differentiable function on 𝑁₁. The warped product of 𝑁₁ and 𝑁₂ is the Riemannian manifold 𝑁₁×_𝜆𝑁₂=(𝑁₁×𝑁₂,𝑔), where

𝑔=𝑔₁+𝜆²𝑔₂. (3.1) A warped product manifold 𝑁₁×_𝜆𝑁₂ is said to be trivial if the warping function 𝜆 is constant. We recall the following general formula on a warped product [4]:

∇_𝑋𝑉=∇_𝑉𝑋=(𝑋ln𝜆)𝑉, (3.2) where 𝑋 is tangent to 𝑁₁ and 𝑉 is tangent to 𝑁₂.

Let 𝑀=𝑁₁×_𝜆𝑁₂ be a warped product manifold then 𝑁₁ is totally geodesic and 𝑁₂ is totally umbilical submanifold of 𝑀, respectively.

Doubly warped product manifolds were introduced as a generalization of warped product manifolds by Ünal [11]. A doubly warped product manifold of 𝑁₁ and 𝑁₂, denoted as _𝑓2𝑁₁×_𝑓1𝑁₂ is the manifold 𝑁₁×𝑁₂ endowed with a metric 𝑔 defined as

𝑔=𝑓²₂𝑔₁+𝑓²₁𝑔₂ (3.3) where 𝑓₁ and 𝑓₂ are positive differentiable functions on 𝑁₁ and 𝑁₂, respectively.

In this case formula (3.2) is generalized as

∇_𝑋𝑍=(𝑋ln𝑓₁)𝑍+(𝑍ln𝑓₂)𝑋 (3.4) for each 𝑋 in 𝑇𝑁₁ and 𝑍 in 𝑇𝑁₂ [7].

If neither 𝑓₁ nor 𝑓₂ is constant we have a nontrivial doubly warped product 𝑀=_𝑓2𝑁₁×_𝑓1𝑁₂. Obviously in this case both 𝑁₁ and 𝑁₂ are totally umbilical submanifolds of 𝑀.

Now, we consider a doubly warped product of two Riemannian manifolds 𝑁₁ and 𝑁₂ embedded into a Sasakian manifold 𝑀 such that the structure vector field 𝜉 is tangent to the submanifold 𝑀=_𝑓2𝑁₁×_𝑓1𝑁₂. Consider 𝜉 is tangent to 𝑁₁, then for any 𝑉∈𝑇𝑁₂ we have

∇_𝑉𝜉=(𝜉ln𝑓₁)𝑉+(𝑉ln𝑓₂)𝜉. (3.5) Thus from (2.4), (2.5), (2.8), and (3.5), we get

∇_𝑉𝜉=(𝜉ln𝑓₁)𝑉+(𝑉ln𝑓₂)𝜉+ℎ(𝑉,𝜉)=−𝑃𝑉−𝐹𝑉. (3.6) On comparing tangential and normal parts and using the fact that 𝜉,𝑉, and 𝑃𝑉 are mutually orthogonal vector fields, (3.6) implies that

𝑉ln𝑓₂=0,𝜉ln𝑓₁=0,ℎ(𝑉,𝜉)=−𝐹𝑉,𝑃𝑉=0. (3.7) This shows that 𝑓₂ is constant and 𝑁₂ is an anti-invariant submanifold of 𝑀, if the structure vector field 𝜉 is tangent to 𝑁₁.

Similarly, if 𝜉 is tangent to 𝑁₂ and for any 𝑈∈𝑇𝑁₁ we have

∇_𝑈𝜉=(𝜉ln𝑓₂)𝑈+(𝑈ln𝑓₁)𝜉+ℎ(𝑈,𝜉)=−𝑃𝑈−𝐹𝑈, (3.8) which gives

𝑈ln𝑓₁=0,𝜉ln𝑓₂=0,𝑃𝑈=0,ℎ(𝑈,𝜉)=−𝐹𝑈. (3.9) That is, 𝑓₁ is constant and 𝑁₁ is an anti-invariant submanifold of 𝑀.

Note. From the above conclusion we see that for warped product submanifolds 𝑀=𝑁₁×_𝜆𝑁₂ of a Sasakian manifold 𝑀, if the structure vector field 𝜉 is tangent to the first factor 𝑁₁ then second factor 𝑁₂ is an anti-invariant submanifold. On the other hand the warped product 𝑀=𝑁₁×_𝜆𝑁₂ is trivial if the structure vector field 𝜉 is tangent to 𝑁₂.

To study the warped product submanifolds 𝑁₁×_𝜆𝑁₂ with structure vector field 𝜉 tangent to 𝑁₁, we have obtained the following lemma.

Lemma 3.1 (see [12]). Let 𝑀=𝑁₁×_𝜆𝑁₂ be a proper warped product submanifold of a Sasakian manifold 𝑀, with 𝜉∈𝑇𝑁₁, where 𝑁₁ and 𝑁₂ are any Riemannian submanifolds of 𝑀. Then (i)𝜉ln𝜆=0, (ii)𝐴_𝐹𝑍𝑋=−𝑡ℎ(𝑋,𝑍), (iii)𝑔(ℎ(𝑋,𝑍),𝐹𝑌)=𝑔(ℎ(𝑋,𝑌),𝐹𝑍), (iv)𝑔(ℎ(𝑋,𝑍),𝐹𝑊)=𝑔(ℎ(𝑋,𝑊),𝐹𝑍) for any 𝑋,𝑌∈𝑇𝑁₁ and 𝑍,𝑊∈𝑇𝑁₂.

4. Warped Product Pseudoslant Submanifolds

The study of semislant submanifolds of almost contact metric manifolds was introduced by Cabrerizo et.al. [2]. A semislant submanifold 𝑀 of an almost contact metric manifold 𝑀 is a submanifold which admits two orthogonal complementary distributions 𝒟 and 𝒟^𝜃 such that 𝒟 is invariant under 𝜙 and 𝒟^𝜃 is slant with slant angle 𝜃≠0, that is, 𝜙𝒟=𝒟 and 𝜙𝑍 makes a constant angle 𝜃 with 𝑇𝑀 for each 𝑍∈𝒟^𝜃. In particular, if 𝜃=𝜋/2, then a semislant submanifold reduces to a contact CR-submanifold. For a semislant submanifold 𝑀 of an almost contact metric manifold, we have

𝑇𝑀=𝒟⊕𝒟^𝜃⊕{𝜉}. (4.1)

Similarly we say that 𝑀 is an pseudo-slant submanifold of 𝑀 if 𝒟 is an anti-invariant distribution of 𝑀, that is, 𝜙𝒟⊆𝑇^⟂𝑀 and 𝒟^𝜃 is slant with slant angle 𝜃≠0. The normal bundle 𝑇^⟂𝑀 of an pseudo-slant submanifold is decomposed as

𝑇^⟂𝑀=𝐹𝑇𝑀⊕𝜇, (4.2) where 𝜇 is an invariant subbundle of 𝑇^⟂𝑀.

From the above note, we see that for warped product submanifolds 𝑁₁×_𝜆𝑁₂ of a Sasakian manifold 𝑀, one of the factors is an anti-invariant submanifold of 𝑀. Thus, if the manifolds 𝑁_𝜃 and 𝑁_⟂ are slant and anti-invariant submanifolds of Sasakian manifold 𝑀, then their possible warped product pseudo-slant submanifolds may be given by one of the following forms:

(a)𝑁_⟂×_𝜆𝑁_𝜃, (b)𝑁_𝜃×_𝜆𝑁_⟂.

The above two types of warped product pseudo-slant submanifolds are trivial if the structure vector field 𝜉 is tangent to 𝑁_𝜃 and 𝑁_⟂, respectively. Here, we are concerned with the other two cases for the above two types of warped product pseudo-slant submanifolds 𝑁_⟂×_𝜆𝑁_𝜃 and 𝑁_𝜃×_𝜆𝑁_⟂ when 𝜉 is in 𝑇𝑁_⟂ and in 𝑇𝑁_𝜃, respectively.

For the warped product of the type (𝑎), we have

Theorem 4.1. There do not exist the warped product Pseudo-slant submanifolds 𝑀=𝑁_⟂×_𝜆𝑁_𝜃 where 𝑁_⟂ is an anti-invariant and 𝑁_𝜃 is a proper slant submanifold of a Sasakian manifold 𝑀 such that 𝜉 is tangent to 𝑁_⟂.

Proof. For any 𝑋∈𝑇𝑁_𝜃 and 𝑍∈𝑇𝑁_⟂, we have (∇_𝑋𝜙)𝑍=∇_𝑋𝜙𝑍−𝜙∇_𝑋𝑍. (4.3) Using (2.3), (2.5), (2.6), and the fact that 𝜉 is tangent to 𝑁_⟂, we obtain −𝜂(𝑍)𝑋=−𝐴_𝐹𝑍𝑋+∇^⟂_𝑋𝐹𝑍−𝑃∇_𝑋𝑍−𝐹∇_𝑋𝑍−𝑡ℎ(𝑋,𝑍)−𝑓ℎ(𝑋,𝑍). (4.4) Comparing tangential and normal parts, we get 𝜂(𝑍)𝑋=𝐴_𝐹𝑍𝑋+𝑃∇_𝑋𝑍+𝑡ℎ(𝑋,𝑍) (4.5) Equation (4.5) takes the form on using (3.2) as 𝜂(𝑍)𝑋=𝐴_𝐹𝑍𝑋+(𝑍ln𝜆)𝑃𝑋+𝑡ℎ(𝑋,𝑍). (4.6) Taking product with 𝑃𝑋, the left hand side of the above equation is zero using the fact that 𝑋 and 𝑃𝑋 are mutually orthogonal vector fields. Then 0=𝑔(𝐴_𝐹𝑍𝑋,𝑃𝑋)+(𝑍ln𝜆)𝑔(𝑃𝑋,𝑃𝑋)+𝑔(𝑡ℎ(𝑋,𝑍),𝑃𝑋). (4.7) Using (2.7), (2.11) and the fact that 𝜉 is tangent to 𝑁_⟂, we get (𝑍ln𝜆)cos²𝜃‖𝑋‖²=𝑔(ℎ(𝑋,𝑍),𝐹𝑃𝑋)−𝑔(ℎ(𝑋,𝑃𝑋),𝐹𝑍). (4.8) As 𝜃≠𝜋/2, then interchanging 𝑋 by 𝑃𝑋 and taking account of (2.10), we obtain (𝑍ln𝜆)cos⁴𝜃‖𝑋‖²=−cos²𝜃𝑔(ℎ(𝑃𝑋,𝑍),𝐹𝑋)+cos²𝜃𝑔(ℎ(𝑋,𝑃𝑋),𝐹𝑍) (4.9) or (𝑍ln𝜆)cos²𝜃‖𝑋‖²=𝑔(ℎ(𝑋,𝑃𝑋),𝐹𝑍)−𝑔(ℎ(𝑃𝑋,𝑍),𝐹𝑋). (4.10) Adding equations (4.8) and (4.10), we get 2(𝑍ln𝜆)cos²𝜃‖𝑋‖²=𝑔(ℎ(𝑋,𝑍),𝐹𝑃𝑋)−𝑔(ℎ(𝑃𝑋,𝑍),𝐹𝑋). (4.11) The right hand side of the above equation is zero by Lemma 3.1(iv); then (𝑍ln𝜆)cos²𝜃‖𝑋‖²=0. (4.12) Since 𝑁_𝜃 is proper slant and 𝑋 is nonnull, then 𝑍ln𝜆=0. (4.13) In particular, for 𝑍=𝜉∈𝑇𝑁_⟂, Lemma 3.1 (i) implies that 𝜉ln𝜆=0. This means that 𝜆 is constant on 𝑁_⟂. Hence the theorem is proved.

Now, the other case is dealt with in the following theorem.

Theorem 4.2. Let 𝑀=𝑁_𝑇×_𝜆𝑁_⟂ be a warped product submanifold of a Sasakian manifold 𝑀 such that 𝑁_𝑇 is an invariant submanifold tangent to 𝜉 and 𝑁_⟂ is an anti-invariant submanifold of 𝑀. Then (∇_𝑋𝐹)𝑍 lies in the invariant normal subbundle for each 𝑋∈𝑇𝑁_𝑇 and 𝑍∈𝑇𝑁_⟂.

Proof. As 𝑀=𝑁_𝑇×_𝜆𝑁_⟂ is a warped product submanifold with 𝜉 tangent to 𝑁_𝑇, then by (2.3), (∇_𝑋𝜙)𝑍=0, (4.14) for any 𝑋∈𝑇𝑁_𝑇 and 𝑍∈𝑇𝑁_⟂. Using this fact in the formula (∇_𝑈𝜙)𝑉=∇_𝑈𝜙𝑉−𝜙∇_𝑈𝑉 (4.15) for each 𝑈,𝑉∈𝑇𝑀, thus, we obtain ∇_𝑋𝜙𝑍=𝜙∇_𝑋𝑍. (4.16) Then from (2.5) and (2.6), we get −𝐴_𝐹𝑍𝑋+∇^⟂_𝑋𝐹𝑍=𝜙(∇_𝑋𝑍+ℎ(𝑋,𝑍)). (4.17) Which on using (2.8) and (2.9) yields −𝐴_𝐹𝑍𝑋+∇^⟂_𝑋𝐹𝑍=𝑃∇_𝑋𝑍+𝐹∇_𝑋𝑍+𝑡ℎ(𝑋,𝑍)+𝑓ℎ(𝑋,𝑍). (4.18) From the normal components of the above equation, formula (3.2) gives ∇^⟂_𝑋𝐹𝑍=(𝑋ln𝜆)𝐹𝑍+𝑓ℎ(𝑋,𝑍). (4.19) Taking the product in (4.19) with 𝐹𝑊₁ for any 𝑊₁∈𝑇𝑁_⟂, we get 𝑔(∇^⟂_𝑋𝐹𝑍,𝐹𝑊₁)=(𝑋ln𝜆)𝑔(𝐹𝑍,𝐹𝑊₁)+𝑔(𝑓ℎ(𝑋,𝑍),𝐹𝑊₁) (4.20) or 𝑔(∇^⟂_𝑋𝐹𝑍,𝐹𝑊₁)=(𝑋ln𝜆)𝑔(𝜙𝑍,𝜙𝑊₁)+𝑔(𝜙ℎ(𝑋,𝑍),𝜙𝑊₁). (4.21) Then from (2.2), we have 𝑔(∇^⟂_𝑋𝐹𝑍,𝐹𝑊₁)=(𝑋ln𝜆)𝑔(𝑍,𝑊₁). (4.22) On the other hand, we have (∇_𝑋𝐹)𝑍=∇^⟂_𝑋𝐹𝑍−𝐹∇_𝑋𝑍. (4.23) Taking the product in (4.23) with 𝐹𝑊₁ for any 𝑊₁∈𝑇𝑁_⟂ and using (4.22), (2.2), (3.2), and the fact that 𝜉 is tangential to 𝑁_𝑇, we obtain that 𝑔((∇_𝑋𝐹)𝑍,𝐹𝑊₁)=0, (4.24) for any 𝑋∈𝑇𝑁_𝑇 and 𝑍,𝑊₁∈𝑇𝑁_⟂.
Now, if 𝑊₂∈𝑇𝑁_𝑇 then using the formula (4.23), we get
𝑔((∇_𝑋𝐹)𝑍,𝜙𝑊₂)=𝑔(∇^⟂_𝑋𝐹𝑍,𝜙𝑊₂)−𝑔(𝐹∇_𝑋𝑍,𝜙𝑊₂). (4.25) As 𝑁_𝑇 is an invariant submanifold, then 𝜙𝑊₂∈𝑇𝑁_𝑇 for any 𝑊₂∈𝑇𝑁_𝑇, thus using the fact that the product of tangential component with normal is zero, we obtain that 𝑔((∇_𝑋𝐹)𝑍,𝜙𝑊₂)=0, (4.26) for any 𝑋,𝑊₂∈𝑇𝑁_𝑇 and 𝑍∈𝑇𝑁_⟂. Thus from (4.24) and (4.26), it follows that (∇_𝑋𝐹)𝑍∈𝜇. Thus the proof is complete.

Acknowledgment

The authors are thankful to the referee for his valuable suggestion and comments which have improved this paper.