# On Lightlike Geometry of Para-Sasakian Manifolds

Authors: Bilal Eftal Acet; Selcen Yüksel Perktaş; Erol Kılıç
Journal: The Scientific World Journal
 (2014)

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## Abstract

We study lightlike hypersurfaces of para-Sasakian manifolds tangent to the characteristic vector field. In particular, we define invariant lightlike hypersurfaces and screen semi-invariant lightlike hypersurfaces, respectively, and give examples. Integrability conditions for the distributions on a screen semi-invariant lightlike hypersurface of para-Sasakian manifolds are investigated. We obtain a para-Sasakian structure on the leaves of an integrable distribution of a screen semi-invariant lightlike hypersurface.


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## Body

## 1. Introduction

It is well known that the main difference between the geometry of submanifolds in Riemannian manifolds and in semi-Riemannian manifolds is that in the latter case the induced metric tensor field by the semi-Riemannian metric on the ambient space is not necessarily nondegenerate. If the induced metric tensor field is degenerate the classical theory of Riemannian submanifolds fails since the normal bundle and the tangent bundle of the submanifold have a nonzero intersection. In particular, from the point of physics lightlike hypersurfaces are important as they are models of various types of horizons, such as Killing, dynamical and conformal horizons, studied in general relativity.Lightlike submanifolds of semi-Riemannian manifolds were introduced by Duggal and Bejancu in [1]. Since then many authors studied lightlike hypersurfaces of semi-Riemannian manifolds and especially of indefinite Sasakian manifolds (for differential geometry of lightlike submanifolds we refer to the book [2]).The study of paracontact geometry was initiated by Kaneyuki and Konzai in [3]. The authors defined almost paracontact structure on a pseudo-Riemannian manifold M of dimension (2n+1) and constructed the almost paracomplex structure on M2n+1×ℝ. Recently, Zamkovoy [4] studied paracontact metric manifolds and some remarkable subclasses like para-Sasakian manifolds. In particular, in the recent years, many authors [5–9] have pointed out the importance of paracontact geometry and, in particular, of para-Sasakian geometry, by several papers giving the relationships with the theory of para-Kähler manifolds and its role in pseudo-Riemannian geometry and mathematical physics.These circumstances motivated us to initiate the study of lightlike geometry of submanifolds in almost paracontact metric manifolds. As a first step, in the present paper, we study the lightlike hypersurfaces of almost paracontact metric manifolds. We introduce the invariant lightlike hypersurfaces and screen semi-invariant lightlike hypersurfaces of almost paracontact metric manifolds, respectively, and give examples. Moreover, integrability conditions for the distributions involved in the definition of a screen semi-invariant lightlike hypersurface are investigated in case of the ambient manifold being para-Sasakian manifold.


## 2. Preliminaries

### 2.1. Almost Paracontact Metric Manifolds

A differentiable manifoldM- of dimension (2n+1) is called almost paracontact manifold with the almost paracontact structure (ϕ-,ξ,η) if it admits a tensor field ϕ- of type (1,1), a vector field ξ, and a 1-form η satisfying the following conditions [3]:
(1)ϕ-2=I-η⊗ξ,(2)η(ξ)=1,(3)ϕ-ξ=0,(4)η∘ϕ-=0,
where I denotes the identity transformation. Moreover, the tensor field ϕ- induces an almost paracomplex structure on the paracontact distribution D=ker⁡η; that is, the eigen distributions D± corresponding to the eigenvalues ±1 of ϕ- are both n-dimensional.If a(2n+1)-dimensional almost paracontact manifold M- with an almost paracontact structure (ϕ-,ξ,η) admits a pseudo-Riemannian metric g- such that [4]
(5)g-(ϕ-X,ϕ-Y)=-g-(X,Y)+η(X)η(Y),X,Y∈Γ(TM-),
then we say that M- is analmost paracontact metric manifold with analmost paracontact metric structure(ϕ-,ξ,η,g-) and such metric g- is calledcompatible metric. Any compatible metric g- is necessarily of signature (n+1,n).From (5) it can be easily seen that [4]
(6)g-(ϕ-X,Y)=-g-(X,ϕ-Y),(7)g-(X,ξ)=η(X),
for any X,Y∈Γ(TM-). The fundamental 2-form of M- is defined by
(8)Φ(X,Y)=g-(X,ϕ-Y).
An almost paracontact metric structure becomes a paracontact metric structure if g-(X,ϕ-Y)=dη(X,Y), for all vector fields X,Y∈Γ(TM-), where dη(X,Y)=(1/2){Xη(Y)-Yη(X)-η([X,Y])}.For a(2n+1)-dimensional manifold M- with an almost paracontact metric structure (ϕ-,ξ,η,g-) one can also construct a local orthonormal basis which is called ϕ--basis(Xi,ϕ-Xi,ξ)(i=1,2,…,n) [4].An almost paracontact metric structure(ϕ-,ξ,η,g-) is apara-Sasakian manifold if and only if [4]
(9)(∇-Xϕ-)Y=-g-(X,Y)ξ+η(Y)X,X,Y∈Γ(TM-),
where X,Y∈Γ(TM-) and ∇- is a Levi-Civita connection on M-.From (9), it can be seen that
(10)∇-Xξ=-ϕ-X.Example 1.
LetM-=ℝ2n+1 be the (2n+1)-dimensional real number space with standard coordinate system (x1,y1,x2,y2,…,xn,yn,z). Defining
(11)ϕ¯∂∂xα=∂∂yα,ϕ¯∂∂yα=∂∂xα,ϕ¯∂∂z=0,ξ=∂∂z,η¯=dz,g¯=η⊗η+∑α=1n‍dxα⊗dxα-∑α=1n‍dyα⊗dyα,
where α=1,2,…,n, the set (ϕ-,ξ,η,g-) is an almost paracontact metric structure on ℝ2n+1.


### 2.2. Lightlike Hypersurfaces

In this section, we recall some basic results about lightlike hypersurfaces of a semi-Riemannian manifold(M-,g-) [1].Let(M-,g-) be a (n+1)-dimensional semi-Riemannian manifold with index q  (0<q<n+1) and M a hypersurface of M-. Assume that the induced metric g=g-|M on the hypersurface is degenerate on M. Then there exists a vector field E≠0 on M such that
(12)g(E,X)=0,∀X∈Γ(TM).
The radical space [10] of TpM is defined by
(13)RadTpM={E∈RadTpM:g(E,X)=0,∀X∈Γ(TpM)},
whose dimension is called the nullity degree of g and (M,g) is calleda lightlike hypersurface of (M-,g-). Since g is degenerate and any null vector is perpendicular to itself, TpM⊥ is also degenerate and
(14)RadTpM=TpM∩TpM⊥.
For a lightlike hypersurface M, dim⁡TpM⊥=1 implies that
(15)dim⁡(RadTpM)=1,RadTpM=TpM⊥.
We call RadTpM the radical distribution and it is spanned by the null vector field E.Consider complementary vector bundleS(TM) of RadTM in TM. This means that
(16)TM=S(TM)⊥RadTM,
where ⊥ denotes the orthogonal direct sum. S(TM) is called thescreen distribution on M. Since the screen distribution S(TM) is nondegenerate, there exists a complementary orthogonal vector subbundle S(TM)⊥ to S(TM) in TM- which is calledscreen transversal subbundle; that is,
(17)TM-=S(TM)⊥S(TM)⊥.
The rank of S(TM)⊥ is 2.Theorem 2 (see [1]).
Let(M,g,S(TM)) be a lightlike hypersurface of an almost paracontact manifold (M-,ϕ-,ξ,η,g-). Then there exists a unique rank one vector subbundle ltr(TM) of  TM-, with base space M, such that, for any nonzero section E of RadTM on a coordinate neighborhood U⊂M, there exists a unique section N of ltr(TM) on U satisfying
(18)g-(N,E)=1,g-(N,N)=0,g-(N,W)=0,
for all W∈Γ(S(TM))|U. ltr(TM) is called the lightlike transversal vector bundle of M with respect to S(TM).One can consider the following decompositions:(19)S(TM)⊥=RadTM⊕ltr(TM),(20)TM-|M=S(TM)⊥{RadTM⊕ltr(TM)}=TM⊕ltr(TM).
Let ∇- be the Levi-Civita connection on M-. Using (20) we deduce
(21)∇-XY=∇XY+h(X,Y),(22)∇-XN=-ANX+∇XtN,
for any X,Y∈Γ(TM) and N∈Γ(ltr(TM)). Then ∇- and ∇t are called the induced connection on M and ltr(TM), respectively, and, as in the classical theory of Riemannian hypersurfaces, h and AN are called the second fundamental form and the shape operator, respectively. The above equations are cited as the Gauss and Weingarten equation, respectively [1].Locally, letE, N, and U be as in Theorem 2. Then, for any X,Y∈Γ(TM|U), putting
(23)B(X,Y)=g-(h(X,Y),E),τ(X)=g-(∇XtN,E),
we can write
(24)∇-XY=∇XY+B(X,Y)N,(25)∇-XN=-ANX+τ(X)N.B is called the local second fundamental form of M, because it determines h on U. Moreover, B is degenerate and
(26)B(X,E)=0,
for any X∈Γ(TM|U).The decomposition (16) allows to define a canonical projection P:Γ(TM)→Γ(S(TM)). For each X∈Γ(TM), we may write
(27)X=PX+θ(X)E,
where θ is a 1-form given by
(28)θ(X)=g-(X,N).
From (24), for all X,Y,Z∈Γ(TM), we get
(29)(∇Xg)(Y,Z)=B(X,Y)θ(Z)+B(X,Z)θ(Y),
which implies that the induced connection ∇ is a nonmetric connection on M.Then for anyX,Y∈Γ(TM) and E∈Γ(TM⊥) we can write
(30)∇XPY=∇X*PY+h*(X,PY),(31)∇XE=-AE*X+∇X*tE,
where ∇* and ∇*t are linear connections on the bundles S(TM) and Rad(TM), respectively. Further, h* and A* are called the second fundamental form and the shape operator of the screen distribution, respectively. Locally, let U be a coordinate neighborhood of M and E, N sections on U, as in Theorem 2. Then, putting C(X,PY)=g-(h*(X,PY),N), for any X,Y∈Γ(TM|U), one has
(32)h*(X,PY)=C(X,PY)E,g-(∇X*tE,N)=-τ(X),
and, locally on U, (30) and (31) become
(33)∇XPY=∇X*PY+C(X,PY)E,∇XE=-AE*X-τ(X)E.The local second fundamental formsB and C, respectively, of M and on S(TM) are related to their shape operators by
(34)g(AE*X,PY)=B(X,PY),g(AE*X,N)=0,(35)g(ANX,PY)=C(X,PY),g(ANX,N)=0.
Furthermore, one has AE*X=0, ∇-EE=∇EE=-τ(E)E. (For more details we refer to [1, 2].)


## 2.1. Almost Paracontact Metric Manifolds

A differentiable manifoldM- of dimension (2n+1) is called almost paracontact manifold with the almost paracontact structure (ϕ-,ξ,η) if it admits a tensor field ϕ- of type (1,1), a vector field ξ, and a 1-form η satisfying the following conditions [3]:
(1)ϕ-2=I-η⊗ξ,(2)η(ξ)=1,(3)ϕ-ξ=0,(4)η∘ϕ-=0,
where I denotes the identity transformation. Moreover, the tensor field ϕ- induces an almost paracomplex structure on the paracontact distribution D=ker⁡η; that is, the eigen distributions D± corresponding to the eigenvalues ±1 of ϕ- are both n-dimensional.If a(2n+1)-dimensional almost paracontact manifold M- with an almost paracontact structure (ϕ-,ξ,η) admits a pseudo-Riemannian metric g- such that [4]
(5)g-(ϕ-X,ϕ-Y)=-g-(X,Y)+η(X)η(Y),X,Y∈Γ(TM-),
then we say that M- is analmost paracontact metric manifold with analmost paracontact metric structure(ϕ-,ξ,η,g-) and such metric g- is calledcompatible metric. Any compatible metric g- is necessarily of signature (n+1,n).From (5) it can be easily seen that [4]
(6)g-(ϕ-X,Y)=-g-(X,ϕ-Y),(7)g-(X,ξ)=η(X),
for any X,Y∈Γ(TM-). The fundamental 2-form of M- is defined by
(8)Φ(X,Y)=g-(X,ϕ-Y).
An almost paracontact metric structure becomes a paracontact metric structure if g-(X,ϕ-Y)=dη(X,Y), for all vector fields X,Y∈Γ(TM-), where dη(X,Y)=(1/2){Xη(Y)-Yη(X)-η([X,Y])}.For a(2n+1)-dimensional manifold M- with an almost paracontact metric structure (ϕ-,ξ,η,g-) one can also construct a local orthonormal basis which is called ϕ--basis(Xi,ϕ-Xi,ξ)(i=1,2,…,n) [4].An almost paracontact metric structure(ϕ-,ξ,η,g-) is apara-Sasakian manifold if and only if [4]
(9)(∇-Xϕ-)Y=-g-(X,Y)ξ+η(Y)X,X,Y∈Γ(TM-),
where X,Y∈Γ(TM-) and ∇- is a Levi-Civita connection on M-.From (9), it can be seen that
(10)∇-Xξ=-ϕ-X.Example 1.
LetM-=ℝ2n+1 be the (2n+1)-dimensional real number space with standard coordinate system (x1,y1,x2,y2,…,xn,yn,z). Defining
(11)ϕ¯∂∂xα=∂∂yα,ϕ¯∂∂yα=∂∂xα,ϕ¯∂∂z=0,ξ=∂∂z,η¯=dz,g¯=η⊗η+∑α=1n‍dxα⊗dxα-∑α=1n‍dyα⊗dyα,
where α=1,2,…,n, the set (ϕ-,ξ,η,g-) is an almost paracontact metric structure on ℝ2n+1.


## 2.2. Lightlike Hypersurfaces

In this section, we recall some basic results about lightlike hypersurfaces of a semi-Riemannian manifold(M-,g-) [1].Let(M-,g-) be a (n+1)-dimensional semi-Riemannian manifold with index q  (0<q<n+1) and M a hypersurface of M-. Assume that the induced metric g=g-|M on the hypersurface is degenerate on M. Then there exists a vector field E≠0 on M such that
(12)g(E,X)=0,∀X∈Γ(TM).
The radical space [10] of TpM is defined by
(13)RadTpM={E∈RadTpM:g(E,X)=0,∀X∈Γ(TpM)},
whose dimension is called the nullity degree of g and (M,g) is calleda lightlike hypersurface of (M-,g-). Since g is degenerate and any null vector is perpendicular to itself, TpM⊥ is also degenerate and
(14)RadTpM=TpM∩TpM⊥.
For a lightlike hypersurface M, dim⁡TpM⊥=1 implies that
(15)dim⁡(RadTpM)=1,RadTpM=TpM⊥.
We call RadTpM the radical distribution and it is spanned by the null vector field E.Consider complementary vector bundleS(TM) of RadTM in TM. This means that
(16)TM=S(TM)⊥RadTM,
where ⊥ denotes the orthogonal direct sum. S(TM) is called thescreen distribution on M. Since the screen distribution S(TM) is nondegenerate, there exists a complementary orthogonal vector subbundle S(TM)⊥ to S(TM) in TM- which is calledscreen transversal subbundle; that is,
(17)TM-=S(TM)⊥S(TM)⊥.
The rank of S(TM)⊥ is 2.Theorem 2 (see [1]).
Let(M,g,S(TM)) be a lightlike hypersurface of an almost paracontact manifold (M-,ϕ-,ξ,η,g-). Then there exists a unique rank one vector subbundle ltr(TM) of  TM-, with base space M, such that, for any nonzero section E of RadTM on a coordinate neighborhood U⊂M, there exists a unique section N of ltr(TM) on U satisfying
(18)g-(N,E)=1,g-(N,N)=0,g-(N,W)=0,
for all W∈Γ(S(TM))|U. ltr(TM) is called the lightlike transversal vector bundle of M with respect to S(TM).One can consider the following decompositions:(19)S(TM)⊥=RadTM⊕ltr(TM),(20)TM-|M=S(TM)⊥{RadTM⊕ltr(TM)}=TM⊕ltr(TM).
Let ∇- be the Levi-Civita connection on M-. Using (20) we deduce
(21)∇-XY=∇XY+h(X,Y),(22)∇-XN=-ANX+∇XtN,
for any X,Y∈Γ(TM) and N∈Γ(ltr(TM)). Then ∇- and ∇t are called the induced connection on M and ltr(TM), respectively, and, as in the classical theory of Riemannian hypersurfaces, h and AN are called the second fundamental form and the shape operator, respectively. The above equations are cited as the Gauss and Weingarten equation, respectively [1].Locally, letE, N, and U be as in Theorem 2. Then, for any X,Y∈Γ(TM|U), putting
(23)B(X,Y)=g-(h(X,Y),E),τ(X)=g-(∇XtN,E),
we can write
(24)∇-XY=∇XY+B(X,Y)N,(25)∇-XN=-ANX+τ(X)N.B is called the local second fundamental form of M, because it determines h on U. Moreover, B is degenerate and
(26)B(X,E)=0,
for any X∈Γ(TM|U).The decomposition (16) allows to define a canonical projection P:Γ(TM)→Γ(S(TM)). For each X∈Γ(TM), we may write
(27)X=PX+θ(X)E,
where θ is a 1-form given by
(28)θ(X)=g-(X,N).
From (24), for all X,Y,Z∈Γ(TM), we get
(29)(∇Xg)(Y,Z)=B(X,Y)θ(Z)+B(X,Z)θ(Y),
which implies that the induced connection ∇ is a nonmetric connection on M.Then for anyX,Y∈Γ(TM) and E∈Γ(TM⊥) we can write
(30)∇XPY=∇X*PY+h*(X,PY),(31)∇XE=-AE*X+∇X*tE,
where ∇* and ∇*t are linear connections on the bundles S(TM) and Rad(TM), respectively. Further, h* and A* are called the second fundamental form and the shape operator of the screen distribution, respectively. Locally, let U be a coordinate neighborhood of M and E, N sections on U, as in Theorem 2. Then, putting C(X,PY)=g-(h*(X,PY),N), for any X,Y∈Γ(TM|U), one has
(32)h*(X,PY)=C(X,PY)E,g-(∇X*tE,N)=-τ(X),
and, locally on U, (30) and (31) become
(33)∇XPY=∇X*PY+C(X,PY)E,∇XE=-AE*X-τ(X)E.The local second fundamental formsB and C, respectively, of M and on S(TM) are related to their shape operators by
(34)g(AE*X,PY)=B(X,PY),g(AE*X,N)=0,(35)g(ANX,PY)=C(X,PY),g(ANX,N)=0.
Furthermore, one has AE*X=0, ∇-EE=∇EE=-τ(E)E. (For more details we refer to [1, 2].)


## 3. Lightlike Hypersurfaces of Para-Sasakian Manifolds

Let(M-,ϕ-,ξ,η,g-) be a (2n+1)-dimensional para-Sasakian manifold and M a lightlike hypersurface of M- such that the structure vector field ξ is tangent to M. For local sections E and N of Rad(TM) and ltr(TM), respectively, in view of (7) we have
(36)η(E)=0,η(N)=0.
From (6) it is easy to see that ϕ-E and ϕ-N are lightlike vector fields and
(37)ϕ-2E=E,ϕ-2N=N.Now, for anyX∈Γ(TM), we write
(38)ϕ-X=ϕX+u(X)N,
where ϕX∈Γ(TM) and
(39)u(X)=g-(ϕ-X,E)=-g-(X,ϕ-E).Proposition 3.
Let(M-,ϕ-,ξ,η,g-) be a (2n+1)-dimensional para-Sasakian manifold and M a lightlike hypersurface of M- such that the structure vector field ξ is tangent to M. Then one has
(40)g-(ϕ-E,N)=-g(ANE,ξ),
where E is any local section of Rad(TM) and N is any local section of ltr(TM).Proof.
From (10) and (18) we have
(41)g-(ϕ-E,N)=-g-(∇-Eξ,N)=g-(ξ,∇-EN),
which gives (40) by virtue of (22).Remark 4.
From (6) we get g-(ϕ-E,E)=0, which implies that there is no component of ϕ-E in ltr(TM) and so ϕ-E∈Γ(TM). Moreover, (40) implies that there may be a component of ϕ-E in Rad(TM). Thus, in view of (27), we observe that
(42)ϕ-E=ϕE=Pϕ-E+θ(ϕ-E)E.Proposition 5.
Let(M-,ϕ-,ξ,η,g-) be a (2n+1)-dimensional para-Sasakian manifold and M a lightlike hypersurface of M- such that the structure vector field ξ is tangent to M. Then one has
(43)g(X,ϕY)=-g(ϕX,Y)-(u⊗θ+θ⊗u)(X,Y),(44)g(ϕX,ϕY)=-g(X,Y)+η(X)η(Y)-u(X)θ(ϕY)-u(Y)θ(ϕX),
for any X,Y∈Γ(TM).Proof.
By using (38) and (39) we obtain
(45)g-(ϕ-X,Y)=g(X,ϕY)+u(Y)g(X,N).
Hence in view of (6) we get (43). From (38) we have
(46)g-(ϕ-X,ϕ-Y)=g(ϕX,ϕY)+u(Y)θ(ϕX)+u(X)θ(ϕY).
Thus, by using the last equation above and (5), we complete the proof.Corollary 6.
LetM be a lightlike hypersurface of a para-Sasakian manifold such that the structure vector field ξ is tangent to M. Then, for all X∈Γ(TM), one has g(ξ,ϕX)=0.Proposition 7.
LetM be a lightlike hypersurface of a (2n+1)-dimensional para-Sasakian manifold (M-,ϕ-,ξ,η,g-) such that the structure vector field ξ is tangent to M. Then, for any X∈Γ(TM), one has
(47)ϕ2X=X-η(X)ξ-u(ϕX)N-u(X)ϕ-N,(48)∇Xξ=-ϕX,(49)B(X,ξ)=-u(X).Proof.
From (1) and (38), we get (47). Next, by using (10), (24), and (38), we have
(50)∇Xξ+B(X,ξ)=-ϕX-u(X)N.
Then by equating the tangential and transversal parts in the previous equation we get (48) and (49), respectively.


## 4. Invariant Lightlike Hypersurfaces of Almost Paracontact Metric Manifolds

We begin with the following.Definition 8.
Let(M-,ϕ-,ξ,η,g-) be a (2n+1)-dimensional para-Sasakian manifold and M a lightlike hypersurface of M- such that the structure vector field ξ is tangent to M. If ϕ-(S(TM))⊆S(TM), then M will be called an invariant lightlike hypersurface of M-.Example 9.
LetM-=ℝ5 be the 5-dimensional real number space with a coordinate system (x1,y1,x2,y2,z). Define a frame of vector fields on M- given by Weƚyczko [11]:
(51)e1=∂∂x1,e2=∂∂x2,e3=11+y12∂∂x1+∂∂y1-2x1∂∂z,e4=11+y12∂∂x2+∂∂y2-2x2∂∂z,e5=∂∂z.
Defining
(52)η=2x1dy1+2x2dy2+dz,ξ=e5,ϕ-e1=-e1,ϕ-e2=-e2,ϕ-e3=e3,ϕ-e4=e4,ϕ-e5=e5,g¯(e1,e3)=g¯(e3,e1)=g¯(e2,e4)=g¯(e4,e2)=g¯(e5,e5)=1,g¯(ei,ej)=0,otherwise,
the set (ϕ-,ξ,η,g-) is an almost paracontact metric structure on M- with index (g-)=2. Consider a hypersurface M of M- given by
(53)x1=arctany1.
It is easy to check that M is a lightlike hypersurface and
(54)Rad(TM)=Span⁡{E=e3}.
Then the lightlike transversal vector bundle ltr(TM) is spanned by
(55)N=e1.
It follows that corresponding screen distribution S(TM) is spanned by
(56){W1=e2,W2=e4,W3=e5}.
We easily check that
(57)ϕ-W1=-W1,ϕ-W2=W2,ϕ-W3=0,which givesϕ-(S(TM))⊆S(TM). Thus M is an invariant lightlike hypersurface of M-.Now we give a characterization of an invariant lightlike hypersurface.Theorem 10.
Let(M-,ϕ-,ξ,η,g-) be a (2n+1)-dimensional para-Sasakian manifold and M a lightlike hypersurface of M- such that the structure vector field ξ is tangent to M. Then M is an invariant lightlike hypersurface of M- if and only if
(58)ϕ-RadTM=RadTM,ϕ-ltrTM=ltrTM.Proof.
LetM be an invariant lightlike hypersurface of M-. From (27) and (42), for any X∈Γ(TM), we get g(Pϕ-E,PX)=0; that is, there is no component of ϕ-E in S(TM). Moreover, it is obvious from (6) that ϕ-E has no component in ltrTM and so ϕ-RadTM=RadTM.
On the other hand, for any local sectionN of ltrTM, we can write
(59)ϕ-N=Pϕ-N+g-(ϕ-N,E)N.
By using the previous equation we have g(Pϕ-N,PX)=0, for any X∈Γ(TM), which implies that ϕ-N has no component in S(TM). Since g¯(ϕ-N,N)=0, then it is also seen that there is no component of ϕ-N in RadTM. Hence ϕ-ltrTM=ltrTM.
Conversely, letϕ-RadTM=RadTM and ϕ-ltrTM=ltrTM. For any X∈Γ(S(TM)) we have
(60)g¯(ϕ-X,E)=-g¯(X,ϕ-E)=0,
which implies that ϕ-X has no component in ltrTM. Similarly, we get
(61)g¯(ϕ-X,N)=-g¯(X,ϕ-N)=0;
thus there is no component of ϕ-X in RadTM. The proof is completed.Corollary 11.
Let(M-,ϕ-,ξ,η,g-) be a (2n+1)-dimensional para-Sasakian manifold and M a lightlike hypersurface of M- such that the structure vector field ξ is tangent to M. Then M is an invariant lightlike hypersurface of M- if and only if
(62)ϕ-E=±E,ϕ-N=±N.Theorem 12.
Let(M-,ϕ-,ξ,η,g-) be a (2n+1)-dimensional almost paracontact metric manifold and M an invariant lightlike hypersurface of M-. Then (M,ϕ,ξ,η,g) is an almost paracontact metric manifold.Proof.
LetM be an invariant lightlike hypersurface of M¯. For any X,Y∈Γ(TM), from (38), we get
(63)ϕ-X=ϕX.
Using (1) and (63), we have
(64)ϕ2X=X-η(X)ξ.
Also from (63), it follows that
(65)ϕξ=0.
Next, in view of (64) and (65) one can easily see that
(66)η∘ϕ=0,η(ξ)=1.
Moreover, from (44), we have
(67)g(ϕX,ϕY)=-g(X,Y)+η(X)η(Y).
From (64)-(67) we complete the proof.Proposition 13.
LetM be an invariant lightlike hypersurface of a para-Sasakian manifold (M-,ϕ-,ξ,η,g-). Then we have
(68)g(ANX,ξ)=0,
for any X∈Γ(TM).Proof.
Sinceg(ξ,N)=0, using (10), we get
(69)g¯(∇¯XN,ξ)=g¯(N,ϕ¯X).
From (25), we have the assertion of the proposition.Theorem 14.
An invariant lightlike hypersurface of a para-Sasakian manifold is always para-Sasakian. Moreover,(70)B(X,ϕY)N-B(X,Y)ϕN=0,(71)ϕ(ANX)=AϕNX-θ(X)ξ,
for any X,Y∈Γ(TM).Proof.
We have(72)(∇¯Xϕ¯)Y=∇¯Xϕ¯Y-ϕ¯(∇¯XY)=∇¯XϕY-ϕ¯(∇XY+B(X,Y)N)=∇XϕY+B(X,ϕY)N-ϕ∇XY-B(X,Y)ϕN=(∇Xϕ)Y+B(X,ϕY)N-B(X,Y)ϕN,
which in view of (9) gives
(73)-g(X,Y)ξ+η(Y)X=(∇Xϕ)Y+B(X,ϕY)N-B(X,Y)ϕN.
Equating tangential parts in (73) provides
(74)(∇Xϕ)Y=-g(X,Y)ξ+η(Y)X.
In view of (74) and Theorem 12, we see that M is a para-Sasakian manifold. Equating transversal parts in (73) yields (70). Next, using (9) and (25), we have
(75)-θ(X)ξ=(∇¯Xϕ¯)N=∇¯Xϕ¯N-ϕ¯(∇¯XN)=-Aϕ¯NX+g¯(∇¯Xϕ¯N,E)N+ϕ¯(ANX)-τ(X)ϕ¯N.
In the last equation, if we equate the tangential parts, we get (71).This completes the proof.Remark 15.
It is well known that, if there exists a lightlike hypersurface in an indefinite Sasakian manifold, then the dimension of the indefinite Sasakian manifold must be equal or greater than5. But in a paracontact metric manifold there is not such a restriction in the dimension of the ambient manifold for the existence of lightlike hypersurfaces.Let(M-,ϕ-,ξ,η,g-) be a 3-dimensional almost paracontact metric manifold and M a lightlike surface of M- such that the structure vector field ξ is tangent to M. Since ξ is a nonnull vector field, it belongs to the screen distribution S(TM). Thus, {ξ,E,N} is a quasi-orthonormal frame of TM-. Also with respect to the quasi-orthonormal frame {ξ,E,N} of TM- we can write
(76)ϕ-E=aξ+bE+cN,ϕ-N=dξ+eE+fN,
where a=g¯(ϕ-E,ξ)=0, b=g¯(ϕ-E,N), c=g¯(ϕ-E,E)=0, d=g¯(ϕ-N,ξ)=0, e=g¯(ϕ-N,N)=0, and f=g¯(ϕ-E,N)=b. Thus we have
(77)ϕ-E=bE,ϕ-N=bN.
Hence, a lightlike surface M of M-, tangent to the structure vector field ξ, is always an invariant lightlike surface.Example 16.
LetM-=ℝ3 be a 3-dimensional almost paracontact metric manifold with the structure (ϕ-,ξ,η,g-) given in Example 1. Consider a surface M of M- given by x1=y1. It is easy to check that M is a lightlike surface and Rad(TM) and ltr(TM) are given by
(78)RadTM=Span⁡{E=∂∂x1+∂∂y1},ltrTM=Span⁡{N=12(∂∂x1-∂∂y1)},
respectively. It follows that corresponding screen distribution is spanned by
(79){ξ=∂∂z}.
Then ϕ-E=E and ϕ-N=-N, which imply that M is an invariant lightlike surface.Example 17.
Let(J,G) be the standard flat para-Kähler structure on ℝn2n+2,
(80)J(∂∂xα)=∂∂xα+n+1,J(∂∂xα+n+1)=∂∂xα,G(∂∂xα,∂∂xα)=1,G(∂∂xα+n+1,∂∂xα+n+1)=-1,
for α=1,…,n, where (x1,x2,…,x2n+2) are the Cartesian coordinates on ℝn2n+2. Consider the hypersurface Hn2n+1 in ℝn+12n+2 given by the equation
(81)Hn2n+1={x∈ℝn+12n+2:∑α=1n+1‍xa2-∑α=n+22n+2‍xa2=-1}.
Let N=∑α=n+22n+2‍xi(∂/∂xi) be the normal vector field of Hn2n+1. Then G(N,N)=-1.
Define a vector fieldξ, a tensor field ϕ- of type (1,1), a 1-form η, and a pseudo-Riemannian metric g- on Hn2n+1 by assuming
(82)J(X)=ϕ-X-η(X)N,ξ=-JN,g-=G|Hn2n+1.
Then we get an almost paracontact metric structure (ϕ-,ξ,η,g-) on Hn2n+1. Moreover, this structure is para-Sasakian [11]. Now cut H13 by the hyperplane x1-x3=0 and obtain a lightlike surface M of H13 with
(83)RadTM=Span⁡{E=U1},S(TM)=Span⁡{W=-x1U1-U2},
where U1=(∂/∂x1)+(∂/∂x3), U2=x4(∂/∂x2)+x2(∂/∂x4)∈Γ(TM). It follows that the lightlike transversal bundle ltr(TM) is spanned by
(84)N=12{∂∂x1-2x1x2+x4∂∂x2-∂∂x3+2x1x2+x4∂∂x4}.
Furthermore,
(85)ϕ-E=E,ϕ-N=-N.
Thus M is an invariant surface of H13.


## 5. Screen Semi-Invariant Lightlike Hypersurfaces of Almost Paracontact Metric Manifolds

IfE is local section of Γ(Rad(TM)), one has g-(ϕ-E,E)=0; therefore ϕ-E∈Γ(TM) and we get a 1-dimensional distribution ϕ-(Rad(TM)) on M.Definition 18.
Let(M-,ϕ-,ξ,η,g-) be a (2n+1)-dimensional almost paracontact metric manifold and M a lightlike hypersurface of M-. If
(86)ϕ-(Rad(TM))⊂S(TM),ϕ-(ltr(TM))⊂S(TM),
then M will be called a screen semi-invariant lightlike hypersurface of M-.Example 19.
LetM-=ℝ7 be a 7-dimensional almost paracontact metric manifold with the structure (ϕ-,ξ,η,g-) given in Example 1. Consider a hypersurface M of M- given by
(87)x1=y1+x2+y2+x3+y3.
Then the tangent bundle TM of M is spanned by
(88){U1=∂∂x1+∂∂y1,U2=∂∂x1+∂∂x2,U3=∂∂x1+∂∂y2,U4=∂∂x1+∂∂x3,U5=∂∂x1+∂∂y3,U6=∂∂z∂∂x1+∂∂y1}.
The radical distribution RadTM and the lightlike transversal bundle ltr(TM) are given by
(89)Rad(TM)=Span⁡{E=U1-U2+U3-U4+U5},ltr(TM)=Span⁡{N=2∂∂x1+32∂∂y1+∂∂y2-12∂∂x3-∂∂y3}.
It follows that the screen distribution S(TM) is spanned by {W1,W2,W3,W4,W5}, where
(90)W1=U6,W2=U1+U2-U3+U4-U5,W3=U1-U3+U5,W4=-2U1-U2+U4+12U5,W5=4U1+U2-U4.
Furthermore,
(91)ϕ-E=W2∈Γ(S(TM)),ϕ-N=-W4∈Γ(S(TM)).Thus,M is a screen semi-invariant lightlike hypersurface of M-.From now on, we will write(M,g,S(TM)) to denote a screen semi-invariant lightlike hypersurface, together with the choices of a fixed nonzero section E of RadTM, a fixed screen distribution S(TM), ltr(TM), and N as in Theorem 2.SinceM is a screen semi-invariant lightlike hypersurface, then we have ϕ-N∈S(TM) and
(92)g-(ϕ-N,E)=-g-(N,ϕ-E)=0,g-(ϕ-N,N)=0,
which imply that ϕ-N is orthogonal to S(TM)⊥ by virtue of (19). Also, from (5), we obtain
(93)g-(ϕ-N,ϕ-E)=-1.
Therefore, ϕ-(Rad(TM))⊕ϕ-(ltr(TM)) is a nondegenerate vector subbundle of S(TM) of rank 2.In the following, beingS(TM) and ϕ-(Rad(TM))⊕ϕ-(ltr(TM)) nondegenerate, we can define the unique nondegenerate distribution D0 such that [2]
(94)S(TM)={ϕ-(Rad(TM))⊕ϕ-(ltr(TM))}⊥D0.
Then ξ∈Γ(D0) and D0 is invariant under ϕ-; that is, ϕ-(D0)=D0. Moreover, from (16), (20), and (94) we write
(95)TM={ϕ-(Rad(TM))⊕ϕ-(ltr(TM))}⊥D0⊥Rad(TM),(96)TM-={ϕ-(Rad(TM))⊕ϕ-(ltr(TM))}⊥D0⊥{Rad(TM)⊕ltr(TM)}.For an almost paracontact metric manifold(M-,ϕ-,ξ,η,g-), we construct a useful local orthonormal basis. Let U be a coordinate neighborhood on M- and e1 any unit vector field on U orthogonal to ξ. Then ϕ-e1 is a vector field orthogonal to both e1 and ξ, and g-(ϕ-e1,ϕ-e1)=-1. Choose a unit vector field e2 orthogonal to ξ, e1, and ϕ-e1. Then ϕ-e2 is also orthogonal to ξ, e1,e2, and ϕ-e1 and g-(ϕ-e2,ϕ-e2)=-1. Proceeding in this way we obtain a set of local orthonormal vector fields {ei,ϕ-ei,ξ}(i=1,2,…,n-2). Now construct the unit vector field en-1=(E+N)/2 orthogonal to ξ, ei, and ϕ-ei  (i=1,2,…,n-2). Then ϕ-en-1=(ϕ-E+ϕ-N)/2 is also orthogonal to ξ, ei, and ϕ-ei  (i=1,2,…,n-2) and g-(ϕ-en-1,ϕ-en-1)=-1. By a similar way set a unit vector field en=(ϕ-E-ϕ-N)/2 orthogonal to ξ, ei, and ϕ-ei(i=1,2,…,n-1). It is easy to see that ϕ-en=(E-N)/2 is also orthogonal to ξ, ei, ϕ-ei(i=1,2,…,n-1) and g-(ϕ-en,ϕ-en)=-1. Hence, from a quasi-orthonormal basis {ei,ϕ-ei,E,N,ϕ-E,ϕ-N,ξ}(i=1,2,…,n-2) of M-, we obtain a local orthonormal basis
(97){ei,en-1=E+N2,en=ϕ-E-ϕ-N2,ϕ-ei,ϕ-en-1=ϕ-E+ϕ-N2,ϕ-en=E-N2,ξ},
where i=1,2,…,n-2, called ϕ--basis. Thus we have the following.Proposition 20.
Let(M-,ϕ-,ξ,η,g-) be a (2n+1)-dimensional almost paracontact metric manifold and M a screen semi-invariant lightlike hypersurface of M-. Then there always exists a ϕ--basis on M- generated from a quasi-orthonormal basis {ei,ϕ-ei,E,N,ϕ-E,ϕ-N,ξ},i=1,2,…,n-2, of M-.Now, we consider the distributionsD=Rad(TM)⊥ϕ-(Rad(TM))⊥D0, D′=ϕ-(ltr(TM)) on M. Then D is a ϕ--invariant distribution and we have
(98)TM=D⊕D′.
Thus, every X∈Γ(TM) can be expressed by
(99)X=RX+QX,
where R and Q are the projections of TM into D and D′, respectively. Hence, we may write ϕX=ϕ¯RX, for any X∈Γ(TM). Let us consider the local lightlike vector fields U=ϕ-N∈Γ(ϕ-(ltr(TM)))and V=ϕ-E∈ΓD. From (1), (38), and (39), we obtain
(100)ϕ¯2X=ϕ2X+u(X)U+u(ϕX)N.
By comparing the tangential and transversal parts in (100) we get
(101)ϕ2=I-η⊗ξ-u⊗U,(102)u∘ϕ=0,
respectively. Next, from (3) one can easily see that
(103)ϕξ=0,u(ξ)=0.
Since ϕ¯2N=N, by using (38), we also have
(104)ϕU=0,u(U)=1.
Furthermore, from (4), we have
(105)η(U)=0.
Finally, we get
(106)(η∘ϕ)X=η(ϕ¯X-u(X)N),
which gives
(107)η∘ϕ=0.Thus we have the following.Proposition 21.
LetM be a screen semi-invariant lightlike hypersurface of an almost paracontact metric manifold (M-,ϕ-,ξ,η,g-). Then M possesses a para (ϕ,ξ,η,U,u)-structure; that is,
(108)ϕ2=I-η⊗ξ-u⊗U,ϕξ=0,ϕU=0,η∘ϕ=0,u∘ϕ=0,η(ξ)=1,u(U)=1,η(U)=0,u(ξ)=0.Theorem 22.
LetM be a screen semi-invariant lightlike hypersurface of a para-Sasakian manifold (M-,ϕ-,ξ,η,g-). Then one has
(109)(∇Xϕ)Y=-g(X,Y)ξ+η(Y)X+u(Y)ANX+B(X,Y)U,(110)(∇Xu)Y=-B(X,ϕY)-u(Y)τ(X),
for all X,Y∈Γ(TM).Proposition 23.
Let(M-,ϕ-,ξ,η,g-) be a para-Sasakian manifold and (M,g,S(TM)) a screen semi-invariant lightlike hypersurface of M-. Then M is totally geodesic if and only if, for any X∈Γ(TM) and for Y∈Γ(D),
(111)(∇Xϕ)Y=-g(X,Y)ξ+η(Y)X,(112)ANX=-ϕ∇Xu+g(X,U)ξ.Proof.
Let assume thatM is totally geodesic; that is, for any X,Y∈Γ(TM), B(X,Y)=0. Then, for Y∈Γ(D) using u(Y)=0 in (109) we have
(113)(∇Xϕ)Y=-g(X,Y)ξ+η(Y)X.
Similarly, using (109) we have
(114)(∇Xϕ)U=ANX-g(X,U)ξ+η(U)X,
which gives
(115)ANX=-ϕ∇XU+g(X,U)ξ,
by virtue of ϕU=0 and η(U)=0.
Conversely, suppose that the conditions (111) and (112) hold and we will prove that B vanishes. If Y∈Γ(TM), using the decomposition (98), there exists α∈ℑ(U) such that
(116)Y=YD+αU
and for any X∈Γ(TM) we obtain
(117)B(X,Y)=B(X,YD)+αB(X,U).
For Y=YD, using (109) and (111), we find
(118)B(X,YD)=u(YD)ANX=0,
which implies that B(X,YD)=0.
Also, for anyY=U from (109), using (112), we get
(119)B(X,U)U=0,
which implies B(X,U)=0. This completes the proof.Proposition 24.
Let(M-,ϕ-,ξ,η,g-) be a para-Sasakian manifold and (M,g,S(TM)) a screen semi-invariant lightlike hypersurface of M-. Then one has, for any X∈Γ(TM),(i)
if the vector fieldUis parallel, then(120)ANX=η(ANX)ξ+u(ANX)U,τ(X)=0;(ii)
if the vector fieldVis parallel, then(121)AE*X=η(AE*X)ξ+u(AE*X)U,τ(X)=0,where U=ϕ-N and V=ϕ-E.


## 6. Integrability of Distributions on a Screen Semi-Invariant Lightlike Hypersurface of a Para-Sasakian Manifold

### 6.1. The DistributionD0

Firstly, we consider the distributionD0, defined in (94). Using (95) and putting μ={ϕ-(Rad(TM))⊕ϕ-(ltr(TM))}⊥Rad(TM), for any X∈Γ(TM), Y∈Γ(D0), and Z∈Γ(μ) we have
(122)∇XY=∇°XY+h°(X,Y),(123)∇XZ=-A°ZX+∇XμZ,
where ∇° is a linear connection on the bundle D0, h°:Γ(TM)×Γ(D0)→Γ(μ) is an ℑ(M) bilinear, A° is an ℑ(M) linear operator on Γ(D0), respectively, and ∇μ is a linear connection on μ.Lemma 25.
Let(M,g,S(TM)) be a screen semi-invariant lightlike hypersurface of a para-Sasakian manifold (M-,ϕ-,ξ,η,g-) and U⊂M a coordinate neighborhood as fixed in Theorem 2. Then, for any X,Y∈Γ(D0), one has
(124)g-((∇-Xϕ-)Y,E)=0,(125)g-((∇-Xϕ-)Y,N)=0.Proof.
Calculation is straightforward by using (9).LetU⊂M be a coordinate neighborhood as fixed in Theorem 2. Then according to decomposition given by (95) we set
(126)α1(X,Y)=-g(h°(X,Y),ϕ-N),α2(X,Y)=-g(h°(X,Y),ϕ-E),α3(X,Y)=g(h°(X,Y),N),
for any X,Y∈Γ(D0|U). So (122) can be written locally as
(127)∇XY=∇°XY+α1(X,Y)ϕ-E+α2(X,Y)ϕ-N+α3(X,Y)E.
We will express α1, α2, and α3 in terms of B and C. Firstly, we compute
(128)g(∇XY,ϕ-N)=g(∇°XY+α1(X,Y)ϕ-E+α2(X,Y)ϕ-N+α3(X,Y)E,ϕ-N∇°XY)=-α1(X,Y).
Then, by using (125), (24) and (25), we get
(129)g(∇XY,ϕ-N)=-g-(ϕ-∇XY,N)=-g-(ϕ-∇-XY,N)=g-((∇-Xϕ-)Y,N)-g-(∇-Xϕ-Y,N)=-g-(∇-Xϕ-Y,N)=g-(ϕ-Y,∇-XN)=-g-(ANX,ϕ-Y)=-C(X,ϕ-Y),
in view of D0 being ϕ--invariant and ∇- being a metric connection.Next we have(130)g(∇XY,ϕ-N)=g(∇°XY+α1(X,Y)ϕ-E+α2(X,Y)ϕ-N+α3(X,Y)E,ϕ-E∇°X)=-α2(X,Y),
And, using (124), (24), and (31) in the previous equation, we obtain
(131)g(∇XY,ϕ-E)=-g-(ϕ-∇XY,E)=-g-(ϕ-∇-XY,E)=g-((∇-Xϕ-)Y,E)-g-(∇-Xϕ-Y,E)=-g-(∇-Xϕ-Y,E)=g-(ϕ-Y,∇-XE)=-B(X,ϕ-Y).By a similar way, we compute(132)g(∇XY,N)=g(∇°XY+α1(X,Y)ϕ-E+α2(X,Y)ϕ-N+α3(X,Y)E,N∇°X)=α3(X,Y).Since∇XY=∇X*Y+C(X,Y)E, then we get
(133)g(∇XY,N)=C(X,Y).
Therefore using the expressions of α1,α2, and α3 in (127) we write
(134)∇XY=∇°XY+C(X,ϕ-Y)ϕ-E+B(X,ϕ-Y)ϕ-N+C(X,Y)E,
which implies that
(135)h°(X,Y)=C(X,ϕ-Y)ϕ-E+B(X,ϕ-Y)ϕ-N+C(X,Y)E.Theorem 26.
Let(M,g,S(TM)) be a screen semi-invariant lightlike hypersurface of a para-Sasakian manifold (M-,ϕ-,ξ,η,g-). Then the distribution D0 is integrable if and only if
(136)C(X,Y)=C(Y,X),B(X,ϕ-Y)=B(ϕ-X,Y),C(X,ϕ-Y)=C(ϕ-X,Y),
where X,Y∈Γ(D0).Proof.
Since∇ is torsion free connection, by using (134), we have
(137)[X,Y]=∇°XY-∇°YX+(C(X,ϕ-Y)-C(ϕ-X,Y))ϕ-E+(B(X,ϕ-Y)-B(ϕ-X,Y))ϕ-N+(C(X,Y)-C(Y,X))E,
for any X,Y∈Γ(D0). Now suppose that D0 is integrable. Then the components of [X,Y] with respect to ϕ-E, ϕ-N, and E vanish. So, we get (136).
Conversely, if (136) is satisfied, then we get (137)
(138)[X,Y]=∇°XY-∇°YX,
for any X,Y∈Γ(D0), which implies that [X,Y]∈Γ(D0). This completes the proof.Corollary 27.
Let(M-,ϕ-,ξ,η,g-) be a para-Sasakian manifold and (M,g,S(TM)) a screen semi-invariant lightlike hypersurface of M-. Then h° is symmetric on D0 if and only if D0 is integrable.Theorem 28.
Let(M,g,S(TM)) be a screen semi-invariant lightlike hypersurface of a para-Sasakian manifold (M-,ϕ-,ξ,η,g-) and the distribution D0 on M is integrable. Then D0 is minimal with respect to the symmetric connection ∇ on M; that is, trace(h°)=0, if and only if
(139)C(X,Y)=C(ϕ-X,ϕ-Y),
for any X,Y∈Γ(D0).Proof.
Using decomposition (94), we find rank⁡D0=2n-3. Now, from Proposition 20, consider an orthonormal ϕ--basis {ei,ϕ-ei,ξ} of D0, i=1,2,…,n-2. Then we have
(140)trace(h°)=∑i=1n-2‍h°(ei,ei)-∑i=1n-2‍h°(ϕ-ei,ϕ-ei)+h°(ξ,ξ).
Also from (25) and (35) we have
(141)h°(ξ,ξ)=C(ξ,ϕ-ξ)ϕ-E+B(ξ,ϕ-ξ)ϕ-N+C(ξ,ξ)E=0.
Hence, using integrability condition of D0 we get
(142)h°(ei,ei)-h°(ϕ-ei,ϕ-ei)=(C(ei,ei)-C(ϕ-ei,ϕ-ei))E=0,
which completes the proof.Now, using the decomposition (96) and putting ω={ϕ-(Rad(TM))⊕ϕ-(ltr(TM))}⊥{Rad(TM)⊕ltr(TM)}, for any X∈Γ(TM), Y∈Γ(D0), and Z∈Γ(ω), we write
(143)∇-XY=∇^XY+h^(X,Y),(144)∇-XZ=-A^ZX+∇XωZ,
where ∇^ is a linear connection on the bundle D0, h^:Γ(TM)×Γ(D0)→Γ(ω) is ℑ(M) bilinear and A^ is an ℑ(M) linear operator on Γ(D0), and ∇ω is a linear connection on ω.LetU⊂M be a coordinate neighborhood as fixed in Theorem 2. Then, by using (96), we set
(145)β1(X,Y)=-g(h^(X,Y),ϕ-N),β2(X,Y)=-g(h^(X,Y),ϕ-E),β3(X,Y)=g(h^(X,Y),N),β4(X,Y)=g(h^(X,Y),E),
for any X,Y∈Γ(D0|U). Hence, (143) can be written locally:
(146)∇-XY=∇^XY+β1(X,Y)ϕ-E+β2(X,Y)ϕ-N+β3(X,Y)E+β4(X,Y)N.
From the expression of βi, i=1,2,3,4, in terms of B and C, we obtain
(147)∇-XY=∇^XY+C(X,ϕ-Y)ϕ-E+B(X,ϕ-Y)ϕ-N+C(X,Y)E+B(X,Y)N,(148)h^(X,Y)=C(X,ϕ-Y)ϕ-E+B(X,ϕ-Y)ϕ-N+C(X,Y)E+B(X,Y)N.Theorem 29.
Let(M,g,S(TM)) be a screen semi-invariant lightlike hypersurface of a para-Sasakian manifold (M-,ϕ-,ξ,η,g-) with the integrable distribution D0. Then D0 is minimal; that is, trace(h^)=0, if and only if
(149)C(X,Y)=C(ϕ-X,ϕ-Y),B(X,Y)=B(ϕ-X,ϕ-Y),
for any X,Y∈Γ(D0).Proof.
Let{ei,ϕ-ei,ξ}, i=1,2,…,n-2, be an orthonormal ϕ--basis. Then, we have
(150)trace(h^)=∑i=1n-2‍h^(ei,ei)-∑i=1n-2‍h^(ϕ-ei,ϕ-ei)+h^(ξ,ξ).
It is easy to see that h^(ξ,ξ)=h°(ξ,ξ)+B(ξ,ξ)N=0. Also, using (147), we get
(151)h^(ei,ei)-h^(ϕ-ei,ϕ-ei)=(C(ei,ei)-C(ϕ-ei,ϕ-ei))E+(B(ei,ei)-B(ϕ-ei,ϕ-ei))N=0,
by virtue of integrability condition of D0. This completes the proof.Theorem 30.
Let(M-,ϕ-,ξ,η,g-) be a para-Sasakian manifold and (M,g,S(TM)) a screen semi-invariant lightlike hypersurface of M-. If D0 is integrable, then the leaves of D0 have a para-Sasakian structure.Proof.
Let(M,g,S(TM)) be a screen semi-invariant lightlike hypersurface and M° a leaf of D0. Then, for any p∈M°, we have TpM°=(D0)p. Since S:Γ(TM)→Γ(D) and D=D0⊥ϕ-(Rad(TM))⊥Rad(TM), we get
(152)ϕX0=ϕ-SX0=ϕ-X0,
for any X0∈TM°.
By puttingϕ°=ϕ|D0 and η°=η|D0, one can see that ϕ° defines an (1,1)-tensor field on M° because D0 is ϕ--invariant. From (101), we get
(153)ϕ°X0=X0-η°(X0)ξ,η°(ξ)=1,
for any X0∈TM°. Hence (M°,ϕ°,ξ,η°) is an almost paracontact manifold.
Next, by using (38), for any X0,Y0∈Γ(TM°), we have
(154)g(ϕ°X0,ϕ°Y0)=g-(ϕ-X0-u(X0)N,ϕ-Y0-u(Y0)N)=-g(X0,Y0)+η°(X0)η°(Y0),
which implies that (M°,ϕ°,ξ,η°,g) is an almost paracontact metric manifold.
Moreover, for anyX0,Y0∈Γ(TM°), dη(X0,Y0)=dη°(X0,Y0), and Φ°(X0,Y0)=g(X0,ϕ°Y0)= g-(X0,ϕ°Y0)=dη(X0,Y0)=dη°(X0,Y0). Also Nϕ°+2dη°⊗ξ and Nϕ-+2dη-⊗ξ coincide on D0.
Finally, we have(155)(∇°X0g)(Y0,Z0)=X0g(Y0,Z0)-g(∇°X0Y0,Z0)-g(Y0,∇°X0Z0)=X0g(Y0,Z0)-g(∇X0Y0,Z0)+g(h°(X0,Y0),Z0)-g(Y0,∇X0Z0)+g(Y0,h°(X0,Z0))=(∇X0g)(Y0,Z0)=0,
for any X0,Y0,Z0∈Γ(TM°), which implies that ∇° is a Levi-Civita connection and using (5) we get
(156)(∇°X0ϕ°)Y0=-g(X0,Y0)ξ+η(Y0)X0.
This completes the proof.


### 6.2. Integrability ofD

In this section, we consider the distributionD, which is defined by D=Rad(TM)⊥ϕ-(Rad(TM))⊥D0. Firstly we have the following.Lemma 31.
Let(M-,ϕ-,ξ,η,g-) be a para-Sasakian manifold and(M,g,S(TM))a screen semi-invariant lightlike hypersurface of M-. Then, for any X∈Γ(TM) and Y∈Γ(TM-), one has
(157)g-((∇-Xϕ-)Y,E)=0;
that is, the component of (∇-Xϕ-)Y along ltr(TM) vanishes.Proposition 32.
Let(M-,ϕ-,ξ,η,g-) be a para-Sasakian manifold and(M,g,S(TM))a screen semi-invariant lightlike hypersurface of M-. Then the distribution D is integrable if and only if B satisfies the following conditions: (i)
B(X,ϕ-Y)=B(ϕ-X,Y),for anyX,Y∈Γ(D0),(ii)
B(X,V)=0,for anyX∈Γ(D0),(iii)
B(V,V)=0,
whereV=ϕ-E.Proof.
For anyX,Y∈Γ(D), we obtain the component of [X,Y] along ϕ-ltr(TM) as
(158)g-([X,Y],ϕ-E)=g(ϕ-X,AE*Y)-g(ϕ-Y,AE*X).
From the definition of the distribution D, we set
(159)X=X0+σ1E+σ2ϕ-E,Y=Y0+ρ1E+ρ2ϕ-E.
Since D is ϕ--invariant, from the previous expressions of X,Y∈Γ(D) and from (26), we have
(160)g-([X,Y],ϕ-E)=(σ1ρ2-σ2ρ1)B(V,V)-ρ2B(ϕ-X0,V)+σ2B(ϕ-Y0,V)-σ1B(Y0,V)+ρ1B(X0,V)-B(X0,ϕ-Y0)+B(Y0,ϕ-X0),
for any X,Y∈Γ(TM).
Now assume thatD is integrable. Since ϕ-E, E, X0, and Y0 are sections of D, then we get
(161)0=g-([ϕ-E,E],ϕ-E)=-g-(ϕ-E,ϕ-E)=-B(V,V).
If X∈Γ(D0), we find
(162)0=g-([X,E],ϕ-E)=B(E,ϕ-X)-B(X,ϕ-E)=-B(X,V),
and if X,Y∈Γ(D0) we get
(163)0=g-([X,Y],ϕ-E)=B(ϕ-Y,X)-B(ϕ-X,Y).
Consequently using (160) with (i), (ii), and (iii), it is easy to check that [X,Y] in Γ(D). This completes the proof.Proposition 33.
Let(M-,ϕ-,ξ,η,g-) be a para-Sasakian manifold and(M,g,S(TM))be a screen semi-invariant lightlike hypersurface of M-. If (M,g,S(TM)) is totally geodesic, then the following statements hold. (i)
The distributionDis integrable.(ii)
The distributionDis parallel with respect to the induced connection∇.Proof.
(i) Assume that(M,g,S(TM)) is totally geodesic. Then we can state that the distribution D is integrable from Proposition 32.
(ii) For anyX∈Γ(TM) and Y∈Γ(D0), using (157), we get
(164)g(∇XE,ϕ-E)=g-(∇-XE,ϕ-E)=-g-(E,∇-Xϕ-E)=B(X,ϕ-E)=0,g(∇Xϕ-E,ϕ-E)=g-(∇-Xϕ-E,ϕ-E)=-g-(ϕ-∇-Xϕ-E,E)=g-((∇-Xϕ-)ϕ-E,E)-g-(∇-Xϕ-2E,E)=g(∇XE,E)=0,g(∇XY0,ϕ-E)=g-(∇-XY0,ϕ-E)=-g-(ϕ-∇-XY0,E)=g-((∇-Xϕ-)Y0,E)-g-(∇-Xϕ-Y0,E)=-g-(∇-Xϕ-Y0,E)=-B(X,ϕ-Y0)=0.
This completes the proof.


## 6.1. The DistributionD0

Firstly, we consider the distributionD0, defined in (94). Using (95) and putting μ={ϕ-(Rad(TM))⊕ϕ-(ltr(TM))}⊥Rad(TM), for any X∈Γ(TM), Y∈Γ(D0), and Z∈Γ(μ) we have
(122)∇XY=∇°XY+h°(X,Y),(123)∇XZ=-A°ZX+∇XμZ,
where ∇° is a linear connection on the bundle D0, h°:Γ(TM)×Γ(D0)→Γ(μ) is an ℑ(M) bilinear, A° is an ℑ(M) linear operator on Γ(D0), respectively, and ∇μ is a linear connection on μ.Lemma 25.
Let(M,g,S(TM)) be a screen semi-invariant lightlike hypersurface of a para-Sasakian manifold (M-,ϕ-,ξ,η,g-) and U⊂M a coordinate neighborhood as fixed in Theorem 2. Then, for any X,Y∈Γ(D0), one has
(124)g-((∇-Xϕ-)Y,E)=0,(125)g-((∇-Xϕ-)Y,N)=0.Proof.
Calculation is straightforward by using (9).LetU⊂M be a coordinate neighborhood as fixed in Theorem 2. Then according to decomposition given by (95) we set
(126)α1(X,Y)=-g(h°(X,Y),ϕ-N),α2(X,Y)=-g(h°(X,Y),ϕ-E),α3(X,Y)=g(h°(X,Y),N),
for any X,Y∈Γ(D0|U). So (122) can be written locally as
(127)∇XY=∇°XY+α1(X,Y)ϕ-E+α2(X,Y)ϕ-N+α3(X,Y)E.
We will express α1, α2, and α3 in terms of B and C. Firstly, we compute
(128)g(∇XY,ϕ-N)=g(∇°XY+α1(X,Y)ϕ-E+α2(X,Y)ϕ-N+α3(X,Y)E,ϕ-N∇°XY)=-α1(X,Y).
Then, by using (125), (24) and (25), we get
(129)g(∇XY,ϕ-N)=-g-(ϕ-∇XY,N)=-g-(ϕ-∇-XY,N)=g-((∇-Xϕ-)Y,N)-g-(∇-Xϕ-Y,N)=-g-(∇-Xϕ-Y,N)=g-(ϕ-Y,∇-XN)=-g-(ANX,ϕ-Y)=-C(X,ϕ-Y),
in view of D0 being ϕ--invariant and ∇- being a metric connection.Next we have(130)g(∇XY,ϕ-N)=g(∇°XY+α1(X,Y)ϕ-E+α2(X,Y)ϕ-N+α3(X,Y)E,ϕ-E∇°X)=-α2(X,Y),
And, using (124), (24), and (31) in the previous equation, we obtain
(131)g(∇XY,ϕ-E)=-g-(ϕ-∇XY,E)=-g-(ϕ-∇-XY,E)=g-((∇-Xϕ-)Y,E)-g-(∇-Xϕ-Y,E)=-g-(∇-Xϕ-Y,E)=g-(ϕ-Y,∇-XE)=-B(X,ϕ-Y).By a similar way, we compute(132)g(∇XY,N)=g(∇°XY+α1(X,Y)ϕ-E+α2(X,Y)ϕ-N+α3(X,Y)E,N∇°X)=α3(X,Y).Since∇XY=∇X*Y+C(X,Y)E, then we get
(133)g(∇XY,N)=C(X,Y).
Therefore using the expressions of α1,α2, and α3 in (127) we write
(134)∇XY=∇°XY+C(X,ϕ-Y)ϕ-E+B(X,ϕ-Y)ϕ-N+C(X,Y)E,
which implies that
(135)h°(X,Y)=C(X,ϕ-Y)ϕ-E+B(X,ϕ-Y)ϕ-N+C(X,Y)E.Theorem 26.
Let(M,g,S(TM)) be a screen semi-invariant lightlike hypersurface of a para-Sasakian manifold (M-,ϕ-,ξ,η,g-). Then the distribution D0 is integrable if and only if
(136)C(X,Y)=C(Y,X),B(X,ϕ-Y)=B(ϕ-X,Y),C(X,ϕ-Y)=C(ϕ-X,Y),
where X,Y∈Γ(D0).Proof.
Since∇ is torsion free connection, by using (134), we have
(137)[X,Y]=∇°XY-∇°YX+(C(X,ϕ-Y)-C(ϕ-X,Y))ϕ-E+(B(X,ϕ-Y)-B(ϕ-X,Y))ϕ-N+(C(X,Y)-C(Y,X))E,
for any X,Y∈Γ(D0). Now suppose that D0 is integrable. Then the components of [X,Y] with respect to ϕ-E, ϕ-N, and E vanish. So, we get (136).
Conversely, if (136) is satisfied, then we get (137)
(138)[X,Y]=∇°XY-∇°YX,
for any X,Y∈Γ(D0), which implies that [X,Y]∈Γ(D0). This completes the proof.Corollary 27.
Let(M-,ϕ-,ξ,η,g-) be a para-Sasakian manifold and (M,g,S(TM)) a screen semi-invariant lightlike hypersurface of M-. Then h° is symmetric on D0 if and only if D0 is integrable.Theorem 28.
Let(M,g,S(TM)) be a screen semi-invariant lightlike hypersurface of a para-Sasakian manifold (M-,ϕ-,ξ,η,g-) and the distribution D0 on M is integrable. Then D0 is minimal with respect to the symmetric connection ∇ on M; that is, trace(h°)=0, if and only if
(139)C(X,Y)=C(ϕ-X,ϕ-Y),
for any X,Y∈Γ(D0).Proof.
Using decomposition (94), we find rank⁡D0=2n-3. Now, from Proposition 20, consider an orthonormal ϕ--basis {ei,ϕ-ei,ξ} of D0, i=1,2,…,n-2. Then we have
(140)trace(h°)=∑i=1n-2‍h°(ei,ei)-∑i=1n-2‍h°(ϕ-ei,ϕ-ei)+h°(ξ,ξ).
Also from (25) and (35) we have
(141)h°(ξ,ξ)=C(ξ,ϕ-ξ)ϕ-E+B(ξ,ϕ-ξ)ϕ-N+C(ξ,ξ)E=0.
Hence, using integrability condition of D0 we get
(142)h°(ei,ei)-h°(ϕ-ei,ϕ-ei)=(C(ei,ei)-C(ϕ-ei,ϕ-ei))E=0,
which completes the proof.Now, using the decomposition (96) and putting ω={ϕ-(Rad(TM))⊕ϕ-(ltr(TM))}⊥{Rad(TM)⊕ltr(TM)}, for any X∈Γ(TM), Y∈Γ(D0), and Z∈Γ(ω), we write
(143)∇-XY=∇^XY+h^(X,Y),(144)∇-XZ=-A^ZX+∇XωZ,
where ∇^ is a linear connection on the bundle D0, h^:Γ(TM)×Γ(D0)→Γ(ω) is ℑ(M) bilinear and A^ is an ℑ(M) linear operator on Γ(D0), and ∇ω is a linear connection on ω.LetU⊂M be a coordinate neighborhood as fixed in Theorem 2. Then, by using (96), we set
(145)β1(X,Y)=-g(h^(X,Y),ϕ-N),β2(X,Y)=-g(h^(X,Y),ϕ-E),β3(X,Y)=g(h^(X,Y),N),β4(X,Y)=g(h^(X,Y),E),
for any X,Y∈Γ(D0|U). Hence, (143) can be written locally:
(146)∇-XY=∇^XY+β1(X,Y)ϕ-E+β2(X,Y)ϕ-N+β3(X,Y)E+β4(X,Y)N.
From the expression of βi, i=1,2,3,4, in terms of B and C, we obtain
(147)∇-XY=∇^XY+C(X,ϕ-Y)ϕ-E+B(X,ϕ-Y)ϕ-N+C(X,Y)E+B(X,Y)N,(148)h^(X,Y)=C(X,ϕ-Y)ϕ-E+B(X,ϕ-Y)ϕ-N+C(X,Y)E+B(X,Y)N.Theorem 29.
Let(M,g,S(TM)) be a screen semi-invariant lightlike hypersurface of a para-Sasakian manifold (M-,ϕ-,ξ,η,g-) with the integrable distribution D0. Then D0 is minimal; that is, trace(h^)=0, if and only if
(149)C(X,Y)=C(ϕ-X,ϕ-Y),B(X,Y)=B(ϕ-X,ϕ-Y),
for any X,Y∈Γ(D0).Proof.
Let{ei,ϕ-ei,ξ}, i=1,2,…,n-2, be an orthonormal ϕ--basis. Then, we have
(150)trace(h^)=∑i=1n-2‍h^(ei,ei)-∑i=1n-2‍h^(ϕ-ei,ϕ-ei)+h^(ξ,ξ).
It is easy to see that h^(ξ,ξ)=h°(ξ,ξ)+B(ξ,ξ)N=0. Also, using (147), we get
(151)h^(ei,ei)-h^(ϕ-ei,ϕ-ei)=(C(ei,ei)-C(ϕ-ei,ϕ-ei))E+(B(ei,ei)-B(ϕ-ei,ϕ-ei))N=0,
by virtue of integrability condition of D0. This completes the proof.Theorem 30.
Let(M-,ϕ-,ξ,η,g-) be a para-Sasakian manifold and (M,g,S(TM)) a screen semi-invariant lightlike hypersurface of M-. If D0 is integrable, then the leaves of D0 have a para-Sasakian structure.Proof.
Let(M,g,S(TM)) be a screen semi-invariant lightlike hypersurface and M° a leaf of D0. Then, for any p∈M°, we have TpM°=(D0)p. Since S:Γ(TM)→Γ(D) and D=D0⊥ϕ-(Rad(TM))⊥Rad(TM), we get
(152)ϕX0=ϕ-SX0=ϕ-X0,
for any X0∈TM°.
By puttingϕ°=ϕ|D0 and η°=η|D0, one can see that ϕ° defines an (1,1)-tensor field on M° because D0 is ϕ--invariant. From (101), we get
(153)ϕ°X0=X0-η°(X0)ξ,η°(ξ)=1,
for any X0∈TM°. Hence (M°,ϕ°,ξ,η°) is an almost paracontact manifold.
Next, by using (38), for any X0,Y0∈Γ(TM°), we have
(154)g(ϕ°X0,ϕ°Y0)=g-(ϕ-X0-u(X0)N,ϕ-Y0-u(Y0)N)=-g(X0,Y0)+η°(X0)η°(Y0),
which implies that (M°,ϕ°,ξ,η°,g) is an almost paracontact metric manifold.
Moreover, for anyX0,Y0∈Γ(TM°), dη(X0,Y0)=dη°(X0,Y0), and Φ°(X0,Y0)=g(X0,ϕ°Y0)= g-(X0,ϕ°Y0)=dη(X0,Y0)=dη°(X0,Y0). Also Nϕ°+2dη°⊗ξ and Nϕ-+2dη-⊗ξ coincide on D0.
Finally, we have(155)(∇°X0g)(Y0,Z0)=X0g(Y0,Z0)-g(∇°X0Y0,Z0)-g(Y0,∇°X0Z0)=X0g(Y0,Z0)-g(∇X0Y0,Z0)+g(h°(X0,Y0),Z0)-g(Y0,∇X0Z0)+g(Y0,h°(X0,Z0))=(∇X0g)(Y0,Z0)=0,
for any X0,Y0,Z0∈Γ(TM°), which implies that ∇° is a Levi-Civita connection and using (5) we get
(156)(∇°X0ϕ°)Y0=-g(X0,Y0)ξ+η(Y0)X0.
This completes the proof.


## 6.2. Integrability ofD

In this section, we consider the distributionD, which is defined by D=Rad(TM)⊥ϕ-(Rad(TM))⊥D0. Firstly we have the following.Lemma 31.
Let(M-,ϕ-,ξ,η,g-) be a para-Sasakian manifold and(M,g,S(TM))a screen semi-invariant lightlike hypersurface of M-. Then, for any X∈Γ(TM) and Y∈Γ(TM-), one has
(157)g-((∇-Xϕ-)Y,E)=0;
that is, the component of (∇-Xϕ-)Y along ltr(TM) vanishes.Proposition 32.
Let(M-,ϕ-,ξ,η,g-) be a para-Sasakian manifold and(M,g,S(TM))a screen semi-invariant lightlike hypersurface of M-. Then the distribution D is integrable if and only if B satisfies the following conditions: (i)
B(X,ϕ-Y)=B(ϕ-X,Y),for anyX,Y∈Γ(D0),(ii)
B(X,V)=0,for anyX∈Γ(D0),(iii)
B(V,V)=0,
whereV=ϕ-E.Proof.
For anyX,Y∈Γ(D), we obtain the component of [X,Y] along ϕ-ltr(TM) as
(158)g-([X,Y],ϕ-E)=g(ϕ-X,AE*Y)-g(ϕ-Y,AE*X).
From the definition of the distribution D, we set
(159)X=X0+σ1E+σ2ϕ-E,Y=Y0+ρ1E+ρ2ϕ-E.
Since D is ϕ--invariant, from the previous expressions of X,Y∈Γ(D) and from (26), we have
(160)g-([X,Y],ϕ-E)=(σ1ρ2-σ2ρ1)B(V,V)-ρ2B(ϕ-X0,V)+σ2B(ϕ-Y0,V)-σ1B(Y0,V)+ρ1B(X0,V)-B(X0,ϕ-Y0)+B(Y0,ϕ-X0),
for any X,Y∈Γ(TM).
Now assume thatD is integrable. Since ϕ-E, E, X0, and Y0 are sections of D, then we get
(161)0=g-([ϕ-E,E],ϕ-E)=-g-(ϕ-E,ϕ-E)=-B(V,V).
If X∈Γ(D0), we find
(162)0=g-([X,E],ϕ-E)=B(E,ϕ-X)-B(X,ϕ-E)=-B(X,V),
and if X,Y∈Γ(D0) we get
(163)0=g-([X,Y],ϕ-E)=B(ϕ-Y,X)-B(ϕ-X,Y).
Consequently using (160) with (i), (ii), and (iii), it is easy to check that [X,Y] in Γ(D). This completes the proof.Proposition 33.
Let(M-,ϕ-,ξ,η,g-) be a para-Sasakian manifold and(M,g,S(TM))be a screen semi-invariant lightlike hypersurface of M-. If (M,g,S(TM)) is totally geodesic, then the following statements hold. (i)
The distributionDis integrable.(ii)
The distributionDis parallel with respect to the induced connection∇.Proof.
(i) Assume that(M,g,S(TM)) is totally geodesic. Then we can state that the distribution D is integrable from Proposition 32.
(ii) For anyX∈Γ(TM) and Y∈Γ(D0), using (157), we get
(164)g(∇XE,ϕ-E)=g-(∇-XE,ϕ-E)=-g-(E,∇-Xϕ-E)=B(X,ϕ-E)=0,g(∇Xϕ-E,ϕ-E)=g-(∇-Xϕ-E,ϕ-E)=-g-(ϕ-∇-Xϕ-E,E)=g-((∇-Xϕ-)ϕ-E,E)-g-(∇-Xϕ-2E,E)=g(∇XE,E)=0,g(∇XY0,ϕ-E)=g-(∇-XY0,ϕ-E)=-g-(ϕ-∇-XY0,E)=g-((∇-Xϕ-)Y0,E)-g-(∇-Xϕ-Y0,E)=-g-(∇-Xϕ-Y0,E)=-B(X,ϕ-Y0)=0.
This completes the proof.



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*Source: 696231-2014-04-28.xml*