Review Article | Open Access
A Review on Unique Existence Theorems in Lightlike Geometry
This is a review paper of up-to-date research done on the existence of unique null curves, screen distributions, Levi-Civita connection, symmetric Ricci tensor, and scalar curvature for a large variety of lightlike submanifolds of semi-Riemannian (in particular, Lorentzian) manifolds, supported by examples and an extensive bibliography. We also propose some open problems.
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 , Beem and Ehrlich , and O’Neill . Berger’s book  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 , where , with for all nonzero , and with for all nonzero . Kupeli  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 and the canonical projection for the study of intrinsic geometry of , where 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 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 in ; that is, where is a symbol for orthogonal direct sum. The technique of using a nondegenerate was first introduced by Bejancu  for null curves and then by Bejancu and Duggal  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 book , nothing much on the above anomalies was available. In , I published a report with limited information available on how to deal with this nonuniqueness problem for null curves and hypersurfaces . 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 .
2. Canonical or Unique Nongeodesic Null Curves
Let be a smooth curve immersed in an -dimensional proper semi-Riemannian manifold of a constant index . By proper we mean that is nonzero. With respect to a local coordinate neighborhood on and a parameter , is given by where is an open interval of a real line and we denote each by . The nonzero tangent vector field on is given by .
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 . The normal bundle of is given by However, null curves behave differently compared to the nonnull curves as follows: (1) is also a null bundle subspace of ,(2).
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  in ) 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 ), respectively.
2.1. Null Curves in Lorentzian Manifolds
The main idea (first used by Cartan  in followed by Bonnor  in ) 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-arc-parameter ,(b)choose such that its first curvature function is of unit length.
Let be a nongeodesic null curve of a Minkowski spacetime with a Frenet frame where is a pseudo-arc-parameter and To deal with the problem of nonuniqueness, Cartan constructed the following, called Cartan Frenet frame, for by using the above two conditions: where the torsion function is invariant up to a sign, under Lorentzian transformations. The above frame is now called the null Cartan frame and the corresponding curve is the null Cartan curve (also, see Bonnor  for the -dimensional case using Cartan method). In Bejancu  proved the following fundamental existence and uniqueness theorem for null curves of an -dimensional Minkowski space . First we define in a quasiorthonormal basis where are null vectors such that and are orthonormal spacelike vectors. It is easy to see that for any , where we put
Theorem 1 (see ). Let be everywhere continuous functions, let be a fixed point of , let and be the quasiorthonormal basis in (6). Then, there exists a unique null curve of given by the equations , , where is a distinguished parameter on , such that and are the curvature functions of with respect to a Frenet frame that satisfies
Proof. We denote by . Using the general Frenet equations [6, page 55], consider the system of differential equations Then there exists a unique solution satisfying the initial conditions , . Furthermore, is a quasiorthonormal basis such that and are lightlike and spacelike, respectively, for each . Following Bonnor , it is proved that is a Frenet frame for with curvature functions and is the distinguished parameter on , which completes the proof.
In year 2001, the above theorem was generalized by Ferrández et al.  by constructing a canonical representation for nongeodesic null Cartan curves in a general -dimensional Lorentzian manifold as follows.
Theorem 2 (see ). Let be a null curve of an orientable Lorentzian manifold with a pseudo-arc-parameter such that a basis of for all is given by . Then there exists exactly one Frenet frame , satisfying and fulfilling the following two conditions: (i) and have the same orientation for , (ii) is positively oriented and , for all .
The above Frenet frame of the equations, its curvature functions, and the corresponding curve are called the Cartan frame, the Cartan curvatures, and the null Cartan curve, respectively.
Corollary 3. The Cartan curvatures of a curve in are invariant under Lorentzian transformations.
Example 4. Let = , , , , ) be a null curve of with vector fields of its Frenet frame given by 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 , null Cartan helices, and relativistic particles involving the curvature of and dimensional null curves and their geometric/physical applications, we refer to website of Lucas  and a Duggal and Jin book .
2.2. Null Bertrand Curves
A curve in , parameterized by the arclength, is a Bertrand curve [18, page 41] if and only if is a plane curve or its curvature and torsion are in a linear relation, with constant coefficients. In , Honda and Inoguchi  studied a pair of null curves , called a null Bertrand pair which is defined as follows. Let and be the Frenet frame of and , where and are their respective pseudo-arc-parameters. This pair is said to be a null Bertrand pair (with being a Bertrand mate of and vice versa) if and are linearly dependent. They established a relation of Bertrand pairs with null helices in . Recently, Inoguchi and Lee have published the following result on the existence of a Bertrand mate.
Theorem 6 (see ). Let be a null Cartan curve in , 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 .
Example 7. Let be a null helix with Frenet frame
It is easy to see that its torsion . Define with . Then, . Therefore, is congruent to the original curve .
In , Çöken and Çiftçi  generalized the case of Bertrand curves for the Minkowski space using the following Frenet equations (see (24)): for a Frenet frame . Following is their characterization theorem.
Theorem 8 (see ). A null Cartan curve in is a Bertrand null curve if and only if is nonzero constant and is zero.
Remark 9. In Theorem 2, Ferrández et al.  assumed the linear independence of the derivative vectors of the curve to obtain a unique Cartan frame. However, in , Sakaki  proved that the assumption in this Theorem 2 can be lessened for obtaining a unique Cartan frame in .
Open Problem. We have seen in this section that the study on Bertrand curves is focused on - and -dimensional Minkowski spaces. Theorem 2 of Ferrández et al.  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 -, -, and also -dimensional Lorentzian manifold and their relation with the corresponding unique null Cartan curves.
2.3. Null Curves in Semi-Riemannian Manifolds of Index
Let be a null curve of a semi-Riemannian manifold of index . Its screen distribution is semi-Riemannian of index . Therefore, contrary to the case of , any of its base vector might change its causal character on . This opens the possibility of more than one type of Frenet equations. To deal with this possibility, in 1999 Duggal and Jin  studied the following two types of Frenet frames for .
Type 1 Frenet Frames for . For a null curve of , any of its screen distributions is Lorentzian. Denote its general Frenet frame by when one of is timelike. Call a Frenet frame of Type 1. Similar to the case of , we have the following general Frenet equations of Type 1: where and are smooth functions on , is an orthonormal basis of , and is the signature of the manifold such that . The functions are called curvature functions of with respect to .
Example 10. Let be a null curve in given by Choose the following general Type 1 frame : Using the general Frenet equations, we obtain
Type 2 Frenet Frames for . Construct a quasiorthonormal basis consisting of the two null vector fields and and another two null vector fields and such that where and are timelike and spacelike, respectively, all taken from . The remaining subset of has all spacelike vector fields. There are choices for for a Frenet frame of the form Denote Frame by Type 2. Following exactly as in the previous case, we have the following general Frenet equations of Type 2:
Example 11. Let be a null curve in given by Choose a Frenet frame of Type 2 as follows: It is easy to obtain the following Frenet equations for the above frame :
Remark 12. Note that there are and 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 . Also, see  on null curves of .
Frenet Frames of Type . Using the above procedure, we first construct Frenet frames of null curves in . Their screen distribution is of index . Therefore, we have timelike vector fields in . To understand this, take a case when are timelike. The construction of Type 1 and Type 2 frames is exactly the same as that in the case so we give details for Type 3. Transform the Frenet frame of Type 1 into another frame which consists of two null vector fields and and additional four null vector fields , , , and such that The remaining vector fields of subset of are all spacelike. In this case, we have a Frenet frame of the form Denote frame by Type 3 which is preferable choice as any of its vector fields will not change its causal character on . In this way one can use all possible choices of two timelike vector fields from and construct corresponding forms of Frenet frames of Type 3.
The above procedure can be easily generalized to show that the null curves of have Frenet frames of Type 1, Type , Type . Also, there are a variety of each of such type and their corresponding Frenet equations.
Precisely, if , then has Type 1 of Frenet frames; if , then has two types of Frenet frames, labeled Type 1 and Type 2, up to the signs of , and if , then have -types, labeled Type 1, Type , Type , up to the signs of . However, for each , only one frame of Type will be a preferable frame as it is invariant with respect to the change of its causal character on .
Example 13. Let be a null curve in given by with Type 3 Frenet frame given as follows: Then one can calculate its following Frenet equations: Related to the focus of this paper, we now discuss the issue of unique existence of null curves in a semi-Euclidean space .
Fundamental Theorems of Unique Null Curves in . Suppose is a null curve in locally given by with a semi-Euclidean metric Let be its general Frenet frame of Type 1. Take timelike and the rest vector fields spacelike. There is one fundamental theorem for each of -types. We give details for the last Type whose Frenet frame is given by where its vector fields are defined by where are null vector fields such that In this case, we find for any , where we put Now we state the following fundamental existence and uniqueness theorem for null curves of (which also includes Theorem 1 for the case ).
Theorem 14 (see ). Let be everywhere continuous functions, a fixed point of , and , ; the quasiorthonormal basis of a Frenet frame as displayed above. Then there exists a unique null curve such that , , and are curvature and torsion functions with respect to this Frenet frame of Type satisfying
We refer to [7, Chapters 2–4] for proofs of the fundamental theorems, the geometry of all possible types of null curves in , 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 , )? 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 .
3. Unique or Canonical Theorems in Lightlike Hypersurfaces
Let be a hypersurface of a prope -dimensional semi-Riemannian manifold of constant index . Suppose is degenerate on . Then, there exists a vector field on such that , for all . The radical subspace of , at each point , is defined by and is called a lightlike hypersurface of . We call 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, in 1991, Bejancu and Duggal  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 in . This means that 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 as its sub bundle. We need the following taken from [6, Chapter 4].
There exists a unique vector bundle of over , such that for any nonzero section of on a coordinate neighborhood we have a unique section of on satisfying It follows that is lightlike such that for any . Moreover, we have the following decompositions: Hence for any screen distribution there is a unique 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 , for all . 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.
Let denote the projection morphism of on . We obtain where is the screen fundamental form of . The two second fundamental forms of and are related to their shape operators by
3.1. 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 [26–28]; Bonnor ; Leistner ; Bolós  are samples of many more authors in the literature. Also, Kupeli  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 . 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 , Bejancu  constructed a canonical for lightlike hypersurfaces of semi-Euclidean spaces . Then, in [10, Chapter 4] it was proved that such a canonical screen distribution is integrable on any lightlike hypersurface of and on any lightlike cone of . This information was used by Bejancu et al.  in showing some interested geometric results. Later on, Akivis and Goldberg  pointed out that such a canonical construction was neither invariant nor intrinsically connected with the geometry of . Therefore, in the same paper , 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 , be a quasiorthonormal basis of along , where , , and are null basis of , , and orthonormal basis of , respectively. For the same , consider other quasiorthonormal frames fields induced on by , . It is easy to obtain where are signatures of orthonormal basis and , , and are smooth functions on such that is semiorthogonal matrices. Computing and we get . 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 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 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  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 ). 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 in .
To avoid trivial ambiguities, we take connected and maximal in the sense that there is no larger domain 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 Denote by the first derivative of given by Let and be two screen distributions on , , and their second fundamental forms with respect to and , respectively, for the same . Denote by the dual -form of the vector field with respect to . Following is a unique existence theorem.
Theorem 16 (see , page 61). Let be a screen conformal lightlike hypersurface of a semi-Riemannian manifold , with 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 ;(3)if 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, is its subbundle. Assume . Then, it is easy to show that vanishes on , which implies that the functions of the transformation equations vanish. Thus, the transformation equation (51) becomes , 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].
There exists a canonical screen distribution for any lightlike hypersurface of a semi-Euclidean space ; however, only the canonical screen distribution on any lightlike hypersurface of 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 .
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 ). Let () be a lightlike hypersurface of a semi-Riemannian manifold , with a complementary vector bundle of in such that admits a covariant constant timelike vector field. Then, with respect to a section of , 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.
Example 19 (see , page 62). Consider a smooth function , where is an open set of . Then is a Monge hypersurface. The natural parameterization on is Hence, the natural frames field on is globally defined by Then spans . Therefore, is lightlike (i.e., ), if and only if the global vector field is spanned by which means, if and only if, is a solution of the partial differential equation Along consider the constant timelike section of . Then implies that is not tangent to . Therefore, the vector bundle is nondegenerate on . The complementary orthogonal vector bundle to in is a nondegenerate distribution on and is complementary to . Thus is a screen distribution on . The transversal bundle is spanned by and for any . Indeed, . The Weingarten equations reduce to and , which implies Hence, any lightlike Monge hypersurface of is screen globally conformal with . Therefore, it can admit a unique screen distribution.
3.2. 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 . 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 ). Let be a lightlike hypersurface of a semi-Riemannian manifold . Then the following assertions are equivalent:(a) is totally geodesic in ;(b) vanishes identically on ;(c) vanish on for any ;(d)There exists a unique torsion-free metric connection on ;(e)Rad TM is a Killing distribution;(f)Rad TM is a parallel distribution with respect to .
On the issue of obtaining an induced symmetric Ricci tensor of , we proceed as follows. Consider a type induced tensor on given by Let be an induced quasiorthonormal frame on , where and and let be the corresponding frames field on . Then, we obtain where denotes the causal character of respective vector field . Using Gauss-Codazzi equations, we obtain Substituting this into the previous equation and using the relations (49), we obtain where 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 ). 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.
Proof. Using the curvature identity