A Review on Unique Existence Theorems in Lightlike Geometry

The theory of Riemannian and semi-Riemannian manifolds (M, g) and their submanifold is one of the most interesting areas of research in differential geometry. Most of the work on the Riemannian, semi-Riemannian, and Lorentzian manifolds has been described in the standard books by Chen [1], Beem and Ehrlich [2], and O’Neill [3]. Berger’s book [4] includes the major developments of Riemannian geometry since 1950, covering the works of differential geometers of that time andmany cited therein. In general, an inner product g on a vector space V is of type (r, l, m), where r = dim{u ∈ V | g(u, V) = 0 for all V ∈ V}, l = sup{dimW | W ⊂ V with g(w,w) < 0 for all nonzero w ∈ W}, and m = sup{dimW | W ⊂ V with g(w,w) > 0 for all nonzero w ∈ W}. Kupeli [5] called a manifold (M, g) of this type a singular semi-Riemannian manifold if M admits a Koszul derivative; that is, g is Lie parallel along the degenerate vector fields on M. Based on this, he studied the intrinsic geometry of such degenerate manifolds. On the other hand, a degenerate submanifold (M, g) of a semi-Riemannian manifold (M, g) may not be studied intrinsically since due to the degenerate tensor field g on M one cannot use, in general, the geometry of M. To overcome this difficulty, Kupeli used the quotient space TM∗ = TM/Rad(TM) and the canonical projection P : TM → TM∗ for the study of intrinsic geometry of M, where Rad(TM) is its radical distribution. For a general study of extrinsic geometry of degenerate submanifolds (popularly known as lightlike submanifolds) of a semi-Riemannian manifold, we refer to three books [6–8] published in 1996, 2007, and 2010, respectively. A submanifold (M, g, S(TM)) of a semi-Riemannian manifold (M, g) is called lightlike submanifold if it is a lightlike manifold with respect to the degenerate metric g induced from g and S(TM) is a nondegenerate screen distribution which is complementary of the radical distribution Rad(TM) in TM; that is,


Introduction
The theory of Riemannian and semi-Riemannian manifolds (, ) and their submanifold is one of the most interesting areas of research in differential geometry.Most of the work on the Riemannian, semi-Riemannian, and Lorentzian manifolds has been described in the standard books by Chen [1], Beem and Ehrlich [2], and O'Neill [3].Berger's book [4] includes the major developments of Riemannian geometry since 1950, covering the works of differential geometers of that time and many cited therein.In general, an inner product  on a vector space V is of type (, ℓ, ), where  = dim{ ∈ V | (, V) = 0 for all V ∈ V}, ℓ = sup{dim  |  ⊂ V with (, ) < 0 for all nonzero  ∈ }, and  = sup{dim  |  ⊂ V with (, ) > 0 for all nonzero  ∈ }.Kupeli [5] called a manifold (, ) of this type a singular semi-Riemannian manifold if  admits a Koszul derivative; that is,  is Lie parallel along the degenerate vector fields on .Based on this, he studied the intrinsic geometry of such degenerate manifolds.On the other hand, a degenerate submanifold (, ) of a semi-Riemannian manifold (, ) may not be studied intrinsically since due to the degenerate tensor field  on  one cannot use, in general, the geometry of .To overcome this difficulty, Kupeli used the quotient space  * = /Rad() and the canonical projection  :  →  * for the study of intrinsic geometry of , where Rad() is its radical distribution.
For a general study of extrinsic geometry of degenerate submanifolds (popularly known as lightlike submanifolds) of a semi-Riemannian manifold, we refer to three books [6][7][8] published in 1996, 2007, and 2010, respectively.A submanifold (, , ()) of a semi-Riemannian manifold (, ) is called lightlike submanifold if it is a lightlike manifold with respect to the degenerate metric  induced from  and () is a nondegenerate screen distribution which is complementary of the radical distribution Rad() in ; that is, where ⊕ orth is a symbol for orthogonal direct sum.The technique of using a nondegenerate () was first introduced by Bejancu [9] for null curves and then by Bejancu and Duggal [10] for hypersurfaces to study the induced geometry of lightlike submanifolds.Unfortunately, (i) the induced objects on  depend on () which, in general, is not unique.This raises the question of the existence of unique or canonical null curves and screen distributions in lightlike geometry.
(ii) The induced connection ∇ on  is not a unique metric (Levi-Civita) connection and depends on both the induced metric  and the choice of a screen, which creates a problem in justifying that the induced objects on  are geometrically stable.(iii) The induced Ricci tensor of  is not a symmetric tensor so, in general, it does not have a geometric or physical meaning similar to the Riemannian Ricci tensor, and (iv) since the inverse of degenerate metric  does not exist, one fails to have well-defined concept of a scalar curvature by contracting Ricci tensor.At the time of the 1996 book [6], nothing much on the above anomalies was available.In 2007, I published a report with limited information available on 2 Geometry how to deal with this nonuniqueness problem for null curves and hypersurfaces [11].Since then considerable further work has been done on these issues, in particular reference to all types of null curves and submanifolds, which has provided strong foundation for the lightlike geometry.The objective of this second report is to review up-to-date results on canonical or unique existence of all types of null curves and screen distributions and, then, find those lightlike submanifolds which also admit a unique metric connection, a symmetric Ricci tensor, and how to recover the induced scalar curvature, subject to some reasonable geometric conditions.
We also propose open problems.Our approach is to give brief information on the motivation for dealing with each anomaly, chronological development of the main results and a sketch of their proofs with examples.In order to include a large number of results in one paper, we provide a good bibliography with the aim to encourage those wishing to pursue this subject further.More details on these and related works may be seen in Bibliography of Lightlike Geometry prepared by Sahin [12].

Canonical or Unique Nongeodesic Null Curves
Let  be a smooth curve immersed in an (+2)-dimensional proper semi-Riemannian manifold ( =  +2  , ) of a constant index  ≥ 1.By proper we mean that  is nonzero.With respect to a local coordinate neighborhood U on  and a parameter ,  is given by   =   () ,  ∈ {0, . . .,  + 1} , where  is an open interval of a real line and we denote each   / by    .The nonzero tangent vector field on U is given by   ≡ ( 0  , . . .,  +1  ) ≡ .Suppose the curve  is a null curve which preserves its causal character.Then, all its tangent vectors are null.Thus,  is a null curve if and only if at each point  of  we have (, ) = 0.The normal bundle of  is given by (3) However, null curves behave differently compared to the nonnull curves as follows: (1)  ⊥ is also a null bundle subspace of , Thus, contrary to the case of nonnull curves, since the normal bundle  ⊥ contains the tangent bundle  of , the sum of these two bundles is not the whole of the tangent bundle .In other words, a vector of    cannot be decomposed uniquely into a component tangent to  and a component perpendicular to .Moreover, since the length of any arc of a null curve is zero, arc-length parameter makes no sense for null curves.For these reasons, in general, a Frenet frame (constructed by Bejancu [9] in 1994) on a Lorentzian manifold  along a null curve  depends on the choice of a pseudoparameter on  and a complementary (but not orthogonal) vector bundle ( ⊥ ) to  in  ⊥ , calling its screen distribution.In the following, we review how one can generate a canonical or unique set of Frenet equations subject to reasonable geometric conditions.We discuss this in two subsections of null curves in Lorentzian and semi-Riemannian manifolds (of index  > 1), respectively.

Null Curves in Lorentzian
Manifolds ( = 1).The main idea (first used by Cartan [13] in 1937 followed by Bonnor [14] in 1969) is to choose minimum number of curvature functions in the Frenet equations.We need the following two geometric conditions on a curve (): (a) () is nongeodesic with respect to a pseudo-arcparameter , (b) choose () such that its first curvature function is of unit length.
Theorem 2 (see [15]).Let () be a null curve of an orientable Lorentzian manifold ( +2 1 , ) with a pseudo-arcparameter  such that a basis of  ()  for all  is given by {  (),   (), . . .,  (+2) ()}.Then there exists exactly one Frenet frame  = {, ,  1 , . . .,   }, satisfying and fulfilling the following two conditions: Then, it is easy to obtain the following Frenet equation: Remark 5.Each null Cartan curve is a canonical representation for nongeodesic null curves.For a collection of papers on the use of null Cartan curves, soliton solutions [16], null Cartan helices, and relativistic particles involving the curvature of 3 and 4 dimensional null curves and their geometric/physical applications, we refer to website of Lucas [17] and a Duggal and Jin book [7].Theorem 6 (see [20]).Let () be a null Cartan curve in R 3  1 , where  is a pseudo-arc-parameter. Then  admits a Bertrand mate  if and only if  and  have same nonzero constant curvatures.Moreover,  is congruent to .
Theorem 8 (see [21]).A null Cartan curve in R 4  1 is a Bertrand null curve if and only if  1 is nonzero constant and  2 is zero.
In a recent paper, Mehmet and Sadik [22] have shown that a null Cartan curve in R 4  1 is not a Bertrand curve if the derivative vectors {  ,   ,  (3) ,  (4) } (see Theorem 2) of the curve are linearly independent.
Remark 9.In Theorem 2, Ferrández et al. [15] assumed the linear independence of the derivative vectors of the curve to obtain a unique Cartan frame.However, in 2010, Sakaki [23] proved that the assumption in this Theorem 2 can be lessened for obtaining a unique Cartan frame in R  1 .
Open Problem.We have seen in this section that the study on Bertrand curves is focused on 3and 4-dimensional Minkowski spaces.Theorem 2 of Ferrández et al. [15] on unique existence of null Cartan curves in a Lorentzian manifold has opened the possibility of research on null Bertrand curves and null Bertrand mates in a 3-, 4-, and also -dimensional Lorentzian manifold and their relation with the corresponding unique null Cartan curves.

Null Curves in Semi-Riemannian Manifolds of Index
Let  be a null curve of a semi-Riemannian manifold Therefore, contrary to the case of  = 1, any of its base vector { 1 , . . .,   } might change its causal character on U ⊂ .This opens the possibility of more than one type of Frenet equations.To deal with this possibility, in 1999 Duggal and Jin [24] studied the following two types of Frenet frames for  = 2.
Type 1 Frenet Frames for  = 2.For a null curve  of  +2 2 , any of its screen distributions is Lorentzian.Denote its general Frenet frame by when one of   is timelike.Call  1 a Frenet frame of Type 1. Similar to the case of  = 1, we have the following general Frenet equations of Type 1: where ℎ and { 1 , .Example 10.Let  be a null curve in R 6  2 given by  : (cosh , Choose the following general Type Using the general Frenet equations, we obtain Type 2 Frenet Frames for  = 2. Construct a quasiorthonormal basis consisting of the two null vector fields  and  and another two null vector fields   and  +1 such that where   and  +1 are timelike and spacelike, respectively, all taken from  1 .The remaining ( − 2) subset {  } of  1 has all spacelike vector fields.There are ( − 1) choices for   for a Frenet frame of the form Denote  2 Frame by Type 2. Following exactly as in the previous case, we have the following general Frenet equations of Type 2:

Geometry
Example 11.Let  be a null curve in R 6  2 given by  : (cosh , Choose a Frenet frame  2 = {, ,  1 ,  2 ,  3 ,  4 } of Type 2 as follows: It is easy to obtain the following Frenet equations for the above frame  2 : Remark 12.Note that there are  and  − 1 different choices of constructing Frenet frames and their Frenet equations of Type 1 and Type 2, respectively.Moreover, Type 2 is preferable as it is invariant with respect to the change of its causal character on U ⊂ .Also, see [25] on null curves of  +2 2 . Frenet Frames of Type (≥ 3).Using the above procedure, we first construct Frenet frames of null curves  in  +2

3
. Their screen distribution ( ⊥ ) is of index 2. Therefore, we have 2 timelike vector fields in { 1 , . . .,   }.To understand this, take a case when { 1 ,  2 } are timelike.The construction of Type 1 and Type 2 frames is exactly the same as that in the case  = 2 so we give details for Type 3. Transform the Frenet frame  1 of Type 1 into another frame which consists of two null vector fields  and  and additional four null vector fields  1 ,  2 ,  3 , and  4 such that The remaining ( − 4) vector fields of subset {  } of  1 are all spacelike.In this case, we have a Frenet frame of the form Denote  3 frame by Type 3 which is preferable choice as any of its vector fields will not change its causal character on U ⊂ .
In this way one can use all possible choices of two timelike vector fields from  1 and construct corresponding forms of Frenet frames of Type 3.
The above procedure can be easily generalized to show that the null curves of  +2  have   Frenet frames of Type 1, Type 2, . .., Type .Also, there are a variety of each of such type and their corresponding Frenet equations.
Precisely, if  = 1, then  +2 1 has Type 1 of Frenet frames; if  = 2, then  +2 2 has two types of Frenet frames, labeled Type 1 and Type 2, up to the signs of   , and if  = , then  +2  have -types, labeled Type 1, Type 2, . .., Type , up to the signs of   .However, for each  =  > 1, only one frame of Type  will be a preferable frame as it is invariant with respect to the change of its causal character on U ⊂ .
Theorem 14 (see [7]).Let   ,   , The construction of the Frenet equations is similar to Frenet equations (10) of Theorem 1 and the rest of the proof easily follows.
We refer to [7, for proofs of the fundamental theorems, the geometry of all possible types of null curves in  +2  , and many examples.
Open Problem.In previous presentation, we have seen that, contrary to the nondegenerate case, the uniqueness of any type of general Frenet equations cannot be assured even if one chooses a pseudo-arc-parameter.Each type depends on the parameter of  and the choice of a screen distribution.However, for a null curve in a Lorentzian manifold, using the natural Frenet equations we found a unique Cartan Frenet frame whose Frenet equations have a minimum number of curvature functions which are invariant under Lorentzian transformations.This raises the following question: is there exist any unique Frenet frame for null curves in a general semi-Riemannian manifold (M +2  , )? We, therefore, invite the readers to work on the following research problem.
Find condition(s) for the existence of unique Frenet frames of nongeodesic null curves in a semi-Riemannian manifold of index , where  > 2.

Unique or Canonical Theorems in Lightlike Hypersurfaces
Let (, ) be a hypersurface of a prope ( + 2)-dimensional semi-Riemannian manifold (, ) of constant index Geometry  ∈ {1, . . .,  + 1}.Suppose  is degenerate on .Then, there exists a vector field  ̸ = 0 on  such that (, ) = 0, for all  ∈ Γ().The radical subspace Rad   of   , at each point  ∈ , is defined by where () is called a screen distribution on  which is nondegenerate.Thus, along  we have the following decomposition: that is, () ⊥ is orthogonal complement to () in  | which is also nondegenerate, but it includes Rad as its sub bundle.We need the following taken from [6, Chapter 4].There exists a unique vector bundle tr() of rank 1 over , such that for any nonzero section  of Rad on a coordinate neighborhood U ⊂  we have a unique section  of tr() on U satisfying It follows that tr() is lightlike such that tr()  ∩    = {0} for any  ∈ .Moreover, we have the following decompositions: Hence for any screen distribution () there is a unique tr() which is complementary vector bundle to  in  | , called the lightlike transversal vector bundle of  with respect to ().Denote by (, , ()) a lightlike hypersurface of (, ).The Gauss and Weingarten type equations are respectively, where  is the local second fundamental form of  and   is its shape operator.It is easy to see that (, ) = 0, for all  ∈ Γ( |U ).Therefore,  is degenerate with respect to .Moreover, the connection ∇ on  is not a metric connection and satisfies (∇  ) (, ) =  (, )  () +  (, )  () .
In the lightlike case, we also have another second fundamental form and its corresponding shape operator which we now explain as follows.

Unique or Canonical Screen Distributions.
Unfortunately, the induced objects (second fundamental forms, induced connection, structure equations, etc.) depend on the choice of a screen () which, in general, is not unique.This raises the question of finding those lightlike hypersurfaces which admit a unique or canonical (), needed in the lightlike geometry.Although a positive answer to this question for an arbitrary  is not possible, due to the degenerate induced metric, considerable progress has been made to heal this essential anomaly for specific classes.There are several approaches in dealing with this nonuniqueness problem.Some authors have used specific methods suitable for their problems.For example, Akivis and Goldberg [26][27][28]; Bonnor [29]; Leistner [30]; Bolós [31] are samples of many more authors in the literature.Also, Kupeli [5] has shown that () is canonically isometric to the factor vector bundle  * = / ⊥ and used canonical projection  :  →  * in studying the intrinsic geometry of degenerate semi-Riemannian manifolds, where our review in this paper is focused on the extrinsic geometry which is in line with the classical theory of submanifolds [1].Consequently, although specific techniques are suitable for good applicable results, nevertheless, for the fundamental deeper study of extrinsic geometry of lightlike spaces, one must look for a canonical or a unique screen distribution.For this purpose, we first start with a chronological history of some isolated results and then quote two main theorems.
In 1993, Bejancu [32] constructed a canonical () for lightlike hypersurfaces  of semi-Euclidean spaces  +1  .Then, in [10,Chapter 4] it was proved that such a canonical screen distribution is integrable on any lightlike hypersurface of   1 and on any lightlike cone ∧  −1 of  +1  .This information was used by Bejancu et al. [33] in showing some interested Geometry 9 geometric results.Later on, Akivis and Goldberg [28] pointed out that such a canonical construction was neither invariant nor intrinsically connected with the geometry of .Therefore, in the same paper [28], they constructed invariant normalizations intrinsically connected with the geometry of  and investigated induced linear connections by these normalizations, using relative and absolute invariant defined by the first and second fundamental forms of .
Let  = {, ,   },  ∈ {1, . . ., } be a quasiorthonormal basis of  along , where {}, {}, and {  } are null basis of Γ(Rad |U ), Γ(tr() |U ), and orthonormal basis of Γ(() |U ), respectively.For the same , consider other quasiorthonormal frames fields   = {,   ,    } induced on U ⊂  by {  (), (tr)  ()}.It is easy to obtain where {  } are signatures of orthonormal basis {  } and    , f, and f  are smooth functions on U such that [   ] is  ×  semiorthogonal matrices.Computing (  ,   ) = 0 and (  ,   ) = 1 we get 2f + ∑  =1   (f  ) 2 = 0. Using this in the second relation of the above two equations, we get The above two relations are used to investigate the transformation of the induced objects when the pair {(), tr()} changes with respect to a change in the basis.To look for a condition so that a chosen screen is invariant with respect to a change in the basis, in 2004 Atindogbe and Duggal observed that a nondegenerate hypersurface has only one fundamental form where as a lightlike hypersurface admits an additional fundamental form of its screen distribution and their two respective shape operators.Moreover, we know [1] that the fundamental form and its shape operator of a nondegenerate hypersurface are related by the metric tensor.Contrary to this, we see from the two equations of (49) that in the lightlike case there are interrelations between its two second fundamental forms.Because of the above differences, Atindogbe and Duggal were motivated to connect the two shape operators by a conformal factor as follows.
Definition 15 (see [34]).A lightlike hypersurface (, , ()) of a semi-Riemannian manifold is called screen locally conformal if the shape operators   and  *  of  and (), respectively, are related by where  is a nonvanishing smooth function on a neighborhood U in .
To avoid trivial ambiguities, we take U connected and maximal in the sense that there is no larger domain U  ⊃ U on which the above relation holds.It is easy to show that two second fundamental forms  and  of a screen conformal lightlike hypersurface  and its (), respectively, are related by  (, ) =  (, ) , ∀, ∈ Γ( |U ) . (53) Denote by S 1 the first derivative of () given by Let () and ()  be two screen distributions on , , and   their second fundamental forms with respect to tr() and tr ()  , respectively, for the same  ∈ Γ( ⊥ | U ). Denote by  the dual 1-form of the vector field  = ∑  =1 f    with respect to .Following is a unique existence theorem.
Theorem 16 (see [8], page 61).Let (, , ()) be a screen conformal lightlike hypersurface of a semi-Riemannian manifold (, ), with S 1 the first derivative of () given by (54).Then, (1) a choice of the screen () of  satisfying (52) is integrable; (2) the one form  vanishes identically on S 1 ; (3) if S 1 coincides with (), then  can admit a unique screen distribution up to an orthogonal transformation and a unique lightlike transversal vector bundle.Moreover, for this class of hypersurfaces, the screen second fundamental form  is independent of its choice.
Proof.It follows from the screen conformal condition (52) that the shape operator  *  of () is symmetric with respect to .Therefore, a result [6, page 89] says that a choice of screen distribution of a screen conformal lightlike hypersurface  is integrable, which proves (1).
As () is integrable, S 1 is its subbundle.Assume S 1 = ().Then, it is easy to show that  vanishes on (), which implies that the functions f  of the transformation equations vanish.Thus, the transformation equation (51) becomes    = ∑  =1      , (1 ≤  ≤ ) and   = , where (   ) is an orthogonal matrix of (  ) at any point  of , which proves the first part of (3).Then independence of  follows which completes the proof.
Remark 17.Based on the above theorem, one may ask the following converse question.Does the existence of a canonical or a unique distribution () of a lightlike hypersurface imply that () is integrable?Unfortunately, the answer, in general, is negative, which we support by recalling the following known results from [6, pages 114-117].

Geometry
There exists a canonical screen distribution for any lightlike hypersurface of a semi-Euclidean space  +2  ; however, only the canonical screen distribution on any lightlike hypersurface of  +2 1 is integrable.Therefore, although any screen conformal lightlike hypersurface admits an integrable screen distribution, the above results say that not every such integrable screen coincides with the corresponding canonical screen; that is, there are cases for which S 1 ̸ = ().Now, one may ask whether there is a class of semi-Riemannian manifolds which admit screen conformal lightlike hypersurfaces and, therefore, can admit a unique screen distribution.This question has been answered as follows.
Theorem 18 (see [11]).Let (, , ()) be a lightlike hypersurface of a semi-Riemannian manifold ( +2  , ), with  a complementary vector bundle of  ⊥ in () ⊥ such that  admits a covariant constant timelike vector field.Then, with respect to a section  of Rad,  is screen conformal.Thus,  can admit a unique screen distribution.
To get a better idea of the proof of this theorem, we give the following example.

Unique Metric Connection and Symmetric Ricci Tensor.
We know from (47) that the induced connection ∇ on a lightlike submanifold (, ) is a metric (Levi-Civita) connection if and only if the second fundamental form  vanishes on .The issue is to find conditions on the induced objects of a lightlike hypersurface which admit such a unique Levi-Civita connection.First, we recall the following definitions.
In case any geodesic of  with respect to an induced connection ∇ is a geodesic of  with respect to ∇, we say that  is a totally geodesic lightlike hypersurface of .Also, note that a vector field  on a lightlike manifold (, ) is said to be a Killing vector field if £   = 0.A distribution  on  is called a Killing distribution if each vector field of  is Killing.Now we quote the following theorem on the existence of a unique metric connection on , which also shows, from the Gauss equation, that the definition of totally geodesic  does not depend on the choice of a screen.
Theorem 20 (see [6]).Let (, , ()) be a lightlike hypersurface of a semi-Riemannian manifold (, ).Then the following assertions are equivalent: where Ric is the Ricci tensor of .This shows that (, ) is not symmetric.Therefore, in general, it has no geometric or physical meaning similar to the symmetric Ricci tensor of .Thus, this (, ) can be called an induced Ricci tensor of  only if it is symmetric.Thus, one may ask the following question: are there any lightlike hypersurfaces with symmetric Ricci tensor?The answer is affirmative for which we quote the following result.
Theorem 21 (see [34]).Let (, , ()) be a locally (or globally) screen conformal lightlike hypersurface of a semi-Riemannian manifold ((), ) of constant sectional curvature .Then,  admits an induced symmetric Ricci tensor.(65) Then using the screen conformal relation (52) mentioned above it is easy to show that (, ) is symmetric and, therefore, it is an induced Ricci tensor of .
In particular, if  is totally geodesic in , then using the curvature identity and proceeding similarly to what is mentioned above one can show that  (, ) = Ric (, ) −  (, ) . (66) Since Ric and  are symmetric we conclude that any totally geodesic lightlike hypersurface of () admits an induced symmetric Ricci tensor.
Finally, we quote a general result on the induced symmetric Ricci tensor.
Remark 23.The symmetry property of the Ricci tensor on a manifold  equipped with an affine connection has also been studied by Nomizu-Sasaki.In fact, we quote the following result (Proposition 3.1, Chapter 1) in their 1994 book.
Proposition 24 (see [35]).Let (, ∇) be a smooth manifold equipped with a torsion-free affine connection.Then the Ricci tensor is symmetric if and only if there exists a volume element  satisfying ∇ = 0.
Open Problem.Give an interpretation of Theorem 22 in terms of affine geometry.

Induced Scalar Curvature.
To introduce a concept of induced scalar curvature for a lightlike hypersurface  we observe that, in general, the nonuniqueness of screen distribution () and its nondegenerate causal structure rule out the possibility of a definition for an arbitrary  of a semi-Riemannian manifold.Although now there are many cases of a canonical or unique screen and canonical transversal vector bundle, the problem of scalar curvature must be classified subject to the causal structure of a screen.For this reason, work has been done on lightlike hypersurfaces  of a Lorentzian manifold (, ) for which we know that any choice of its screen () is Riemannian.This case is also physically useful.In general,  given by the above expression cannot be called a scalar curvature of  since it has been calculated from a tensor quantity (, ).It can only have a geometric meaning if it is symmetric and its value is independent of the screen, its transversal vector bundle, and the null section .Thus to recover a scalar curvature, we recall the following conditions [36] on .
A lightlike hypersurface  (labeled by  0 ) of a Lorentzian manifold (, ) is of genus zero with screen () 0 if (a)  admits a canonical or unique screen distribution () that induces a canonical or unique lightlike transversal vector bundle ; (b)  admits an induced symmetric Ricci tensor, denoted by Ric.
It follows from (a) that () 0 and  0 are either canonical or unique.For the stability of  with respect to a choice of the second fundamental form  and the 1-form , it is easy to show that with canonical or unique  0 , both  and  are independent of the choice of  0 , except for a nonzero constant factor.Finally, we know [8, page 70] that the Ricci tensor does not depend on the choice of  0 .Thus,  is a well-defined induced scalar curvature of a class of lightlike hypersurfaces of genus zero.The following result shows that there exists a variety of Lorentzian manifolds which admit hypersurfaces of class C[] 0 .Theorem 26 (see [36]).Let (, , ()) be a screen conformal lightlike hypersurface of a Lorentzian space form ((), ), with S 1 the first derivative of () given by (54).If S 1 = (), then  belongs to C[] 0 .Consequently, this class of lightlike hypersurfaces admits induced scalar curvature of genus zero.
Since S 1 = (), it follows from Theorem 16 that  admits a unique screen distribution () that induces a unique lightlike transversal vector bundle, which satisfies the condition (a).The condition (b) also holds from Theorem 21.
Moreover, consider a class of Lorentzian manifolds (, ) which admit at least one covariant constant timelike vector field.Then, Theorem 18 says that  belongs to C[] 0 and, therefore, it admits an induced scalar curvature.Also see [37,38] for a followup on scalar curvature.Remark 27.We know from Duggal and Bejancu's book [6,Page 111] that any lightlike surface  of a 3-dimensional Lorentz manifold is either totally umbilical or totally geodesic.Moreover, there exists a canonical screen distribution () for a lightlike Monge surface (, , ()) of  3  1 for which the induced linear connection is flat (see Proposition 7.1 on page 126 in [6]).In particular, the null cone of  3  1 has flat induced connection.Readers may get more information on pages 123-138 on lightlike hypersurfaces of  3  1 ,  4 1 , and  4 2 in [6].In general, see Section 9.1 of Chapter 9 in a book by Duggal and Sahin [8] on null surfaces of spacetimes.
We follow [8] for notations.For the case (A) of -lightlike submanifold there exist two complementary nondegenerate distributions () and ( ⊥ ) of Rad() in  and  ⊥ , respectively, called the screen and screen transversal distributions on , such that
, ]  ,   , . . .: [−, ] →  be everywhere continuous functions,   = (   ) a fixed point of  +2  , and {  ,  *  ,   }, 1 ≤  ≤ ;  + 1 ≤  ≤  the quasiorthonormal basis of a Frenet frame   as displayed above.Then there exists a unique null curve  : [−, ] →  +2  such that   =   (), (0) =   , and {  ,   ,   , ]  ,   , . ..} are curvature and torsion functions with respect to this Frenet frame   of Type  satisfying dim(Rad  ) = 1 and (, ) is called a lightlike hypersurface of (, ).We call Rad a radical distribution of .Since  ⊃  ⊥ , contrary to the nondegenerate case, their sum is not the whole of tangent bundle space .In other words, a vector of    cannot be decomposed uniquely into a component tangent to    and a component of    ⊥ .Therefore, the standard text-book definition of the second fundamental form and the Gauss-Weingarten formulas do not work for the lightlike case.To deal with this problem, [10]991, Bejancu and Duggal[10]introduced a geometric technique by splitting the tangent bundle  into two nonintersecting complementary (but not orthogonal) vector bundles (one null and one nonnull) as follows.Consider a complementary vector bundle () of  ⊥ = Rad in .This means that  = Rad⊕ orth  () , For a lightlike  there exists a smooth distribution such that Rad   =    ∩    ⊥ ̸ = {0} , ∀ ∈ .