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

Authors: Siraj Uddin; Meraj Ali Khan; Khushwant Singh
Journal: Mathematical Problems in Engineering
 (2011)

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## 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 submanifoldM 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.


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## Body

## 1. Introduction

A manifoldM¯ 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

LetM¯ 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],
(2.1)ϕ2=X+η(X)ξ,η(ξ)=-1,ϕξ=0,η∘ϕ=0,rank⁡(ϕ)=n-1,(2.2)g(ϕX,ϕY)=g(X,Y)+η(X)η(Y),η(X)=g(X,ξ),
for all X,Y∈TM¯.A Lorentzian paracontact metric structure onM¯ 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.LetM 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
(2.3)∇¯XY=∇XY+h(X,Y),(2.4)∇¯XN=-ANX+∇X⊥N,
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 AN is the Weingarten endomorphism associated with N. It is easy to see that
(2.5)g(ANX,Y)=g(h(X,Y),N).For anyX∈TM, we write
(2.6)ϕX=PX+FX,
where PX is the tangential component and FX is the normal component of ϕX. Similarly for N∈T⊥M, we write
(2.7)ϕN=BN+CN,
where BN is the tangential component and CN is the normal component of ϕN.The covariant derivatives of the tensor fieldsϕ, P, and F are defined as
(2.8)(∇¯Xϕ)Y=∇¯XϕY-ϕ∇¯XY,∀X,Y∈TM¯,(2.9)(∇¯XP)Y=∇XPY-P∇XY,∀X,Y∈TM,(2.10)(∇¯XF)Y=∇X⊥FY-F∇XY,∀X,Y∈TM.
Moreover, for an LP-cosymplectic manifold, one has
(2.11)(∇¯XP)Y=AFYX+Bh(X,Y),(2.12)(∇¯XF)Y=Ch(X,Y)-h(X,PY).A submanifoldM is said to be totally umbilical if
(2.13)h(X,Y)=g(X,Y)H,
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 submanifoldM of a paracontact manifold M¯ is said to be a slant submanifold if for any x∈M and X∈TxM-〈ξ〉, the angle between ϕX and TxM 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
(2.14)TM=D⊕〈ξ〉,
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
(2.15)T⊥M=FTM⊕μ.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.
LetM 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
(2.16)P2=λ(I+η⊗ξ).
Furthermore, if  θ is slant angle of  M, then  λ=cos2θ.
Thus, one has the following consequences of formula (2.16):
(2.17)g(PX,PX)=cos⁡2θ[g(X,Y)+η(X)η(Y)],(2.18)g(FX,FY)=sin2θ[g(X,Y)+η(X)η(Y)],
for any X,Y∈TM.


## 3. Totally Umbilical Proper Slant Submanifold

In this section, we considerM 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 submanifoldM 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 anyX,Y∈TM, (2.11) gives
(3.1)(∇¯XP)Y=AFYX+Bh(X,Y).
Taking the product with ξ and using (2.9), we obtain
(3.2)g(∇XPY,ξ)=g(AFYX,ξ)+g(Bh(X,Y),ξ).
Using (2.5) and the fact that M is totally umbilical, the above equation takes the form
(3.3)-g(PY,∇Xξ)=g(H,FY)η(X)+g(X,Y)g(BH,ξ).
Then, from the characteristic equation of LP-cosymplectic manifold, we obtain
(3.4)0=g(H,FY)η(X).
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),
(3.5)∇¯XϕY=ϕ∇¯XY,
for any X,Y∈TM. From (2.3) and (2.6), we obtain
(3.6)∇¯XPY+∇¯XFY=ϕ(∇XY+h(X,Y)).
Again using (2.3), (2.4), and (2.6), we get
(3.7)∇XPY+h(X,PY)-AFYX+∇X⊥FY=P∇XY+F∇XY+ϕh(X,Y).
As M is totally umbilical, then
(3.8)∇XPY+h(X,PY)-AFYX+∇X⊥FY=P∇XY+F∇XY+g(X,Y)ϕH.
Taking the inner product with ϕH and using the fact that H∈μ, we obtain
(3.9)g(h(X,PY),ϕH)+g(∇X⊥FY,ϕH)=g(F∇XY,ϕH)+g(X,Y)g(ϕH,ϕH).
Then from (2.2) and (2.13), we get
(3.10)g(X,PY)g(H,ϕH)+g(∇X⊥FY,ϕH)=g(F∇XY,ϕH)+g(X,Y)‖H‖2.
Again, using (2.2) and the fact that H∈μ, then ϕH is also lies in μ; thus, we obtain
(3.11)g(∇X⊥FY,ϕH)=g(X,Y)‖H‖2.
Then, from (2.4), we derive
(3.12)g(∇¯XFY,ϕH)=g(X,Y)‖H‖2.
Now, for any X∈TM, one has
(3.13)(∇¯Xϕ)H=∇¯XϕH-ϕ∇¯XH.
Using the fact that as M¯ is an LP-cosymplectic manifold, we obtain
(3.14)∇¯XϕH=ϕ∇¯XH.
Using (2.4), (2.6), and (2.7), we obtain
(3.15)-AϕHX+∇X⊥ϕH=-PAHX-FAHX+B∇X⊥H+C∇X⊥H.
Taking the product in (3.15) with FY for any Y∈TM and using the fact C∇X⊥H∈μ, the above equation gives
(3.16)g(∇X⊥ϕH,FY)=-g(FAHX,FY).
Using (2.18), we obtain
(3.17)g(∇¯XFY,ϕH)=sin2θ[g(AHX,Y)+η(AHX)η(Y)],
then, from (2.5) and (2.13), we get
(3.18)g(∇¯XFY,ϕH)=sin2θ[g(X,Y)+η(X)η(Y)]‖H‖2.
Thus, from (3.12) and (3.18), we derive
(3.19)[cos⁡2θg(X,Y)-sin2⁡θη(X)η(Y)]‖H‖2=0.
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 thatM 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 submanifoldM of an LP-contact manifold M¯ is said to be a hemi-slant submanifold if there exist two orthogonal complementary distributions D1andD2 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.Ifμ 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
(4.1)T⊥M=FD1⊕FD2⊕μ.In the following, we obtain the integrability conditions of involved distributions in the definition of hemi-slant submanifold.Proposition 4.2.
LetM be a hemi-slant submanifold of an LP-cosymplectic manifold  M¯, then the anti-invariant distribution  D2 is integrable if and only if
(4.2)AFZW=AFWZ,
for any Z,W∈D2.Proof.
For anyZ,W∈D2, one has
(4.3)ϕ[Z,W]=ϕ∇¯ZW-ϕ∇¯WZ.
Using (2.8), we obtain
(4.4)ϕ[Z,W]=∇¯ZϕW-∇¯WϕZ.
Then, from (2.4), we derive
(4.5)ϕ[Z,W]=-AFWZ+∇Z⊥FW+AFZW-∇W⊥FZ.
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.
LetM be a hemi-slant submanifold of an LP-cosymplectic manifold  M¯, then the invariant distribution  D1⊕〈ξ〉 is integrable if and only if
(4.6)g(h(X,PY)-h(Y,PX)+∇X⊥FY-∇Y⊥FX,FZ)=0,
for any  X,Y∈D1⊕〈ξ〉 and  Z∈D2.Proof.
For anyX,Y∈D1⊕〈ξ〉, one has
(4.7)ϕ[X,Y]=ϕ∇¯XY-ϕ∇¯YX.
Then, from (2.8) and the fact that M¯ is LP-cosymplectic, we obtain
(4.8)ϕ[X,Y]=∇¯XϕY-∇¯YϕX.
Using (2.6), we get
(4.9)ϕ[X,Y]=∇¯XPY+∇¯XFY-∇¯YPX-∇¯YFX.
Thus, from (2.3) and (2.4), we derive
(4.10)ϕ[X,Y]=∇XPY+h(X,PY)-AFYX+∇X⊥FY-∇YPX-h(Y,PX)+AFXY-∇Y⊥FX.
Taking the product in (4.10) with FZ, for any Z∈D2, we obtain
(4.11)g(ϕ[X,Y],FZ)=g(h(X,PY)+∇X⊥FY-h(Y,PX)-∇Y⊥FX,FZ).
Thus, the assertion follows from (4.11) after using (2.2) and the fact that ξ is tangential to D1.Now, we considerM as a totally umbilical hemi-slant submanifold of an LP-cosymplectic manifold M¯. For any X,Y∈TM, one has
(4.12)∇¯XϕY=ϕ∇¯XY.
Using this fact, if we take for any Z,W∈D2, then from (2.3) and (2.4), the above equation takes the form
(4.13)-AFWZ+∇Z⊥FW=ϕ(∇ZW+h(Z,W)).
Thus, on using (2.6) and (2.7), we obtain
(4.14)-AFWZ+∇Z⊥FW=P∇ZW+F∇ZW+Bh(Z,W)+Ch(Z,W).
Equating the tangential components, we get
(4.15)P∇ZW=-AFWZ-Bh(Z,W).
Taking the product with V∈D2, we obtain
(4.16)g(P∇ZW,V)=-g(AFWZ,V)-g(Bh(Z,W),V).
Using (2.2), (2.5), and the fact that PW=0, for any W∈D2, thus, the above equation takes the form
(4.17)0=g(h(Z,V),FW)+g(Bh(Z,W),V).
As M is totally umbilical, we derive
(4.18)0=g(Z,V)g(H,FW)+g(Z,W)g(BH,V).
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.
LetM 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 vectorH∈μ,(iii)
M is proper slant submanifold of  M¯.



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