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Einstein half lightlike submanifolds of a Lorentzian space form with a semi-symmetric non-metric connection

Abstract

We study screen conformal Einstein half lightlike submanifolds M of a Lorentzian space form M ˜ (c) of constant curvature c admitting a semi-symmetric non-metric connection subject to the conditions; (1) the structure vector field of M ˜ is tangent to M, and (2) the canonical normal vector field of M is conformal Killing. The main result is a characterization theorem for such a half lightlike submanifold.

MSC:53C25, 53C40, 53C50.

Dedication

Dedicated to Professor Hari M Srivastava

1 Introduction

The theory of lightlike submanifolds is used in mathematical physics, in particular, in general relativity as lightlike submanifolds produce models of different types of horizons [1, 2]. Lightlike submanifolds are also studied in the theory of electromagnetism [3]. Thus, large number of applications but limited information available, motivated us to do the research on this subject matter. As for any semi-Riemannian manifold, there is a natural existence of lightlike subspaces, Duggal and Bejancu published their work [3] on the general theory of lightlike submanifolds to fill a gap in the study of submanifolds. Since then, there has been very active study on lightlike geometry of submanifolds (see up-to date results in two books [4, 5]). The class of lightlike submanifolds of codimension 2 is composed of two classes by virtue of the rank of its radical distribution, named by half lightlike and coisotropic submanifolds [6, 7]. Half lightlike submanifold is a special case of general r-lightlike submanifold such that r=1, and its geometry is more general form than that of coisotropic submanifold or lightlike hypersurface. Much of the works on half lightlike submanifolds will be immediately generalized in a formal way to general r-lightlike submanifolds of arbitrary codimension n and arbitrary rank r. For this reason, we study half lightlike submanifold M of a semi-Riemannian manifold M ˜ .

Ageshe and Chafle [8] introduced the notion of a semi-symmetric non-metric connection on a Riemannian manifold. Although now, we have lightlike version of a large variety of Riemannian submanifolds, the theory of lightlike submanifolds of semi-Riemannian manifolds, equipped with semi-symmetric metric connections, has not been introduced until quite recently. Yasar et al. [9] studied lightlike hypersurfaces in a semi-Riemannian manifold admitting a semi-symmetric non-metric connection. Recently, Jin and Lee [10] and Jin [1113] studied half lightlike and r-lightlike submanifolds of a semi-Riemannian manifold with a semi-symmetric non-metric connection.

In this paper, we study the geometry of screen conformal Einstein half lightlike submanifolds M of a Lorentzian space form M ˜ (c) of constant curvature c admitting a semi-symmetric non-metric connection subject to the conditions; (1) the structure vector field of M ˜ is tangent to M, and (2) the canonical normal vector field of M is conformal Killing. The reason for this geometric restriction on M is due to the fact that such a class admits an integrable screen distribution and a symmetric Ricci tensor of M. We prove a characterization theorem for such a half lightlike submanifold.

2 Semi-symmetric non-metric connection

Let ( M ˜ , g ˜ ) be a semi-Riemannian manifold. A connection ˜ on M ˜ is called a semi-symmetric non-metric connection [8] if ˜ and its torsion tensor T ˜ satisfy

( ˜ X g ˜ )(Y,Z)=π(Y) g ˜ (X,Z)π(Z) g ˜ (X,Y),
(2.1)
T ˜ (X,Y)=π(Y)Xπ(X)Y,
(2.2)

for any vector fields X, Y and Z on M ˜ , where π is a 1-form associated with a non-vanishing vector field ζ, which is called the structure vector field of M ˜ , by

π(X)= g ˜ (X,ζ).
(2.3)

In the entire discussion of this article, we shall assume the structure vector field ζ to be unit spacelike, unless otherwise specified.

A submanifold (M,g) of codimension 2 is called half lightlike submanifold if the radical distribution Rad(TM)=TMT M is a subbundle of the tangent bundle TM and the normal bundle T M of rank 1. Therefore, there exist complementary non-degenerate distributions S(TM) and S(T M ) of Rad(TM) in TM and T M respectively, which are called the screen and co-screen distributions of M, respectively, such that

TM=Rad(TM) orth S(TM),T M =Rad(TM) orth S ( T M ) ,
(2.4)

where orth denotes the orthogonal direct sum. We denote such a half lightlike submanifold by M=(M,g,S(TM)). Denote by F(M) the algebra of smooth functions on M and by Γ(E) the F(M) module of smooth sections of a vector bundle E over M. Choose LΓ(S(T M )) as a unit vector field with g ˜ (L,L)=±1. We may assume that L to be unit spacelike vector field without loss of generality, i.e., g ˜ (L,L)=1. We call L the canonical normal vector field of M. Consider the orthogonal complementary distribution S ( T M ) to S(TM) in T M ˜ . Certainly, Rad(TM) and S(T M ) are subbundles of S ( T M ) . As S(T M ) is non-degenerate, we have

S ( T M ) =S ( T M ) orth S ( T M ) ,

where S ( T M ) is the orthogonal complementary to S(T M ) in S ( T M ) . For any null section ξ of Rad(TM) on a coordinate neighborhood UM, there exists a uniquely defined lightlike vector bundle ltr(TM) and a null vector field N of ltr ( T M ) | U satisfying

g ˜ (ξ,N)=1, g ˜ (N,N)= g ˜ (N,X)= g ˜ (N,L)=0,XΓ ( S ( T M ) ) .

We call N, ltr(TM) and tr(TM)=S(T M ) orth ltr(TM) the lightlike transversal vector field, lightlike transversal vector bundle and transversal vector bundle of M with respect to the screen distribution, respectively [6]. Then T M ˜ is decomposed as follows:

T M ˜ = T M tr ( T M ) = { Rad ( T M ) tr ( T M ) } orth S ( T M ) = { Rad ( T M ) ltr ( T M ) } orth S ( T M ) orth S ( T M ) .
(2.5)

Given a screen distribution S(TM), there exists a unique complementary vector bundle tr(TM) to TM in T M ˜ | M . Using (2.4) and (2.5), there exists a local quasi-orthonormal frame field of M ˜ along M given by

F={ξ,N,L, W a },a{1,,m},
(2.6)

where { W a } is an orthonormal frame field of S ( T M ) | U .

In the entire discussion of this article, we shall assume that ζ is tangent to M, and we take X,Y,Z,WΓ(TM), unless otherwise specified. Let P be the projection morphism of TM on S(TM) with respect to the first decomposition of (2.4). Then the local Gauss and Weingartan formulas of M and S(TM) are given respectively by

˜ X Y= X Y+B(X,Y)N+D(X,Y)L,
(2.7)
˜ X N= A N X+τ(X)N+ρ(X)L,
(2.8)
˜ X L= A L X+ϕ(X)N,
(2.9)
X PY= X PY+C(X,PY)ξ,
(2.10)
X ξ= A ξ Xτ(X)ξ,
(2.11)

where and are induced linear connections on TM and S(TM), respectively, B and D are called the local lightlike, and screen second fundamental forms of M, respectively, C is called the local second fundamental form on S(TM), A N , A ξ and A L are called the shape operators, and τ, ρ and ϕ are 1-forms on TM. We say that

h(X,Y)=B(X,Y)N+D(X,Y)L

is the second fundamental form tensor of M. Using (2.1), (2.2) and (2.7), we have

( X g)(Y,Z)=B(X,Y)η(Z)+B(X,Z)η(Y)π(Y)g(X,Z)π(Z)g(X,Y),
(2.12)
T(X,Y)=π(Y)Xπ(X)Y,
(2.13)

and B and D are symmetric on TM, where T is the torsion tensor with respect to the induced connection , and η is a 1-form on TM such that

η(X)= g ˜ (X,N).

From the facts B(X,Y)= g ˜ ( ˜ X Y,ξ) and D(X,Y)= g ˜ ( ˜ X Y,L), we know that B and D are independent of the choice of the screen distribution S(TM) and satisfy

B(X,ξ)=0,D(X,ξ)=ϕ(X).
(2.14)

In case ζ is tangent to M, the above three local second fundamental forms on M and S(TM) are related to their shape operators by

g ( A ξ X , Y ) =B(X,Y), g ˜ ( A ξ X , N ) =0,
(2.15)
g( A L X,Y)=D(X,Y)+ϕ(X)η(Y), g ˜ ( A L X,N)=ρ(X),
(2.16)
g( A N X,PY)=C(X,PY)fg(X,PY)η(X)π(PY), g ˜ ( A N X,N)=fη(X),
(2.17)

where f is the smooth function given by f=π(N). From (2.15) and (2.17), we show that A ξ and A N are S(TM)-valued, and A ξ is self-adjoint operator and satisfies

A ξ ξ=0,
(2.18)

that is, ξ is an eigenvector field of A ξ corresponding to the eigenvalue 0.

In general, the screen distribution S(TM) is not necessarily integrable. The following result gives equivalent conditions for the integrability of S(TM).

Theorem 2.1 [10]

Let M be a half lightlike submanifold of a semi-Riemannian manifold M ˜ admitting a semi-symmetric non-metric connection. Then the following assertions are equivalent:

  1. (1)

    The screen distribution S(TM) is an integrable distribution.

  2. (2)

    C is symmetric, i.e., C(X,Y)=C(Y,X) for all X,YΓ(S(TM)).

  3. (3)

    The shape operator A N is a self-adjoint with respect to g, i.e.,

    g( A N X,Y)=g(X, A N Y),X,YΓ ( S ( T M ) ) .

Just as in the well-known case of locally product Riemannian or semi-Riemannian manifolds [24, 7], if S(TM) is an integrable distribution, then M is locally a product manifold M= C 1 × M , where C 1 is a null curve tangent to Rad(TM), and M is a leaf of the integrable screen distribution S(TM).

3 Structure equations

Denote by R ˜ , R and R the curvature tensors of the semi-symmetric non-metric connection ˜ on M ˜ , the induced connection on M and the induced connection on S(TM), respectively. Using the Gauss-Weingarten formulas for M and S(TM), we obtain the Gauss-Codazzi equations for M and S(TM):

R ˜ ( X , Y ) Z = R ( X , Y ) Z + B ( X , Z ) A N Y B ( Y , Z ) A N X R ˜ ( X , Y ) Z = + D ( X , Z ) A L Y D ( Y , Z ) A L X R ˜ ( X , Y ) Z = + { ( X B ) ( Y , Z ) ( Y B ) ( X , Z ) R ˜ ( X , Y ) Z = + B ( Y , Z ) [ τ ( X ) π ( X ) ] B ( X , Z ) [ τ ( Y ) π ( Y ) ] R ˜ ( X , Y ) Z = + D ( Y , Z ) ϕ ( X ) D ( X , Z ) ϕ ( Y ) } N R ˜ ( X , Y ) Z = + { ( X D ) ( Y , Z ) ( Y D ) ( X , Z ) + B ( Y , Z ) ρ ( X ) R ˜ ( X , Y ) Z = B ( X , Z ) ρ ( Y ) D ( Y , Z ) π ( X ) + D ( X , Z ) π ( Y ) } L ,
(3.1)
R ˜ ( X , Y ) N = X ( A N Y ) + Y ( A N X ) + A N [ X , Y ] R ˜ ( X , Y ) N = + τ ( X ) A N Y τ ( Y ) A N X + ρ ( X ) A L Y ρ ( Y ) A L X R ˜ ( X , Y ) N = + { B ( Y , A N X ) B ( X , A N Y ) + 2 d τ ( X , Y ) R ˜ ( X , Y ) N = + ϕ ( X ) ρ ( Y ) ϕ ( Y ) ρ ( X ) } N R ˜ ( X , Y ) N = + { D ( Y , A N X ) D ( X , A N Y ) + 2 d ρ ( X , Y ) R ˜ ( X , Y ) N = + ρ ( X ) τ ( Y ) ρ ( Y ) τ ( X ) } L ,
(3.2)
R ˜ ( X , Y ) L = X ( A L Y ) + Y ( A L X ) + A L [ X , Y ] R ˜ ( X , Y ) L = + ϕ ( X ) A N Y ϕ ( Y ) A N X R ˜ ( X , Y ) L = + { B ( Y , A L X ) B ( X , A L Y ) + 2 d ϕ ( X , Y ) R ˜ ( X , Y ) L = + τ ( X ) ϕ ( Y ) τ ( Y ) ϕ ( X ) } N R ˜ ( X , Y ) L = + { D ( Y , A L X ) D ( X , A L Y ) + ρ ( X ) ϕ ( Y ) ρ ( Y ) ϕ ( X ) } L ,
(3.3)
R ( X , Y ) P Z = R ( X , Y ) P Z + C ( X , P Z ) A ξ Y C ( Y , P Z ) A ξ X R ( X , Y ) P Z = + { ( X C ) ( Y , P Z ) ( Y C ) ( X , P Z ) R ( X , Y ) P Z = + C ( X , P Z ) [ τ ( Y ) + π ( Y ) ] C ( Y , P Z ) [ τ ( X ) + π ( X ) ] } ξ ,
(3.4)
R ( X , Y ) ξ = X ( A ξ Y ) + Y ( A ξ X ) + A ξ [ X , Y ] + τ ( Y ) A ξ X R ( X , Y ) ξ = τ ( X ) A ξ Y + { C ( Y , A ξ X ) C ( X , A ξ Y ) 2 d τ ( X , Y ) } ξ .
(3.5)

A semi-Riemannian manifold M ˜ of constant curvature c is called a semi-Riemannian space form and denote it by M ˜ (c). The curvature tensor R ˜ of M ˜ (c) is given by

R ˜ (X,Y)Z=c { g ˜ ( Y , Z ) X g ˜ ( X , Z ) Y } ,X,Y,ZΓ(T M ˜ ).
(3.6)

Taking the scalar product with ξ and L to (3.6), we obtain g ˜ ( R ˜ (X,Y)Z,ξ)=0 and g ˜ ( R ˜ (X,Y)Z,L)=0 for any X,Y,ZΓ(TM). From these equations and (3.1), we get

R ˜ ( X , Y ) Z = R ( X , Y ) Z + B ( X , Z ) A N Y B ( Y , Z ) A N X + D ( X , Z ) A L Y D ( Y , Z ) A L X , X , Y , Z Γ ( T M ) .
(3.7)

4 Screen conformal half lightlike submanifolds

Definition 1 A half lightlike submanifold M of a semi-Riemannian manifold M ˜ is said to be irrotational [14] if ˜ X ξΓ(TM) for any XΓ(TM).

From (2.7) and (2.14), we show that the above definition is equivalent to

D(X,ξ)=0=ϕ(X),XΓ(TM).

Theorem 4.1 Let M be an irrotational half lightlike submanifold of a semi-Riemannian manifold M ˜ admitting a semi-symmetric non-metric connection such that ζ is tangent to M. Then ζ is conjugate to any vector field X on M, i.e., ζ satisfies h(X,ζ)=0.

Proof Taking the scalar product with ξ to (3.2) and N to (3.1) such that Z=ξ by turns and using (2.14), (3.5) and the fact that ϕ=0, we obtain

g ˜ ( R ˜ ( X , Y ) ξ , N ) = B ( X , A N Y ) B ( Y , A N X ) 2 d τ ( X , Y ) = C ( Y , A ξ X ) C ( X , A ξ Y ) 2 d τ ( X , Y ) .

From these two representations, we obtain

B(X, A N Y)B(Y, A N X)=C ( Y , A ξ X ) C ( X , A ξ Y ) .

Using (2.15)1, (2.17)2 and the fact that A ξ is self-adjoint, we have

π ( A ξ X ) η(Y)=π ( A ξ Y ) η(X).

Replacing Y by ξ to this equation and using (2.18), we have

B(X,ζ)=π ( A ξ X ) =0.
(4.1)

As D is symmetric and ϕ=0, we show that A L is self-adjoint. Taking the scalar product with L to (3.2) and N to (3.3) with ϕ=0 by turns, we obtain

g ˜ ( R ˜ ( X , Y ) N , L ) = g ˜ ( X ( A L Y ) Y ( A L X ) A L [ X , Y ] , N ) = D ( Y , A N X ) D ( X , A N Y ) + 2 d ρ ( X , Y ) + ρ ( X ) τ ( Y ) ρ ( Y ) τ ( X ) .

Using these two representations and (2.16)2, we show that

D ( Y , A N X ) D ( X , A N Y ) + 2 d ρ ( X , Y ) + ρ ( X ) τ ( Y ) ρ ( Y ) τ ( X ) = g ˜ ( X ( A L Y ) , N ) g ˜ ( Y ( A L X ) , N ) ρ ( [ X , Y ] ) .

Applying ˜ X to g ˜ ( A L Y,N)=ρ(Y) and using (2.1), (2.7) and (2.8), we have

g ˜ ( X ( A L Y ) , N ) = X ( ρ ( Y ) ) + π ( A L Y ) η ( X ) + f g ( X , A L Y ) + g ( A L Y , A N X ) τ ( X ) ρ ( Y ) .

Substituting this equation into the last equation and using (2.16)1, we have

π( A L X)η(Y)=π( A L Y)η(X).

Replacing Y by ξ to this equation, we have

π( A L X)=π( A L ξ)η(X).

Taking X=ξ and Y=ζ to (2.16)1, we get π( A L ξ)=0. Therefore, we have

D(X,ζ)=π( A L X)=0.
(4.2)

From (4.1) and (4.2), we show that h(X,ζ)=0 for all XΓ(TM). □

Definition 2 A half lightlike submanifold M of a semi-Riemannian manifold M ˜ is screen conformal [4, 5, 7] if the second fundamental forms B and C satisfy

C(X,PY)=φB(X,Y),X,YΓ(TM),
(4.3)

where φ is a non-vanishing function on a coordinate neighborhood in M.

Theorem 4.2 Let M be an irrotational half lightlike submanifold of a semi-Riemannian space form M ˜ (c) admitting a semi-symmetric non-metric connection such that ζ is tangent to M. If M is screen conformal, then c=0.

Proof Substituting (3.6) into (3.2) and using the fact that ϕ=0, we have

( X B ) ( Y , Z ) ( Y B ) ( X , Z ) = B ( Y , Z ) { π ( X ) τ ( X ) } B ( X , Z ) { π ( Y ) τ ( Y ) } .
(4.4)

Taking the scalar product with N to (3.1) and (3.4) by turns and using (2.16)2, (2.17)2 and (3.6), we have the following two forms of g ˜ (R(X,Y)PZ,N):

{ c g ( Y , P Z ) f B ( Y , P Z ) } η ( X ) { c g ( X , P Z ) f B ( X , P Z ) } η ( Y ) + ρ ( X ) D ( Y , P Z ) ρ ( Y ) D ( X , P Z ) = ( X C ) ( Y , P Z ) ( Y C ) ( X , P Z ) + C ( X , P Z ) { π ( Y ) + τ ( Y ) } C ( Y , P Z ) { π ( X ) + τ ( X ) } .
(4.5)

Applying X to C(Y,PZ)=φB(Y,PZ), we have

( X C)(Y,PZ)=X[φ]B(Y,PZ)+φ( X B)(Y,PZ).

Substituting this into (4.5) and using (4.4), we obtain

c { g ( Y , P Z ) η ( X ) g ( X , P Z ) η ( Y ) } = { X [ φ ] 2 φ τ ( X ) + f η ( X ) } B ( Y , P Z ) ρ ( X ) D ( Y , P Z ) { Y [ φ ] 2 φ τ ( Y ) + f η ( Y ) } B ( X , P Z ) + ρ ( Y ) D ( X , P Z ) .
(4.6)

Replacing Z by ζ to (4.5) and using (4.1) and (4.2), we have c=0. □

Remark 4.3 If M is screen conformal, then, from (4.3), we show that C is symmetric on S(TM). By Theorem 2.1, S(TM) is integrable and M is locally a product manifold C 1 × M , where C 1 is a null curve tangent to Rad(TM) and M is a leaf of S(TM).

5 Main theorem

Let Ric ˜ be the Ricci curvature tensor of M ˜ and R ( 0 , 2 ) the induced Ricci type tensor on M given respectively by

Ric ˜ ( X , Y ) = trace { Z R ˜ ( Z , X ) Y } , X , Y Γ ( T M ˜ ) , R ( 0 , 2 ) ( X , Y ) = trace { Z R ( Z , X ) Y } , X , Y Γ ( T M ) .

Using the quasi-orthonormal frame field (2.6) on M ˜ , we show [10] that

R ( 0 , 2 ) ( X , Y ) = Ric ˜ ( X , Y ) + B ( X , Y ) tr A N + D ( X , Y ) tr A L g ( A N X , A ξ Y ) g ( A L X , A L Y ) + ρ ( X ) ϕ ( Y ) g ˜ ( R ˜ ( ξ , Y ) X , N ) g ˜ ( R ˜ ( L , X ) Y , L ) ,

where tr A N is the trace of A N . From this, we show that R ( 0 , 2 ) is not symmetric. The tensor field R ( 0 , 2 ) is called the induced Ricci curvature tensor [4, 5] of M, denoted by Ric, if it is symmetric. M is called Ricci flat if its induced Ricci tensor vanishes on M. It is known [10] that R ( 0 , 2 ) is symmetric if and only if the 1-form τ is closed, i.e., dτ=0.

Remark 5.1 If the induced Ricci type tensor R ( 0 , 2 ) is symmetric, then there exists a null pair {ξ,N} such that the corresponding 1-form τ satisfies τ=0 [3, 4], which is called a canonical null pair of M. Although S(TM) is not unique, it is canonically isomorphic to the factor vector bundle S ( T M ) =TM/Rad(TM) [14]. This implies that all screen distribution are mutually isomorphic. For this reason, in case dτ=0, we consider only lightlike hypersurfaces M endow with the canonical null pair such that τ=0.

We say that M is an Einstein manifold if the Ricci tensor of M satisfies

Ric=κg.

It is well known that if dimM>2, then κ is a constant.

Let dim M ˜ =m+3. In case M ˜ is a semi-Riemannian space form M ˜ (c), we have

R ( 0 , 2 ) ( X , Y ) = m c g ( X , Y ) + B ( X , Y ) tr A N + D ( X , Y ) tr A L g ( A N X , A ξ Y ) g ( A L X , A L Y ) + ρ ( X ) ϕ ( Y ) .
(5.1)

Due to (2.15) and (2.17), we show that M is screen conformal if and only if the shape operators A N and A ξ are related by

A N X=φ A ξ XfXη(X)ζ.
(5.2)

Assume that ϕ=0. As D is symmetric, A L is self-adjoint. Using this, (5.1) and (5.2), we show that R ( 0 , 2 ) is symmetric. Thus, we can take τ=0. As τ=0, (4.4) reduce to

( X B)(Y,Z)( Y B)(X,Z)=π(X)B(Y,Z)π(Y)B(X,Z).
(5.3)

Definition 3 A vector field X on M ˜ is said to be conformal Killing [3, 5, 15] if L ˜ X g ˜ =2δ g ˜ for a scalar function δ, where L ˜ denotes the Lie derivative on M ˜ , that is,

( L ˜ X g ˜ )(Y,Z)=X ( g ˜ ( Y , Z ) ) g ˜ ( [ X , Y ] , Z ) g ˜ ( Y , [ X , Z ] ) ,Y,ZΓ(T M ˜ ).

In particular, if δ=0, then X is called a Killing vector field on M ˜ .

Theorem 5.2 Let M be a half lightlike submanifold of a semi-Riemannian manifold M ˜ admitting a semi-symmetric non-metric connection. If the canonical normal vector field L is a conformal Killing one, then L is a Killing vector field.

Proof Using (2.1) and (2.2), for any X,Y,ZΓ(T M ˜ ), we have

( L ˜ X g ˜ )(Y,Z)= g ˜ ( ˜ Y X,Z)+ g ˜ (Y, ˜ Z X)2π(X) g ˜ (Y,Z).

As L is a conformal Killing vector field, we have g ˜ ( ˜ X L,Y)=D(X,Y) by (2.9) and (2.16). This implies ( L ˜ L g ˜ )(X,Y)=2D(X,Y) for any X,YΓ(TM). Thus, we have

D(X,Y)=δg(X,Y),X,YΓ(TM).
(5.4)

Taking X=Y=ζ and using (4.2), we get δ=0. Thus, L is a Killing vector field. □

Remark 5.3 Cǎlin [16] proved the following result. For any lightlike submanifolds M of indefinite almost contact metric manifolds M ˜ , if ζ is tangent to M, then it belongs to S(TM). Duggal and Sahin also proved this result (see pp.318-319 of [5]). After Cǎlin’s work, many earlier works [1719], which were written on lightlike submanifolds of indefinite almost contact metric manifolds or lightlike submanifolds of semi-Riemannian manifolds, admitting semi-symmetric non-metric connections, obtained their results by using the Cǎlin’s result described in above. However, Jin [12, 13] proved that Cǎlin’s result is not true for any lightlike submanifolds M of a semi-Riemannian space form M ˜ (c), admitting a semi-symmetric non-metric connection.

For the rest of this section, we may assume that the structure vector field ζ of M ˜ belongs to the screen distribution S(TM). In this case, we show that f=0.

Theorem 5.4 Let M be a screen conformal Einstein half lightlike submanifold of a Lorentzian space form M ˜ (c), admitting a semi-symmetric non-metric connection such that ζ belongs to S(TM). If the canonical normal vector field L is conformal Killing, then M is Ricci flat. Moreover, if the mean curvature of M is constant, then M is locally a product manifold C 1 × C 2 × M m 1 , where C 1 and C 2 are null and non-null curves, and M m 1 is an (m1)-dimensional Euclidean space.

Proof As L is conformal Killing vector field, D= A L =0 by (5.4) and Theorem 5.2. Therefore, from (2.14), we show that ϕ=0, i.e., M is irrotational. By Theorem 4.2, we also have c=0. Using (2.15), (4.1) and (5.2) with f=0, from (5.1), we have

g ( A ξ X , A ξ Y ) αg ( A ξ X , Y ) + φ 1 κg(X,Y)=0
(5.5)

due to c=0, where α=tr A ξ . As g( A ξ ζ,X)=B(ζ,X)=0 for all XΓ(TM) and S(TM) is non-degenerate, we show that

A ξ ζ=0.
(5.6)

Taking X=Y=ζ to (5.5) and using (5.6), we have φ 1 κ=0. Thus, (5.5) reduce to

g ( A ξ X , A ξ Y ) αg ( A ξ X , Y ) =0,κ=0.
(5.7)

From the second equation of (5.7), we show that M is Ricci flat.

As M is screen conformal and M ˜ is Lorentzian, S(TM) is an integrable Riemannian vector bundle. Since ξ is an eigenvector field of A ξ , corresponding to the eigenvalue 0 due to (2.15), and A ξ is S(TM)-valued real self-adjoint operator, A ξ has m real orthonormal eigenvector fields in S(TM) and is diagonalizable. Consider a frame field of eigenvectors {ξ, E 1 ,, E m } of A ξ such that { E 1 ,, E m } is an orthonormal frame field of S(TM) and A ξ E i = λ i E i . Put X=Y= E i in (5.7), each eigenvalue λ i is a solution of

x 2 αx=0.

As this equation has at most two distinct solutions 0 and α, there exists p{0,1,,m} such that λ 1 == λ p =0 and λ p + 1 == λ m =α, by renumbering if necessary. As tr A ξ =0p+(mp)α, we have

α=tr A ξ =(mp)α.

So p=m1, i.e.,

A ξ =( 0 0 α ).

Consider two distributions D o and D α on S(TM) given by

D o = { X Γ ( S ( T M ) ) A ξ X = 0  and  X 0 } , D α = { U Γ ( S ( T M ) ) A ξ U = α U  and  U 0 } .

Clearly we show that D o D α ={0} as α0. In the sequel, we take X,YΓ( D o ), U,VΓ( D α ) and Z,WΓ(S(TM)). Since X and U are eigenvector fields of the real self-adjoint operator A ξ , corresponding to the different eigenvalues 0 and α respectively, we have g(X,U)=0. From this and the fact that B(X,U)=g( A ξ X,U)=0, we show that D α g D o and D α B D o , respectively. Since { E i } 1 i m 1 and { E m } are vector fields of D o and D α , respectively, and D o and D α are mutually orthogonal, we show that D o and D α are non-degenerate distributions of rank (m1) and rank 1, respectively. Thus, the screen distribution S(TM) is decomposed as S(TM)= D α orth D o .

From (5.7), we get A ξ ( A α αP)=0. Let WIm A ξ . Then there exists ZΓ(S(TM)) such that W= A ξ Z. Then ( A ξ αP)W=0 and WΓ( D α ). Thus, Im A ξ Γ( D α ). By duality, we have Im( A ξ αP)Γ( D o ).

Applying X to B(Y,U)=0 and using (2.15) and A ξ Y=0, we obtain

( X B)(Y,U)=g ( A ξ X Y , U ) .

Substituting this into (5.3) and using (2.12) and A ξ X= A ξ Y=0, we get

g ( A ξ [ X , Y ] , U ) =0.

As Im A ξ Γ( D α ) and D α is non-degenerate, we get A ξ [X,Y]=0. This implies that [X,Y]Γ( D o ). Thus, D o is an integrable distribution.

Applying U to B(X,Y)=0 and X to B(U,Y)=0, we have

( U B)(X,Y)=0,( X B)(U,Y)=αg( X Y,U).

Substituting this two equations into (5.3), we have αg( X Y,U)=0. As

g ( A ξ X Y , U ) =B( X Y,U)=αg( X Y,U)=0

and Im A ξ Γ( D α ) and D α is non-degenerate, we get A ξ X Y=0. This implies that X YΓ( D o ). Thus, D o is an auto-parallel distribution on S(TM).

As A ξ ζ=0, ζ belongs to D o . Thus, π(U)=0 for any UΓ( D α ). Applying X to g(U,Y)=0 and using (2.12) and the fact that D o is auto-parallel, we get g( X U,Y)=0. This implies that X UΓ( D α ).

Applying U to B(V,X)=0 and using A ξ X=0, we obtain

( U B)(V,X)=αg(V, U X).

Substituting this into (5.3) and using the fact that D o B D α , we get

g(V, U X)=g(U, V X).

Applying U to g(V,X)=0 and using (2.12), we get

g( U V,X)=π(X)g(U,V)g(V, V X).

Taking the skew-symmetric part of this and using (2.13), we obtain

g ( [ U , V ] , X ) =0.

This implies that [U,V]Γ( D α ) and D α is an integrable distribution.

Now we assume that the mean curvature H= 1 m + 2 trB= 1 m + 2 tr A ξ of M is a constant. As tr A ξ =α, we see that α is a constant. Applying X to B(U,V)=αg(U,V) and U to B(X,V)=0 by turns and using the facts that X UΓ(TM), D o g D α , D o B D α and B(X, U V)=g( A ξ X, U V)=0, we have

( X B)(U,V)=0,( U B)(X,V)=αg( U X,V).

Substituting these two equations into (5.3) and using D o B D α , we have

g( U X,V)=π(X)g(U,V).

Applying U to g(X,V)=0 and using (2.12), we obtain

g(X, U V)=π(X)g(U,V)g( U X,V)=0.

Thus, D α is also an integrable and auto-parallel distribution.

Since the leaf M of S(TM) is a Riemannian manifold and S(TM)= D α orth D o , where D α and D o are auto-parallel distributions of M , by the decomposition of the theorem of de Rham [20], we have M = C 2 × M m 1 , where C 2 is a leaf of D α , and M m 1 is a totally geodesic leaf of D o . Consider the frame field of eigenvectors {ξ, E 1 ,, E m } of A ξ such that { E i } i is an orthonormal frame field of S(TM), then B( E i , E j )=C( E i , E j )=0 for 1i<jm and B( E i , E i )=C( E i , E i )=0 for 1im1. From (3.1) and (3.4), we have g ˜ ( R ˜ ( E i , E j ) E j , E i )=g( R ( E i , E j ) E j , E i )=0. Thus, the sectional curvature K of the leaf M m 1 of D o is given by

K( E i , E j )= g ( R ( E i , E j ) E j , E i ) g ( E i , E i ) g ( E j , E j ) g 2 ( E i , E j ) =0.

Thus, M is a local product C 1 × C 2 × M m 1 , where C 1 is a null curve, C 2 is a non-null curve, and M m 1 is an (m1)-dimensional Euclidean space. □

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Jin, D.H. Einstein half lightlike submanifolds of a Lorentzian space form with a semi-symmetric non-metric connection. J Inequal Appl 2013, 403 (2013). https://doi.org/10.1186/1029-242X-2013-403

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Keywords

  • semi-symmetric non-metric connection
  • screen conformal
  • conformal Killing distribution
  • half lightlike submanifold