 Research
 Open access
 Published:
Einstein half lightlike submanifolds of a Lorentzian space form with a semisymmetric nonmetric connection
Journal of Inequalities and Applications volume 2013, Article number: 403 (2013)
Abstract
We study screen conformal Einstein half lightlike submanifolds M of a Lorentzian space form \tilde{M}(c) of constant curvature c admitting a semisymmetric nonmetric connection subject to the conditions; (1) the structure vector field of \tilde{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 semiRiemannian 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 upto 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 rlightlike 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 rlightlike submanifolds of arbitrary codimension n and arbitrary rank r. For this reason, we study half lightlike submanifold M of a semiRiemannian manifold \tilde{M}.
Ageshe and Chafle [8] introduced the notion of a semisymmetric nonmetric connection on a Riemannian manifold. Although now, we have lightlike version of a large variety of Riemannian submanifolds, the theory of lightlike submanifolds of semiRiemannian manifolds, equipped with semisymmetric metric connections, has not been introduced until quite recently. Yasar et al. [9] studied lightlike hypersurfaces in a semiRiemannian manifold admitting a semisymmetric nonmetric connection. Recently, Jin and Lee [10] and Jin [11–13] studied half lightlike and rlightlike submanifolds of a semiRiemannian manifold with a semisymmetric nonmetric connection.
In this paper, we study the geometry of screen conformal Einstein half lightlike submanifolds M of a Lorentzian space form \tilde{M}(c) of constant curvature c admitting a semisymmetric nonmetric connection subject to the conditions; (1) the structure vector field of \tilde{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 Semisymmetric nonmetric connection
Let (\tilde{M},\tilde{g}) be a semiRiemannian manifold. A connection \tilde{\mathrm{\nabla}} on \tilde{M} is called a semisymmetric nonmetric connection [8] if \tilde{\mathrm{\nabla}} and its torsion tensor \tilde{T} satisfy
for any vector fields X, Y and Z on \tilde{M}, where π is a 1form associated with a nonvanishing vector field ζ, which is called the structure vector field of \tilde{M}, by
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)=TM\cap T{M}^{\mathrm{\perp}} is a subbundle of the tangent bundle TM and the normal bundle T{M}^{\mathrm{\perp}} of rank 1. Therefore, there exist complementary nondegenerate distributions S(TM) and S(T{M}^{\mathrm{\perp}}) of Rad(TM) in TM and T{M}^{\mathrm{\perp}} respectively, which are called the screen and coscreen distributions of M, respectively, such that
where {\oplus}_{\mathrm{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 \mathrm{\Gamma}(E) the F(M) module of smooth sections of a vector bundle E over M. Choose L\in \mathrm{\Gamma}(S(T{M}^{\mathrm{\perp}})) as a unit vector field with \tilde{g}(L,L)=\pm 1. We may assume that L to be unit spacelike vector field without loss of generality, i.e., \tilde{g}(L,L)=1. We call L the canonical normal vector field of M. Consider the orthogonal complementary distribution S{(TM)}^{\mathrm{\perp}} to S(TM) in T\tilde{M}. Certainly, Rad(TM) and S(T{M}^{\mathrm{\perp}}) are subbundles of S{(TM)}^{\mathrm{\perp}}. As S(T{M}^{\mathrm{\perp}}) is nondegenerate, we have
where S{(T{M}^{\mathrm{\perp}})}^{\mathrm{\perp}} is the orthogonal complementary to S(T{M}^{\mathrm{\perp}}) in S{(TM)}^{\mathrm{\perp}}. For any null section ξ of Rad(TM) on a coordinate neighborhood \mathcal{U}\subset M, there exists a uniquely defined lightlike vector bundle ltr(TM) and a null vector field N of ltr{(TM)}_{{}_{\mathcal{U}}} satisfying
We call N, ltr(TM) and tr(TM)=S(T{M}^{\mathrm{\perp}}){\oplus}_{\mathrm{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\tilde{M} is decomposed as follows:
Given a screen distribution S(TM), there exists a unique complementary vector bundle tr(TM) to TM in T{\tilde{M}}_{M}. Using (2.4) and (2.5), there exists a local quasiorthonormal frame field of \tilde{M} along M given by
where \{{W}_{a}\} is an orthonormal frame field of S{(TM)}_{{}_{\mathcal{U}}}.
In the entire discussion of this article, we shall assume that ζ is tangent to M, and we take X,Y,Z,W\in \mathrm{\Gamma}(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
where ∇ and {\mathrm{\nabla}}^{\ast} 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}_{\xi}^{\ast} and {A}_{L} are called the shape operators, and τ, ρ and ϕ are 1forms on TM. We say that
is the second fundamental form tensor of M. Using (2.1), (2.2) and (2.7), we have
and B and D are symmetric on TM, where T is the torsion tensor with respect to the induced connection ∇, and η is a 1form on TM such that
From the facts B(X,Y)=\tilde{g}({\tilde{\mathrm{\nabla}}}_{X}Y,\xi ) and D(X,Y)=\tilde{g}({\tilde{\mathrm{\nabla}}}_{X}Y,L), we know that B and D are independent of the choice of the screen distribution S(TM) and satisfy
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
where f is the smooth function given by f=\pi (N). From (2.15) and (2.17), we show that {A}_{\xi}^{\ast} and {A}_{{}_{N}} are S(TM)valued, and {A}_{\xi} is selfadjoint operator and satisfies
that is, ξ is an eigenvector field of {A}_{\xi}^{\ast} 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 semiRiemannian manifold \tilde{M} admitting a semisymmetric nonmetric connection. Then the following assertions are equivalent:

(1)
The screen distribution S(TM) is an integrable distribution.

(2)
C is symmetric, i.e., C(X,Y)=C(Y,X) for all X,Y\in \mathrm{\Gamma}(S(TM)).

(3)
The shape operator {A}_{{}_{N}} is a selfadjoint with respect to g, i.e.,
g({A}_{{}_{N}}X,Y)=g(X,{A}_{{}_{N}}Y),\phantom{\rule{1em}{0ex}}\mathrm{\forall}X,Y\in \mathrm{\Gamma}(S(TM)).
Just as in the wellknown case of locally product Riemannian or semiRiemannian manifolds [2–4, 7], if S(TM) is an integrable distribution, then M is locally a product manifold M={\mathcal{C}}_{1}\times {M}^{\ast}, where {\mathcal{C}}_{1} is a null curve tangent to Rad(TM), and {M}^{\ast} is a leaf of the integrable screen distribution S(TM).
3 Structure equations
Denote by \tilde{R}, R and {R}^{\ast} the curvature tensors of the semisymmetric nonmetric connection \tilde{\mathrm{\nabla}} on \tilde{M}, the induced connection ∇ on M and the induced connection {\mathrm{\nabla}}^{\ast} on S(TM), respectively. Using the GaussWeingarten formulas for M and S(TM), we obtain the GaussCodazzi equations for M and S(TM):
A semiRiemannian manifold \tilde{M} of constant curvature c is called a semiRiemannian space form and denote it by \tilde{M}(c). The curvature tensor \tilde{R} of \tilde{M}(c) is given by
Taking the scalar product with ξ and L to (3.6), we obtain \tilde{g}(\tilde{R}(X,Y)Z,\xi )=0 and \tilde{g}(\tilde{R}(X,Y)Z,L)=0 for any X,Y,Z\in \mathrm{\Gamma}(TM). From these equations and (3.1), we get
4 Screen conformal half lightlike submanifolds
Definition 1 A half lightlike submanifold M of a semiRiemannian manifold \tilde{M} is said to be irrotational [14] if {\tilde{\mathrm{\nabla}}}_{X}\xi \in \mathrm{\Gamma}(TM) for any X\in \mathrm{\Gamma}(TM).
From (2.7) and (2.14), we show that the above definition is equivalent to
Theorem 4.1 Let M be an irrotational half lightlike submanifold of a semiRiemannian manifold \tilde{M} admitting a semisymmetric nonmetric 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,\zeta )=0.
Proof Taking the scalar product with ξ to (3.2) and N to (3.1) such that Z=\xi by turns and using (2.14), (3.5) and the fact that \varphi =0, we obtain
From these two representations, we obtain
Using (2.15)_{1}, (2.17)_{2} and the fact that {A}_{\xi}^{\ast} is selfadjoint, we have
Replacing Y by ξ to this equation and using (2.18), we have
As D is symmetric and \varphi =0, we show that {A}_{{}_{L}} is selfadjoint. Taking the scalar product with L to (3.2) and N to (3.3) with \varphi =0 by turns, we obtain
Using these two representations and (2.16)_{2}, we show that
Applying {\tilde{\mathrm{\nabla}}}_{X} to \tilde{g}({A}_{{}_{L}}Y,N)=\rho (Y) and using (2.1), (2.7) and (2.8), we have
Substituting this equation into the last equation and using (2.16)_{1}, we have
Replacing Y by ξ to this equation, we have
Taking X=\xi and Y=\zeta to (2.16)_{1}, we get \pi ({A}_{{}_{L}}\xi )=0. Therefore, we have
From (4.1) and (4.2), we show that h(X,\zeta )=0 for all X\in \mathrm{\Gamma}(TM). □
Definition 2 A half lightlike submanifold M of a semiRiemannian manifold \tilde{M} is screen conformal [4, 5, 7] if the second fundamental forms B and C satisfy
where φ is a nonvanishing function on a coordinate neighborhood in M.
Theorem 4.2 Let M be an irrotational half lightlike submanifold of a semiRiemannian space form \tilde{M}(c) admitting a semisymmetric nonmetric 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 \varphi =0, we have
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 \tilde{g}(R(X,Y)PZ,N):
Applying {\mathrm{\nabla}}_{X} to C(Y,PZ)=\phi B(Y,PZ), we have
Substituting this into (4.5) and using (4.4), we obtain
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 {\mathcal{C}}_{1}\times {M}^{\ast}, where {\mathcal{C}}_{1} is a null curve tangent to Rad(TM) and {M}^{\ast} is a leaf of S(TM).
5 Main theorem
Let \tilde{\mathit{Ric}} be the Ricci curvature tensor of \tilde{M} and {R}^{(0,2)} the induced Ricci type tensor on M given respectively by
Using the quasiorthonormal frame field (2.6) on \tilde{M}, we show [10] that
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 1form τ is closed, i.e., d\tau =0.
Remark 5.1 If the induced Ricci type tensor {R}^{(0,2)} is symmetric, then there exists a null pair \{\xi ,N\} such that the corresponding 1form τ satisfies \tau =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{(TM)}^{\mathrm{\u266f}}=TM/Rad(TM) [14]. This implies that all screen distribution are mutually isomorphic. For this reason, in case d\tau =0, we consider only lightlike hypersurfaces M endow with the canonical null pair such that \tau =0.
We say that M is an Einstein manifold if the Ricci tensor of M satisfies
It is well known that if dimM>2, then κ is a constant.
Let dim\tilde{M}=m+3. In case \tilde{M} is a semiRiemannian space form \tilde{M}(c), we have
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}_{\xi}^{\ast} are related by
Assume that \varphi =0. As D is symmetric, {A}_{{}_{L}} is selfadjoint. Using this, (5.1) and (5.2), we show that {R}^{(0,2)} is symmetric. Thus, we can take \tau =0. As \tau =0, (4.4) reduce to
Definition 3 A vector field X on \tilde{M} is said to be conformal Killing [3, 5, 15] if {\tilde{\mathcal{L}}}_{X}\tilde{g}=2\delta \tilde{g} for a scalar function δ, where \tilde{\mathcal{L}} denotes the Lie derivative on \tilde{M}, that is,
In particular, if \delta =0, then X is called a Killing vector field on \tilde{M}.
Theorem 5.2 Let M be a half lightlike submanifold of a semiRiemannian manifold \tilde{M} admitting a semisymmetric nonmetric 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,Z\in \mathrm{\Gamma}(T\tilde{M}), we have
As L is a conformal Killing vector field, we have \tilde{g}({\tilde{\mathrm{\nabla}}}_{X}L,Y)=D(X,Y) by (2.9) and (2.16). This implies ({\tilde{\mathcal{L}}}_{{}_{L}}\tilde{g})(X,Y)=2D(X,Y) for any X,Y\in \mathrm{\Gamma}(TM). Thus, we have
Taking X=Y=\zeta and using (4.2), we get \delta =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 \tilde{M}, if ζ is tangent to M, then it belongs to S(TM). Duggal and Sahin also proved this result (see pp.318319 of [5]). After Cǎlin’s work, many earlier works [17–19], which were written on lightlike submanifolds of indefinite almost contact metric manifolds or lightlike submanifolds of semiRiemannian manifolds, admitting semisymmetric nonmetric 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 semiRiemannian space form \tilde{M}(c), admitting a semisymmetric nonmetric connection.
For the rest of this section, we may assume that the structure vector field ζ of \tilde{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 \tilde{M}(c), admitting a semisymmetric nonmetric 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 {\mathcal{C}}_{1}\times {\mathcal{C}}_{2}\times {M}^{m1}, where {\mathcal{C}}_{1} and {\mathcal{C}}_{2} are null and nonnull curves, and {M}^{m1} 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 \varphi =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
due to c=0, where \alpha =tr{A}_{\xi}^{\ast}. As g({A}_{\xi}^{\ast}\zeta ,X)=B(\zeta ,X)=0 for all X\in \mathrm{\Gamma}(TM) and S(TM) is nondegenerate, we show that
Taking X=Y=\zeta to (5.5) and using (5.6), we have {\phi}^{1}\kappa =0. Thus, (5.5) reduce to
From the second equation of (5.7), we show that M is Ricci flat.
As M is screen conformal and \tilde{M} is Lorentzian, S(TM) is an integrable Riemannian vector bundle. Since ξ is an eigenvector field of {A}_{\xi}^{\ast}, corresponding to the eigenvalue 0 due to (2.15), and {A}_{\xi}^{\ast} is S(TM)valued real selfadjoint operator, {A}_{\xi}^{\ast} has m real orthonormal eigenvector fields in S(TM) and is diagonalizable. Consider a frame field of eigenvectors \{\xi ,{E}_{1},\dots ,{E}_{m}\} of {A}_{\xi}^{\ast} such that \{{E}_{1},\dots ,{E}_{m}\} is an orthonormal frame field of S(TM) and {A}_{\xi}^{\ast}{E}_{i}={\lambda}_{i}{E}_{i}. Put X=Y={E}_{i} in (5.7), each eigenvalue {\lambda}_{i} is a solution of
As this equation has at most two distinct solutions 0 and α, there exists p\in \{0,1,\dots ,m\} such that {\lambda}_{1}=\cdots ={\lambda}_{p}=0 and {\lambda}_{p+1}=\cdots ={\lambda}_{m}=\alpha, by renumbering if necessary. As tr{A}_{\xi}^{\ast}=0p+(mp)\alpha, we have
So p=m1, i.e.,
Consider two distributions {D}_{o} and {D}_{\alpha} on S(TM) given by
Clearly we show that {D}_{o}\cap {D}_{\alpha}=\{0\} as \alpha \ne 0. In the sequel, we take X,Y\in \mathrm{\Gamma}({D}_{o}), U,V\in \mathrm{\Gamma}({D}_{\alpha}) and Z,W\in \mathrm{\Gamma}(S(TM)). Since X and U are eigenvector fields of the real selfadjoint operator {A}_{\xi}^{\ast}, 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}_{\xi}^{\ast}X,U)=0, we show that {D}_{\alpha}\phantom{\rule{0.2em}{0ex}}{\mathrm{\perp}}_{{}_{g}}\phantom{\rule{0.2em}{0ex}}{D}_{o} and {D}_{\alpha}\phantom{\rule{0.2em}{0ex}}{\mathrm{\perp}}_{{}_{B}}\phantom{\rule{0.2em}{0ex}}{D}_{o}, respectively. Since {\{{E}_{i}\}}_{1\le i\le m1} and \{{E}_{m}\} are vector fields of {D}_{o} and {D}_{\alpha}, respectively, and {D}_{o} and {D}_{\alpha} are mutually orthogonal, we show that {D}_{o} and {D}_{\alpha} are nondegenerate distributions of rank (m1) and rank 1, respectively. Thus, the screen distribution S(TM) is decomposed as S(TM)={D}_{\alpha}{\oplus}_{\mathrm{orth}}{D}_{o}.
From (5.7), we get {A}_{\xi}^{\ast}({A}_{\alpha}^{\ast}\alpha P)=0. Let W\in Im{A}_{\xi}^{\ast}. Then there exists Z\in \mathrm{\Gamma}(S(TM)) such that W={A}_{\xi}^{\ast}Z. Then ({A}_{\xi}^{\ast}\alpha P)W=0 and W\in \mathrm{\Gamma}({D}_{\alpha}). Thus, Im{A}_{\xi}^{\ast}\subset \mathrm{\Gamma}({D}_{\alpha}). By duality, we have Im({A}_{\xi}^{\ast}\alpha P)\subset \mathrm{\Gamma}({D}_{o}).
Applying {\mathrm{\nabla}}_{X} to B(Y,U)=0 and using (2.15) and {A}_{\xi}^{\ast}Y=0, we obtain
Substituting this into (5.3) and using (2.12) and {A}_{\xi}^{\ast}X={A}_{\xi}^{\ast}Y=0, we get
As Im{A}_{\xi}^{\ast}\subset \mathrm{\Gamma}({D}_{\alpha}) and {D}_{\alpha} is nondegenerate, we get {A}_{\xi}^{\ast}[X,Y]=0. This implies that [X,Y]\in \mathrm{\Gamma}({D}_{o}). Thus, {D}_{o} is an integrable distribution.
Applying {\mathrm{\nabla}}_{U} to B(X,Y)=0 and {\mathrm{\nabla}}_{X} to B(U,Y)=0, we have
Substituting this two equations into (5.3), we have \alpha g({\mathrm{\nabla}}_{X}Y,U)=0. As
and Im{A}_{\xi}^{\ast}\subset \mathrm{\Gamma}({D}_{\alpha}) and {D}_{\alpha} is nondegenerate, we get {A}_{\xi}^{\ast}{\mathrm{\nabla}}_{X}Y=0. This implies that {\mathrm{\nabla}}_{X}Y\in \mathrm{\Gamma}({D}_{o}). Thus, {D}_{o} is an autoparallel distribution on S(TM).
As {A}_{\xi}^{\ast}\zeta =0, ζ belongs to {D}_{o}. Thus, \pi (U)=0 for any U\in \mathrm{\Gamma}({D}_{\alpha}). Applying {\mathrm{\nabla}}_{X} to g(U,Y)=0 and using (2.12) and the fact that {D}_{o} is autoparallel, we get g({\mathrm{\nabla}}_{X}U,Y)=0. This implies that {\mathrm{\nabla}}_{X}U\in \mathrm{\Gamma}({D}_{\alpha}).
Applying {\mathrm{\nabla}}_{U} to B(V,X)=0 and using {A}_{\xi}^{\ast}X=0, we obtain
Substituting this into (5.3) and using the fact that {D}_{o}\phantom{\rule{0.2em}{0ex}}{\mathrm{\perp}}_{{}_{B}}\phantom{\rule{0.2em}{0ex}}{D}_{\alpha}, we get
Applying {\mathrm{\nabla}}_{U} to g(V,X)=0 and using (2.12), we get
Taking the skewsymmetric part of this and using (2.13), we obtain
This implies that [U,V]\in \mathrm{\Gamma}({D}_{\alpha}) and {D}_{\alpha} is an integrable distribution.
Now we assume that the mean curvature H=\frac{1}{m+2}trB=\frac{1}{m+2}tr{A}_{\xi}^{\ast} of M is a constant. As tr{A}_{\xi}^{\ast}=\alpha, we see that α is a constant. Applying {\mathrm{\nabla}}_{X} to B(U,V)=\alpha g(U,V) and {\mathrm{\nabla}}_{U} to B(X,V)=0 by turns and using the facts that {\mathrm{\nabla}}_{X}U\in \mathrm{\Gamma}(TM), {D}_{o}\phantom{\rule{0.2em}{0ex}}{\mathrm{\perp}}_{g}\phantom{\rule{0.2em}{0ex}}{D}_{\alpha}, {D}_{o}\phantom{\rule{0.2em}{0ex}}{\mathrm{\perp}}_{{}_{B}}\phantom{\rule{0.2em}{0ex}}{D}_{\alpha} and B(X,{\mathrm{\nabla}}_{U}V)=g({A}_{\xi}^{\ast}X,{\mathrm{\nabla}}_{U}V)=0, we have
Substituting these two equations into (5.3) and using {D}_{o}\phantom{\rule{0.2em}{0ex}}{\mathrm{\perp}}_{{}_{B}}\phantom{\rule{0.2em}{0ex}}{D}_{\alpha}, we have
Applying {\mathrm{\nabla}}_{U} to g(X,V)=0 and using (2.12), we obtain
Thus, {D}_{\alpha} is also an integrable and autoparallel distribution.
Since the leaf {M}^{\ast} of S(TM) is a Riemannian manifold and S(TM)={D}_{\alpha}{\oplus}_{\mathrm{orth}}{D}_{o}, where {D}_{\alpha} and {D}_{o} are autoparallel distributions of {M}^{\ast}, by the decomposition of the theorem of de Rham [20], we have {M}^{\ast}={\mathcal{C}}_{2}\times {M}^{m1}, where {\mathcal{C}}_{2} is a leaf of {D}_{\alpha}, and {M}^{m1} is a totally geodesic leaf of {D}_{o}. Consider the frame field of eigenvectors \{\xi ,{E}_{1},\dots ,{E}_{m}\} of {A}_{\xi}^{\ast} 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 1\le i<j\le m and B({E}_{i},{E}_{i})=C({E}_{i},{E}_{i})=0 for 1\le i\le m1. From (3.1) and (3.4), we have \tilde{g}(\tilde{R}({E}_{i},{E}_{j}){E}_{j},{E}_{i})=g({R}^{\ast}({E}_{i},{E}_{j}){E}_{j},{E}_{i})=0. Thus, the sectional curvature K of the leaf {M}^{m1} of {D}_{o} is given by
Thus, M is a local product {\mathcal{C}}_{1}\times {\mathcal{C}}_{2}\times {M}^{m1}, where {\mathcal{C}}_{1} is a null curve, {\mathcal{C}}_{2} is a nonnull curve, and {M}^{m1} is an (m1)dimensional Euclidean space. □
References
Hawking SW, Ellis GFR: The Large Scale Structure of SpaceTime. Cambridge University Press, Cambridge; 1973.
O’Neill B: SemiRiemannian Geometry with Applications to Relativity. Academic Press, New York; 1983.
Duggal KL, Bejancu A: Lightlike Submanifolds of SemiRiemannian Manifolds and Applications. Kluwer Academic, Dordrecht; 1996.
Duggal KL, Jin DH: Null Curves and Hypersurfaces of SemiRiemannian Manifolds. World Scientific, Hackensack; 2007.
Duggal KL, Sahin B Frontiers in Mathematics. In Differential Geometry of Lightlike Submanifolds. Birkhäuser, Basel; 2010.
Duggal KL, Bejancu A: Lightlike submanifolds of codimension 2. Math. J. Toyama Univ. 1992, 15: 59–82.
Duggal KL, Jin DH: Halflightlike submanifolds of codimension 2. Math. J. Toyama Univ. 1999, 22: 121–161.
Ageshe NS, Chafle MR: A semisymmetric nonmetric connection on a Riemannian manifold. Indian J. Pure Appl. Math. 1992, 23(6):399–409.
Yasar E, Cöken AC, Yücesan A: Lightlike hypersurfaces in semiRiemannian manifold with semisymmetric nonmetric connection. Math. Scand. 2008, 102: 253–264.
Jin DH, Lee JW: A classification of half lightlike submanifolds of a semiRiemannian manifold with a semisymmetric nonmetric connection. Bull. Korean Math. Soc. 2013, 50(3):705–714. 10.4134/BKMS.2013.50.3.705
Jin DH: Lightlike submanifolds of a semiRiemannian manifold with a semisymmetric nonmetric connection. J. Korean Soc. Math. Edu., Ser. B. Pure Appl. Math. 2012, 19(3):211–228.
Jin, DH: Two characterization theorems for irrotational lightlike geometry. Commun. Korean Math. Soc. (2012, accepted)
Jin, DH: Singularities of lightlike submanifolds in a semiRiemannian space form. J. Korean Math. Soc. J. Appl. Math. (2013, submitted)
Kupeli DN 366. In Singular SemiRiemannian Geometry. Kluwer Academic, Dordrecht; 1996.
Duggal KL, Sharma R 487. In Symmetries of Spacetimes and Riemannian Manifolds. Kluwer Academic, Dordrecht; 1999.
Cǎlin, C: Contributions to geometry of CRsubmanifold. Thesis, University of Iasi, Iasi, Romania (1998)
Duggal KL, Sahin B: Lightlike submanifolds of indefinite Sasakian manifolds. Int. J. Math. Math. Sci. 2007., 2007: Article ID 57585
Duggal KL, Sahin B: Generalized CauchyRiemann lightlike submanifolds of indefinite Sasakian manifolds. Acta Math. Hung. 2009, 122(1–2):45–58. 10.1007/s1047400872218
Kang TH, Jung SD, Kim BH, Pak HK, Pak JS: Lightlike hypersurfaces of indefinite Sasakian manifolds. Indian J. Pure Appl. Math. 2003, 34: 1369–1380.
de Rham G: Sur la réductibilité d’un espace de Riemannian. Comment. Math. Helv. 1952, 26: 328–344. 10.1007/BF02564308
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The author declares that he has no competing interests.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Jin, D.H. Einstein half lightlike submanifolds of a Lorentzian space form with a semisymmetric nonmetric connection. J Inequal Appl 2013, 403 (2013). https://doi.org/10.1186/1029242X2013403
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029242X2013403