Totally Umbilical Proper Slant and Hemislant Submanifolds of an LP-Cosymplectic Manifold

Abstract

In the present note, we study slant and hemislant submanifolds of an LP-cosymplectic manifold which are totally umbilical. We prove that every totally umbilical proper slant submanifold 𝑀 of an LP-cosymplectic manifold 𝑀 is either totally geodesic or if 𝑀 is not totally geodesic in 𝑀 then we derive a formula for slant angle of 𝑀. Also, we obtain the integrability conditions of the distributions of a hemi-slant submanifold, and then we give a result on its classification.

1. Introduction

A manifold 𝑀 with Lorentzian paracontact metric structure (𝜙,𝜉,𝜂,𝑔) satisfying (∇_𝑋𝜙)𝑌=0 is called an LP-cosymplectic manifold, where ∇ is the Levi-Civita connection corresponding to the Lorentzian metric 𝑔 on 𝑀. The study of slant submanifolds was initiated by Chen [1]. Since then, many research papers have appeared in this field. Slant submanifolds are the natural generalization of both holomorphic and totally real submanifolds. Lotta [2] defined and studied these submanifolds in contact geometry. Later on, Cabrerizo et al. studied slant, semi-slant, and bislant submanifolds in contact geometry [3, 4]. In particular, totally umbilical proper slant submanifold of a Kaehler manifold has also been studied in [5]. Recently, Khan et al. [6] studied these submanifolds in the setting of Lorentzian paracontact manifolds.

The idea of hemi-slant submanifolds was introduced by Carriazo as a particular class of bislant submanifolds, and he called them antislant submanifolds [7]. Recently, these submanifolds are studied by Sahin for their warped products [8]. In this paper, we study slant and hemi-slant submanifolds of an LP-cosymplectic manifold. We prove that a totally umbilical proper slant submanifold 𝑀 is either totally geodesic in 𝑀 or if it is not totally geodesic, then the slant angle 𝜃=tan⁻¹(√𝑔(𝑋,𝑌)/𝜂(𝑋)𝜂(𝑌)). Also, we define hemi-slant submanifolds of an LP-contact manifold. After we find integrability conditions of the distributions, we investigate a classification of totally umbilical hemi-slant submanifolds of an LP-cosymplectic manifold.

2. Preliminaries

Let 𝑀 be a 𝑛-dimensional paracontact manifold with the Lorentzian paracontact metric structure (𝜙,𝜉,𝜂,𝑔), that is, 𝜙 is a (1,1) tensor field, 𝜉 is a contravariant vector field, 𝜂 is a 1-form, and 𝑔 is a Lorentzian metric with signature (−,+,+,…,+) on 𝑀, satisfying [9], 𝜙²=𝑋+𝜂(𝑋)𝜉,𝜂(𝜉)=−1,𝜙𝜉=0,𝜂∘𝜙=0,rank(𝜙)=𝑛−1,(2.1)𝑔(𝜙𝑋,𝜙𝑌)=𝑔(𝑋,𝑌)+𝜂(𝑋)𝜂(𝑌),𝜂(𝑋)=𝑔(𝑋,𝜉),(2.2) for all 𝑋,𝑌∈𝑇𝑀.

A Lorentzian paracontact metric structure on 𝑀 is called a Lorentzian para-cosymplectic structure if ∇𝜙=0, where ∇ denotes the Levi-Civita connection with respect to 𝑔. The manifold 𝑀 in this case is called a Lorentzian para-cosymplectic (in brief, an LP-cosymplectic) manifold [10]. From formula ∇𝜙=0, it follows that ∇_𝑋𝜉=0.

Let 𝑀 be a submanifold of a Lorentzian almost paracontact manifold 𝑀 with Lorentzian almost paracontact structure (𝜙,𝜉,𝜂,𝑔). Let the induced metric on 𝑀 also be denoted by 𝑔, then Gauss and Weingarten formulae are given by ∇_𝑋𝑌=∇_𝑋𝑌+ℎ(𝑋,𝑌),(2.3)∇_𝑋𝑁=−𝐴_𝑁𝑋+∇^⟂𝑋𝑁,(2.4) for any 𝑋,𝑌 in 𝑇𝑀 and 𝑁 in 𝑇^⟂𝑀, where 𝑇𝑀 is the Lie algebra of vector field in 𝑀 and 𝑇^⟂𝑀 is the set of all vector fields normal to 𝑀. ∇^⟂ is the connection in the normal bundle, ℎ is the second fundamental form, and 𝐴_𝑁 is the Weingarten endomorphism associated with 𝑁. It is easy to see that 𝑔𝐴_𝑁𝑋,𝑌=𝑔(ℎ(𝑋,𝑌),𝑁).(2.5)

For any 𝑋∈𝑇𝑀, we write 𝜙𝑋=𝑃𝑋+𝐹𝑋,(2.6) where 𝑃𝑋 is the tangential component and 𝐹𝑋 is the normal component of 𝜙𝑋. Similarly for 𝑁∈𝑇^⟂𝑀, we write 𝜙𝑁=𝐵𝑁+𝐶𝑁,(2.7) where 𝐵𝑁 is the tangential component and 𝐶𝑁 is the normal component of 𝜙𝑁.

The covariant derivatives of the tensor fields 𝜙, 𝑃, and 𝐹 are defined as ∇_𝑋𝜙𝑌=∇_𝑋𝜙𝑌−𝜙∇_𝑋𝑌,∀𝑋,𝑌∈𝑇𝑀,(2.8)∇_𝑋𝑃𝑌=∇_𝑋𝑃𝑌−𝑃∇_𝑋𝑌,∀𝑋,𝑌∈𝑇𝑀,(2.9)∇_𝑋𝐹𝑌=∇^⟂𝑋𝐹𝑌−𝐹∇_𝑋𝑌,∀𝑋,𝑌∈𝑇𝑀.(2.10) Moreover, for an LP-cosymplectic manifold, one has ∇_𝑋𝑃𝑌=𝐴_𝐹𝑌𝑋+𝐵ℎ(𝑋,𝑌),(2.11)∇_𝑋𝐹𝑌=𝐶ℎ(𝑋,𝑌)−ℎ(𝑋,𝑃𝑌).(2.12)

A submanifold 𝑀 is said to be totally umbilical if ℎ(𝑋,𝑌)=𝑔(𝑋,𝑌)𝐻,(2.13) where 𝐻 is the mean curvature vector. Furthermore, if ℎ(𝑋,𝑌)=0 for all 𝑋,𝑌∈𝑇𝑀, then 𝑀 is said to be totally geodesic, and if 𝐻=0, then 𝑀 is minimal in 𝑀.

A submanifold 𝑀 of a paracontact manifold 𝑀 is said to be a slant submanifold if for any 𝑥∈𝑀 and 𝑋∈𝑇_𝑥𝑀−⟨𝜉⟩, the angle between 𝜙𝑋 and 𝑇_𝑥𝑀 is constant. The constant angle 𝜃∈[0,𝜋/2] is then called slant angle of 𝑀. The tangent bundle 𝑇𝑀 of 𝑀 is decomposed as 𝑇𝑀=𝐷⊕⟨𝜉⟩,(2.14) where the orthogonal complementary distribution 𝐷 of ⟨𝜉⟩ is known as the slant distribution on 𝑀. If 𝜇 is 𝜙-invariant subspace of the normal bundle 𝑇^⟂𝑀, then 𝑇^⟂𝑀=𝐹𝑇𝑀⊕𝜇.(2.15)

Khan et al. [6] proved the following theorem for a slant submanifold 𝑀 of a Lorentzian paracontact manifold 𝑀 with slant angle 𝜃.

Theorem 2.1. Let 𝑀 be a submanifold of an 𝐿𝑃-contact manifold 𝑀 such that 𝜉∈𝑇𝑀, then 𝑀 is slant submanifold if and only if there exists a constant 𝜆∈[0,1] such that 𝑃²=𝜆(𝐼+𝜂⊗𝜉).(2.16) Furthermore, if 𝜃 is slant angle of 𝑀, then 𝜆=𝑐𝑜𝑠²𝜃.
Thus, one has the following consequences of formula (2.16): 𝑔𝑔((𝑃𝐹𝑋𝑋,,𝑃𝐹𝑋𝑌))==csions²₂𝜃[𝑔(𝑋,𝑌)+𝜂(𝑋)𝜂(𝑌)],(2.17)𝜃[𝑔(𝑋,𝑌)+𝜂(𝑋)𝜂(𝑌)],(2.18) for any 𝑋,𝑌∈𝑇𝑀.

3. Totally Umbilical Proper Slant Submanifold

In this section, we consider 𝑀 as a totally umbilical proper slant submanifold of an LP-cosymplectic manifold 𝑀. Such submanifolds we always consider tangent to the structure vector field 𝜉.

Theorem 3.1. A nontrivial totally umbilical proper slant submanifold 𝑀 of an LP-cosymplectic manifold 𝑀 is either totally geodesic or if it is not totally geodesic in 𝑀, then the slant angle 𝜃=tan⁻¹(√𝑔(𝑋,𝑌)/𝜂(𝑋)𝜂(𝑌)), for any 𝑋,𝑌∈𝑇𝑀.

Proof. For any 𝑋,𝑌∈𝑇𝑀, (2.11) gives ∇_𝑋𝑃𝑌=𝐴_𝐹𝑌𝑋+𝐵ℎ(𝑋,𝑌).(3.1) Taking the product with 𝜉 and using (2.9), we obtain 𝑔∇_𝑋𝑃𝑌,𝜉=𝑔𝐴_𝐹𝑌𝑋,𝜉+𝑔(𝐵ℎ(𝑋,𝑌),𝜉).(3.2) Using (2.5) and the fact that 𝑀 is totally umbilical, the above equation takes the form −𝑔𝑃𝑌,∇_𝑋𝜉=𝑔(𝐻,𝐹𝑌)𝜂(𝑋)+𝑔(𝑋,𝑌)𝑔(𝐵𝐻,𝜉).(3.3) Then, from the characteristic equation of LP-cosymplectic manifold, we obtain 0=𝑔(𝐻,𝐹𝑌)𝜂(𝑋).(3.4) Thus, from (3.4), it follows that either 𝐻∈𝜇 or 𝑀 is trivial.
Now, for an LP-cosymplectic manifold, one has, from (2.8), ∇_𝑋𝜙𝑌=𝜙∇_𝑋𝑌,(3.5) for any 𝑋,𝑌∈𝑇𝑀. From (2.3) and (2.6), we obtain ∇_𝑋𝑃𝑌+∇_𝑋𝐹𝑌=𝜙∇_𝑋𝑌+ℎ(𝑋,𝑌).(3.6) Again using (2.3), (2.4), and (2.6), we get ∇_𝑋𝑃𝑌+ℎ(𝑋,𝑃𝑌)−𝐴_𝐹𝑌𝑋+∇^⟂𝑋𝐹𝑌=𝑃∇_𝑋𝑌+𝐹∇_𝑋𝑌+𝜙ℎ(𝑋,𝑌).(3.7) As 𝑀 is totally umbilical, then ∇_𝑋𝑃𝑌+ℎ(𝑋,𝑃𝑌)−𝐴_𝐹𝑌𝑋+∇^⟂𝑋𝐹𝑌=𝑃∇_𝑋𝑌+𝐹∇_𝑋𝑌+𝑔(𝑋,𝑌)𝜙𝐻.(3.8) Taking the inner product with 𝜙𝐻 and using the fact that 𝐻∈𝜇, we obtain 𝑔(ℎ(𝑋,𝑃𝑌),𝜙𝐻)+𝑔∇^⟂𝑋𝐹𝑌,𝜙𝐻=𝑔𝐹∇_𝑋𝑌,𝜙𝐻+𝑔(𝑋,𝑌)𝑔(𝜙𝐻,𝜙𝐻).(3.9) Then from (2.2) and (2.13), we get 𝑔(𝑋,𝑃𝑌)𝑔(𝐻,𝜙𝐻)+𝑔∇^⟂𝑋𝐹𝑌,𝜙𝐻=𝑔𝐹∇_𝑋𝑌,𝜙𝐻+𝑔(𝑋,𝑌)‖𝐻‖².(3.10) Again, using (2.2) and the fact that 𝐻∈𝜇, then 𝜙𝐻 is also lies in 𝜇; thus, we obtain 𝑔∇^⟂𝑋𝐹𝑌,𝜙𝐻=𝑔(𝑋,𝑌)‖𝐻‖².(3.11) Then, from (2.4), we derive 𝑔∇_𝑋𝐹𝑌,𝜙𝐻=𝑔(𝑋,𝑌)‖𝐻‖².(3.12) Now, for any 𝑋∈𝑇𝑀, one has ∇_𝑋𝜙𝐻=∇_𝑋𝜙𝐻−𝜙∇_𝑋𝐻.(3.13) Using the fact that as 𝑀 is an LP-cosymplectic manifold, we obtain ∇_𝑋𝜙𝐻=𝜙∇_𝑋𝐻.(3.14) Using (2.4), (2.6), and (2.7), we obtain −𝐴_𝜙𝐻𝑋+∇^⟂𝑋𝜙𝐻=−𝑃𝐴_𝐻𝑋−𝐹𝐴_𝐻𝑋+𝐵∇^⟂𝑋𝐻+𝐶∇^⟂𝑋𝐻.(3.15) Taking the product in (3.15) with 𝐹𝑌 for any 𝑌∈𝑇𝑀 and using the fact 𝐶∇^⟂𝑋𝐻∈𝜇, the above equation gives 𝑔∇^⟂𝑋𝜙𝐻,𝐹𝑌=−𝑔𝐹𝐴_𝐻𝑋,𝐹𝑌.(3.16) Using (2.18), we obtain 𝑔∇_𝑋𝐹𝑌,𝜙𝐻=sin²𝜃𝑔𝐴_𝐻𝑋,𝑌+𝜂𝐴_𝐻𝑋𝜂(𝑌),(3.17) then, from (2.5) and (2.13), we get 𝑔∇_𝑋𝐹𝑌,𝜙𝐻=sin²𝜃[𝑔(𝑋,𝑌)+𝜂(𝑋)𝜂(𝑌)]‖𝐻‖².(3.18) Thus, from (3.12) and (3.18), we derive cos²𝜃𝑔(𝑋,𝑌)−sin²𝜃𝜂(𝑋)𝜂(𝑌)‖𝐻‖²=0.(3.19) Hence, (3.19) gives either 𝐻=0 or if 𝐻≠0, then the slant angle of 𝑀 is 𝜃=tan⁻¹(√𝑔(𝑋,𝑌)/𝜂(𝑋)𝜂(𝑌)). This proves the theorem completely.

4. Hemislant Submanifolds

In the following section, we assume that 𝑀 is a hemi-slant submanifold of an LP-cosymplectic manifold 𝑀 such that the structure vector field 𝜉 tangent to 𝑀. First, we define a hemi-slant submanifold, and then we obtain the integrability conditions of the involved distributions 𝐷₁ and 𝐷₂ in the definition of a hemi-slant submanifold 𝑀 of an LP-cosymplectic manifold 𝑀.

Definition 4.1. A submanifold M of an LP-contact manifold 𝑀 is said to be a hemi-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 totally real that is, 𝜙𝐷₂⊆𝑇^⟂𝑀.

If 𝜇 is 𝜙-invariant subspace of the normal bundle 𝑇^⟂𝑀, then in case of hemi-slant submanifold, the normal bundle 𝑇^⟂𝑀 can be decomposed as 𝑇^⟂𝑀=𝐹𝐷₁⊕𝐹𝐷₂⊕𝜇.(4.1)

In the following, we obtain the integrability conditions of involved distributions in the definition of hemi-slant submanifold.

Proposition 4.2. Let 𝑀 be a hemi-slant submanifold of an LP-cosymplectic manifold 𝑀, then the anti-invariant distribution 𝐷₂ is integrable if and only if 𝐴_𝐹𝑍𝑊=𝐴_𝐹𝑊𝑍,(4.2) for any 𝑍,𝑊∈𝐷₂.

Proof. For any 𝑍,𝑊∈𝐷₂, one has 𝜙[𝑍,𝑊]=𝜙∇_𝑍𝑊−𝜙∇_𝑊𝑍.(4.3) Using (2.8), we obtain 𝜙[𝑍,𝑊]=∇_𝑍𝜙𝑊−∇_𝑊𝜙𝑍.(4.4) Then, from (2.4), we derive 𝜙[𝑍,𝑊]=−𝐴_𝐹𝑊𝑍+∇^⟂𝑍𝐹𝑊+𝐴_𝐹𝑍𝑊−∇^⟂𝑊𝐹𝑍.(4.5) As 𝐷₂ is an anti-invariant distribution, then the tangential part of (4.5) should be identically zero; hence, we obtain the required result.

Proposition 4.3. Let 𝑀 be a hemi-slant submanifold of an LP-cosymplectic manifold 𝑀, then the invariant distribution 𝐷₁⊕⟨𝜉⟩ is integrable if and only if 𝑔ℎ(𝑋,𝑃𝑌)−ℎ(𝑌,𝑃𝑋)+∇^⟂𝑋𝐹𝑌−∇^⟂𝑌𝐹𝑋,𝐹𝑍=0,(4.6) for any 𝑋,𝑌∈𝐷₁⊕⟨𝜉⟩ and 𝑍∈𝐷₂.

Proof. For any 𝑋,𝑌∈𝐷₁⊕⟨𝜉⟩, one has 𝜙[𝑋,𝑌]=𝜙∇_𝑋𝑌−𝜙∇_𝑌𝑋.(4.7) Then, from (2.8) and the fact that 𝑀 is LP-cosymplectic, we obtain 𝜙[𝑋,𝑌]=∇_𝑋𝜙𝑌−∇_𝑌𝜙𝑋.(4.8) Using (2.6), we get 𝜙[𝑋,𝑌]=∇_𝑋𝑃𝑌+∇_𝑋𝐹𝑌−∇_𝑌𝑃𝑋−∇_𝑌𝐹𝑋.(4.9) Thus, from (2.3) and (2.4), we derive 𝜙[𝑋,𝑌]=∇_𝑋𝑃𝑌+ℎ(𝑋,𝑃𝑌)−𝐴_𝐹𝑌𝑋+∇^⟂𝑋𝐹𝑌−∇_𝑌𝑃𝑋−ℎ(𝑌,𝑃𝑋)+𝐴_𝐹𝑋𝑌−∇^⟂𝑌𝐹𝑋.(4.10) Taking the product in (4.10) with 𝐹𝑍, for any 𝑍∈𝐷₂, we obtain 𝑔(𝜙[𝑋,𝑌],𝐹𝑍)=𝑔ℎ(𝑋,𝑃𝑌)+∇^⟂𝑋𝐹𝑌−ℎ(𝑌,𝑃𝑋)−∇^⟂𝑌𝐹𝑋,𝐹𝑍.(4.11) Thus, the assertion follows from (4.11) after using (2.2) and the fact that 𝜉 is tangential to 𝐷₁.

Now, we consider 𝑀 as a totally umbilical hemi-slant submanifold of an LP-cosymplectic manifold 𝑀. For any 𝑋,𝑌∈𝑇𝑀, one has ∇_𝑋𝜙𝑌=𝜙∇_𝑋𝑌.(4.12) Using this fact, if we take for any 𝑍,𝑊∈𝐷₂, then from (2.3) and (2.4), the above equation takes the form −𝐴_𝐹𝑊𝑍+∇^⟂𝑍𝐹𝑊=𝜙∇_𝑍𝑊+ℎ(𝑍,𝑊).(4.13) Thus, on using (2.6) and (2.7), we obtain −𝐴_𝐹𝑊𝑍+∇^⟂𝑍𝐹𝑊=𝑃∇_𝑍𝑊+𝐹∇_𝑍𝑊+𝐵ℎ(𝑍,𝑊)+𝐶ℎ(𝑍,𝑊).(4.14) Equating the tangential components, we get 𝑃∇_𝑍𝑊=−𝐴_𝐹𝑊𝑍−𝐵ℎ(𝑍,𝑊).(4.15) Taking the product with 𝑉∈𝐷₂, we obtain 𝑔𝑃∇_𝑍𝑊,𝑉=−𝑔𝐴_𝐹𝑊𝑍,𝑉−𝑔(𝐵ℎ(𝑍,𝑊),𝑉).(4.16) Using (2.2), (2.5), and the fact that 𝑃𝑊=0, for any 𝑊∈𝐷₂, thus, the above equation takes the form 0=𝑔(ℎ(𝑍,𝑉),𝐹𝑊)+𝑔(𝐵ℎ(𝑍,𝑊),𝑉).(4.17) As 𝑀 is totally umbilical, we derive 0=𝑔(𝑍,𝑉)𝑔(𝐻,𝐹𝑊)+𝑔(𝑍,𝑊)𝑔(𝐵𝐻,𝑉).(4.18) Thus, (4.18) has a solution if either 𝑍=𝑊=𝑉=𝜉, that is, dim𝐷₂=1 or 𝐻∈𝜇 or 𝐷₂={0}. Hence, we state the following theorem.

Theorem 4.4. Let 𝑀 be a totally umbilical hemi-slant submanifold of an LP-cosymplectic manifold 𝑀, then at least one of the following statements is true: (i)the dimension of anti-invariant distribution is one, that is, dim𝐷₂=1,(ii)the mean curvature vector 𝐻∈𝜇,(iii)𝑀 is proper slant submanifold of 𝑀.

Acknowledgments

The authors are thankful to the referee for his valuable suggestion and comments. The first author is supported by the research Grant no. RG117/10AFR (University Malaya).