A Review on Unique Existence Theorems in Lightlike Geometry

Abstract

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.

1. Introduction

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 and many cited therein. In general, an inner product 𝑔 on a vector space V is of type (𝑟,,𝑚), where 𝑟=dim{𝑢∈𝐕∣𝑔(𝑢,𝑣)=0forall𝑣∈𝐕}, =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, 𝑇𝑀=Rad(𝑇𝑀)⊕_orth𝑆(𝑇𝑀), (1) where ⊕_orth is a symbol for orthogonal direct sum. The technique of using a nondegenerate 𝑆(𝑇𝑀) 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 𝑀 depend on 𝑆(𝑇𝑀) 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 𝑀 is not a unique metric (Levi-Civita) connection and depends on both the induced metric 𝑔 and the choice of a screen, which creates a problem in justifying that the induced objects on 𝑀 are geometrically stable. (iii) The induced Ricci tensor of 𝑀 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 𝑔 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 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].

2. Canonical or Unique Nongeodesic Null Curves

Let 𝐶 be a smooth curve immersed in an (𝑚+2)-dimensional proper semi-Riemannian manifold (𝑀=𝑀_𝑞^𝑚+2,𝑔) of a constant index 𝑞≥1. By proper we mean that 𝑞 is nonzero. With respect to a local coordinate neighborhood 𝒰 on 𝐶 and a parameter 𝑡, 𝐶 is given by 𝑥^𝑖=𝑥^𝑖(𝑡),𝑖∈{0,…,𝑚+1},rank𝑑𝑥⁰_𝑡⋯𝑑𝑥_𝑡^𝑚+1=1,∀𝑡∈𝐼, (2) where 𝐼 is an open interval of a real line and we denote each 𝑑𝑥^𝑖/𝑑𝑡 by 𝑑𝑥^𝑖_𝑡. The nonzero tangent vector field on 𝒰 is given by 𝑑_𝑡≡(𝑑𝑥⁰_𝑡,…,𝑑𝑥_𝑡^𝑚+1)≡𝜉.

Suppose the curve 𝐶 is a null curve which preserves its causal character. Then, all its tangent vectors are null. Thus, 𝐶 is a null curve if and only if at each point 𝑥 of 𝐶 we have 𝑔(𝜉,𝜉)=0. The normal bundle of 𝑇𝐶 is given by 𝑇𝐶^⟂=𝑋∈Γ(𝑇𝑀)∶𝑔(𝑋,𝜉)=0,dim𝑇𝐶^⟂_𝑥=𝑚+1. (3) However, null curves behave differently compared to the nonnull curves as follows: (1)𝑇𝐶^⟂ is also a null bundle subspace of 𝑇𝑀,(2)𝑇𝐶∩𝑇𝐶^⟂=𝑇𝐶→𝑇𝐶⊕𝑇𝐶^⟂≠𝑇𝑀.

Thus, contrary to the case of nonnull curves, since the normal bundle 𝑇𝐶^⟂ contains the tangent bundle 𝑇𝐶 of 𝐶, the sum of these two bundles is not the whole of the tangent bundle 𝑇𝑀. In other words, a vector of 𝑇_𝑥𝑀 cannot be decomposed uniquely into a component tangent to 𝐶 and a component perpendicular to 𝐶. 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 𝑀 along a null curve 𝐶 depends on the choice of a pseudoparameter on 𝐶 and a complementary (but not orthogonal) vector bundle 𝑆(𝑇𝐶^⟂) to 𝐶 in 𝑇𝐶^⟂, 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 𝑞>1), respectively.

2.1. Null Curves in Lorentzian Manifolds (𝑞=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 𝐶(𝑝):(a)𝐶(𝑝) is nongeodesic with respect to a pseudo-arc-parameter 𝑝,(b)choose 𝐶(𝑝) such that its first curvature function is of unit length.

Let 𝐶(𝑝) be a nongeodesic null curve of a Minkowski spacetime (𝑀=𝐑³₁,𝑔) with a Frenet frame {𝜉,𝑁,𝑊} where 𝑝 is a pseudo-arc-parameter and 𝑔(𝜉,𝑁)=1=𝑔(𝑊,𝑊),𝑔(𝑁,𝑁)=𝑔(𝜉,𝜉)=0. (4) To deal with the problem of nonuniqueness, Cartan constructed the following, called Cartan Frenet frame, for 𝐶(𝑝) by using the above two conditions: ∇_𝜉𝜉=𝑊,∇_𝜉𝑁=𝜏𝑊,∇_𝜉𝑊=−𝜏𝜉−𝑁, (5) where the torsion function 𝜏 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 [14] for the 4-dimensional case using Cartan method). In 1994 Bejancu [9] proved the following fundamental existence and uniqueness theorem for null curves of an (𝑚+2)-dimensional Minkowski space (𝑅₁^𝑚+2,𝑔). First we define in 𝑅₁^𝑚+2 a quasiorthonormal basis 𝑊_𝑜=−1√2,1√2,𝑊,0,…,0₁=1√2,1√2,𝑊,0,…,0₂=()𝑊0,0,1,0,,…,0,…,_𝑚+1=(0,0,…,0,1), (6) where {𝑊_𝑜,𝑊₁} are null vectors such that 𝑔(𝑊_𝑜,𝑊₁)=1 and {𝑊₂,…,𝑊_𝑚+1} are orthonormal spacelike vectors. It is easy to see that 𝑊^𝑖_𝑜𝑊^𝑗₁+𝑊^𝑗_𝑜𝑊^𝑖₁+^𝑚+1_𝛼=2𝑊^𝑖_𝛼𝑊^𝑗_𝛼=ℎ^𝑖𝑗 (7) for any 𝑖,𝑗,∈{0,…,𝑚+1}, where we put ℎ^𝑖𝑗=−1,𝑖=𝑗=01,𝑖=𝑗≠00,𝑖≠𝑗. (8)

Theorem 1 (see [9]). Let 𝑘₁,…,𝑘_2𝑚∶[−𝜖,𝜖]→𝑅 be everywhere continuous functions, let 𝑥_𝑜=(𝑥^𝑖_𝑜) be a fixed point of 𝑅₁^𝑚+2, let and {𝑊_𝑜,…,𝑊_𝑚+1} be the quasiorthonormal basis in (6). Then, there exists a unique null curve 𝐶 of 𝑅₁^𝑚+2 given by the equations 𝑥^𝑖=𝑥^𝑖(𝑝), 𝑝∈[−𝜖,𝜖], where 𝑝 is a distinguished parameter on 𝐶, such that 𝑥^𝑖_𝑜=𝑥^𝑖(0) and 𝑘₁,…,𝑘_2𝑚 are the curvature functions of 𝐶 with respect to a Frenet frame 𝐹={𝑑/𝑑𝑡,𝑁,𝑊₂,…,𝑊_𝑚+1} that satisfies 𝑑(0)=𝑊𝑑𝑝⁰,𝑁(0)=𝑊₁,𝑊_𝛼(0)=𝑊_𝛼,𝛼∈{2,…,𝑚+1}. (9)

Proof. We denote ∇_𝜉𝑋 by 𝑋^′. Using the general Frenet equations [6, page 55], consider the system of differential equations 𝑊^′_𝑜𝑝=𝑘₁𝑊₂𝑊^′₁𝑝=𝑘₂𝑊₂+𝑘₃𝑊₃𝑊^′₂𝑝=−𝑘₂𝑊_𝑜−𝑘₁𝑊₁+𝑘₄𝑊₃+𝑘₅𝑊₄𝑊^′₃𝑝=−𝑘₃𝑊_𝑜−𝑘₄𝑊₂+𝑘₆𝑊₄+𝑘₇𝑊₅⋮𝑊^′_𝑚−1𝑝=−𝑘_2𝑚−5𝑊_𝑚−3−𝑘_2𝑚−4𝑊_𝑚−2+𝑘_2𝑚−3𝑊_𝑚+𝑘_2𝑚−1𝑊_𝑚+1𝑊^′_𝑚𝑝=−𝑘_2𝑚−4𝑊_𝑚−2−𝑘_2𝑚−2𝑊_𝑚−1+𝑘_2𝑚𝑊_𝑚+1𝑊^′_𝑚+1𝑝=𝑘_2𝑚−1𝑊_𝑚−1−𝑘_2𝑚𝑊_𝑚. (10) Then there exists a unique solution {𝑊_𝑜,…,𝑊_𝑚+1} satisfying the initial conditions 𝑊_𝑖(0)=𝑊_𝑖,  𝑖∈{0,…,𝑚+1}. Furthermore, {𝑊_𝑜(𝑝),…,𝑊_𝑚+1(𝑝)} is a quasiorthonormal basis such that {𝑊_𝑜(𝑝),𝑊₁(𝑝)} and {𝑊₂(𝑝),…,𝑊_𝑚+1(𝑝)} are lightlike and spacelike, respectively, for each 𝑝∈[−𝜖,𝜖]. Following Bonnor [14], it is proved that 𝑑𝐹=𝑑𝑝=𝑊_𝑜𝑝,𝑁𝑝=𝑊₁𝑝,𝑊₂𝑝,…,𝑊_𝑚+1𝑝 (11) is a Frenet frame for 𝐶 with curvature functions {𝑘₁,…,𝑘_2𝑚} and 𝑝 is the distinguished parameter on 𝐶, which completes the proof.

In year 2001, the above theorem was generalized by Ferrández et al. [15] by constructing a canonical representation for nongeodesic null Cartan curves in a general (𝑚+2)-dimensional Lorentzian manifold as follows.

Theorem 2 (see [15]). Let 𝐶(𝑝) be a null curve of an orientable Lorentzian manifold (𝑀₁^𝑚+2,𝑔) with a pseudo-arc-parameter 𝑝 such that a basis of 𝑇_𝐶(𝑝)𝑀 for all 𝑝 is given by {𝐶^′(𝑝),𝐶^′′(𝑝),…,𝐶^(𝑚+2)(𝑝)}. Then there exists exactly one Frenet frame 𝐹={𝜉,𝑁,𝑊₁,…,𝑊_𝑚}, satisfying ∇_𝜉𝜉=𝑊₁,∇_𝜉𝑁=𝜅₁𝑊₁+𝜅₂𝑊₂,∇_𝜉𝑊₁=−𝜅₁∇𝜉−𝑁,_𝜉𝑊₂=−𝜅₂𝜉+𝜅₃𝑊₃,⋮∇_𝜉𝑊_𝑖=−𝜅_𝑖𝑊_𝑖−1+𝜅_𝑖+1𝑊_𝑖+1{},∇,𝑖∈3,…,𝑚−1_𝜉𝑊_𝑚=−𝜅_𝑚𝑊_𝑚−1 (12) and fulfilling the following two conditions: (i){𝐶^′(𝑝),𝐶^′′(𝑝),…,𝐶^(𝑖+2)(𝑝)} and {𝜉,𝑁,𝑊₁,…,𝑊_𝑖} have the same orientation for 2≤𝑖≤𝑚−1, (ii){𝜉,𝑁,𝑊₁,…,𝑊_𝑚} is positively oriented and 𝜅_𝑖>0, for all 𝑖≥2.

The above Frenet frame of the equations, its curvature functions, and the corresponding curve 𝐶 are called the Cartan frame, the Cartan curvatures, and the null Cartan curve, respectively.

Corollary 3. The Cartan curvatures of a curve 𝐶 in 𝑀₁^𝑚+2 are invariant under Lorentzian transformations.

Example 4. Let 𝐶(𝑝) = (1/√3)(√2sinh𝑝, √2cosh𝑝, sin𝑝, cos𝑝, 𝑝) be a null curve of 𝑅⁵₁ with vector fields of its Frenet frame 𝐹 given by 1𝜉=√3√2cosh𝑝,√,12sinh𝑝,cos𝑝,−sin𝑝,1𝑁=−6√35√2cosh𝑝,5√,𝑊2sinh𝑝,−7cos𝑝,7sin𝑝,−1₁=1√3√2sinh𝑝,√,𝑊2cosh𝑝,−sin𝑝,−cos𝑝,0₂=1√6√2sinh𝑝,√,𝑊2cosh𝑝,2sin𝑝,2cos𝑝,0₃1=−3√2√2cosh𝑝,√2sinh𝑝,−2cos𝑝,2sin𝑝,4. (13) Then, it is easy to obtain the following Frenet equation: ∇_𝜉𝜉=𝑊₁,∇_𝜉1𝑁=−6𝑊₁−43√2𝑊₂,∇_𝜉𝑊₁=16𝜉−𝑁,∇_𝜉𝑊₂=43√2√𝜉+23𝑊₃,∇_𝜉𝑊₃√=−23𝑊₂. (14)

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].

2.2. Null Bertrand Curves

A curve 𝐶 in 𝐑^𝟑, parameterized by the arclength, is a Bertrand curve [18, page 41] if and only if 𝐶 is a plane curve or its curvature and torsion are in a linear relation, with constant coefficients. In 2003, Honda and Inoguchi [19] studied a pair of null curves (𝐶(𝑝),𝐶(𝑝)), called a null Bertrand pair which is defined as follows. Let {𝜉,𝑁,𝑊₁} and {𝜉,𝑁,𝑊₁} be the Frenet frame of 𝐶(𝑝) and 𝐶(𝑝), where 𝑝 and 𝑝 are their respective pseudo-arc-parameters. This pair (𝐶,𝐶) is said to be a null Bertrand pair (with 𝐶 being a Bertrand mate of 𝐶 and vice versa) if 𝑊₁ and 𝑊₁ are linearly dependent. They established a relation of Bertrand pairs with null helices in 𝐑³₁. Recently, Inoguchi and Lee have published the following result on the existence of a Bertrand mate.

Theorem 6 (see [20]). Let 𝐶(𝑝) be a null Cartan curve in 𝐑³₁, where 𝑝 is a pseudo-arc-parameter. Then 𝐶 admits a Bertrand mate 𝐶 if and only if 𝐶 and 𝐶 have same nonzero constant curvatures. Moreover, 𝐶 is congruent to 𝐶.

Example 7. Let 𝐶(𝑝)=(𝑝,cos𝑝,sin𝑝) be a null helix with Frenet frame 𝐹=𝐶^′1,𝑁=2,.−1,−sin𝑝,−cos𝑝𝑊=0,−cos𝑝,−sin𝑝 (15) It is easy to see that its torsion 𝜏=1/2. Define 𝐶=𝐶+2𝑊 with 𝑝=𝑝. Then, 𝐶=(𝑝,−cos𝑝,−sin𝑝). Therefore, 𝐶 is congruent to the original curve 𝐶.
In 2005, Çöken and Çiftçi [21] generalized the 𝐑³₁ case of Bertrand curves for the Minkowski space 𝐑⁴₁ using the following Frenet equations (see (24)): ∇_𝜉𝜉=𝑊₁,∇_𝜉𝑁=𝜅₁𝑊₁+𝜅₂𝑊₂,∇_𝜉𝑊₁=−𝜅₁∇𝜉−𝑁,_𝜉𝑊₂=−𝜅₂𝜉 (16) for a Frenet frame {𝜉,𝑁,𝑊₁,𝑊₂}. Following is their characterization theorem.

Theorem 8 (see [21]). A null Cartan curve in 𝐑⁴₁ is a Bertrand null curve if and only if 𝜅₁ is nonzero constant and 𝜅₂ is zero.

In a recent paper, Mehmet and Sadik [22] have shown that a null Cartan curve in 𝐑⁴₁ is not a Bertrand curve if the derivative vectors {𝐶^′,𝐶^′′,𝐶⁽³⁾,𝐶⁽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^𝑛₁.

Open Problem. We have seen in this section that the study on Bertrand curves is focused on 3- and 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 𝑛-dimensional Lorentzian manifold and their relation with the corresponding unique null Cartan curves.

2.3. Null Curves in Semi-Riemannian Manifolds of Index 𝑞>1

Let 𝐶 be a null curve of a semi-Riemannian manifold (𝑀_𝑞^𝑚+2,𝑔) of index 𝑞>1. Its screen distribution 𝑆(𝑇𝐶^⟂) is semi-Riemannian of index 𝑞−1. Therefore, contrary to the case of 𝑞=1, any of its base vector {𝑊₁,…,𝑊_𝑚} might change its causal character on 𝒰⊂𝐶. 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 𝑞=2.

Type 1 Frenet Frames for 𝑞=2. For a null curve 𝐶 of 𝑀₂^𝑚+2, any of its screen distributions is Lorentzian. Denote its general Frenet frame by 𝐹₁=𝜉,𝑁,𝑊₁,…,𝑊_𝑚, (17) when one of 𝑊_𝑖 is timelike. Call 𝐹₁ a Frenet frame of Type 1. Similar to the case of 𝑞=1, we have the following general Frenet equations of Type 1: ∇_𝜉𝜉=ℎ𝜉+𝜅₁𝑊₁,∇_𝜉𝑁=−ℎ𝑁+𝜅₂𝑊₁+𝜅₃𝑊₂,𝜖₁∇_𝜉𝑊₁=−𝜅₂𝜉−𝜅₁𝑁+𝜅₄𝑊₂+𝜅₅𝑊₃,𝜖₂∇_𝜉𝑊₂=−𝜅₃𝜉−𝜅₄𝑊₁+𝜅₆𝑊₃+𝜅₇𝑊₄,𝜖₃∇_𝜉𝑊₃=−𝜅₅𝑊₁−𝜅₆𝑊₂+𝜅₈𝑊₄+𝜅₉𝑊₅,⋮𝜖_𝑚−1∇_𝜉𝑊_𝑚−1=−𝜅_2𝑚−3𝑊_𝑚−3−𝜅_2𝑚−2𝑊_𝑚−2+𝜅_2𝑚𝑊_𝑚,𝜖_𝑚∇_𝜉𝑊_𝑚=−𝜅_2𝑚−1𝑊_𝑚−2−𝜅_2𝑚𝑊_𝑚−1, (18) where ℎ and {𝜅₁,…,𝜅_2𝑚} are smooth functions on 𝒰, {𝑊₁,…,𝑊_𝑚} is an orthonormal basis of Γ(𝑆(𝑇𝐶^⟂)_|𝒰), and (𝜖₁,…,𝜖_𝑚) is the signature of the manifold 𝑀₂^𝑚+2 such that 𝜖_𝑖𝛿_𝑖𝑗=𝑔(𝑊_𝑖,𝑊_𝑗). The functions {𝜅₁,…,𝜅_2𝑚} are called curvature functions of 𝐶 with respect to 𝐹₁.

Example 10. Let 𝐶 be a null curve in 𝐑⁶₂ given by 1𝐶∶cosh𝑡,𝑡27³1+2𝑡,3𝑡²,1𝑡27³,√3𝑡,sinh𝑡,𝑡∈𝐑. (19) Choose the following general Type 1 frame 𝐹={𝜉,𝑁,𝑊₁,𝑊₂,𝑊₃,𝑊₄}: 𝑡𝜉=sinh𝑡,²92+2,3𝑡𝑡,²9,√,13,cosh𝑡𝑁=2𝑡sinh𝑡,−²92−2,−3𝑡𝑡,−²9,−√,𝑊3,cosh𝑡₁3=−√52cosh𝑡,92𝑡,3,29,𝑊𝑡,0,sinh𝑡₂1=−√5(),𝑊2cosh𝑡,𝑡,3,𝑡,0,2sinh𝑡₃𝑡=−0,²9+32,23𝑡𝑡,²9−12,√,𝑊3,0₄=√0,32√,0,32,1,0. (20) Using the general Frenet equations, we obtain ∇_𝜉√𝜉=−53𝑊₁,∇_𝜉𝑁=−136√5𝑊₁+2√5𝑊₂,𝜖₁∇_𝜉𝑊₁=136√5√𝜉+534𝑁+3√5𝑊₃,𝜖₂∇_𝜉𝑊₂2=−√52𝜉−√5𝑊₃,𝜖₃∇_𝜉𝑊₃4=−3√5𝑊₁+2√5𝑊₂,𝜖₃∇_𝜉𝑊₄𝜅=0,ℎ=0.₁√=−53,𝜅₂=−136√5,𝜅₃=2√5,𝜅₄=0,𝜅₅=43√5,𝜅₆2=−√5,𝜅₇=0,𝜅₈=0. (10)

Type 2 Frenet Frames for 𝑞=2. Construct a quasiorthonormal basis consisting of the two null vector fields 𝜉 and 𝑁 and another two null vector fields 𝐿_𝑖 and 𝐿_𝑖+1 such that 𝐿_𝑖=𝑊_𝑖+𝑊_𝑖+1√2,𝐿_𝑖+1=𝑊_𝑖+1−𝑊_𝑖√2,𝑔𝐿_𝑖,𝐿_𝑖+1=1, (22) where 𝑊_𝑖 and 𝑊_𝑖+1 are timelike and spacelike, respectively, all taken from 𝐹₁. The remaining (𝑚−2) subset {𝑊_𝛼} of 𝐹₁ has all spacelike vector fields. There are (𝑚−1) choices for 𝐿_𝑖 for a Frenet frame of the form 𝐹₂=𝜉,𝑁,𝑊₁,…,𝐿_𝑖,𝐿_𝑖+1,…,𝑊_𝑚. (23)Denote 𝐹₂ Frame by Type 2. Following exactly as in the previous case, we have the following general Frenet equations of Type 2: ∇_𝜉𝜉=ℎ𝜉+𝜅₁𝑊₁+𝜏₁𝑊₂,∇_𝜉𝑁=−ℎ𝑁+𝜅₂𝑊₁+𝜅₃𝑊₂+𝜏₂𝑊₃,∇_𝜉𝑊₁=𝜅₂𝜉+𝜅₁𝑁−𝜅₄𝑊₂−𝜅₅𝑊₃−𝜏₃𝑊₄,∇_𝜉𝑊₂=−𝜅₃𝜉−𝜏₁𝑁−𝜅₄𝑊₁+𝜅₆𝑊₃+𝜅₇𝑊₄+𝜏₄𝑊₅,∇_𝜉𝑊₃=−𝜏₂𝜉−𝜅₅𝑊₁−𝜅₆𝑊₂+𝜅₈𝑊₄+𝜅₉𝑊₅+𝜏₅𝑊₆,∇_𝜉𝑊₄=−𝜏₃𝑊₁−𝜅₇𝑊₂−𝜅₈𝑊₃+𝜅₁₀𝑊₅+𝜅₁₁𝑊₆+𝜏₆𝑊₇⋮∇_𝜉𝑊_𝑚−1=−𝜏_𝑚−2𝑊_𝑚−4−𝑘_2𝑚−3𝑊_𝑚−3−𝑘_2𝑚−2𝑊_𝑚−2+𝑘_2𝑚𝑊_𝑚,∇_𝜉𝑊_𝑚=−𝜏_𝑚−1𝑊_𝑚−3−𝑘_2𝑚−1𝑊_𝑚−2−𝑘_2𝑚𝑊_𝑚−1. (24)

Example 11. Let 𝐶 be a null curve in 𝐑⁶₂ given by 1𝐶∶cosh𝑡,𝑡12³1+2𝑡,2𝑡²,1𝑡12³,√3𝑡,sinh𝑡,𝑡∈𝑅. (25) Choose a Frenet frame 𝐹₂={𝜉,𝑁,𝐿₁,𝐿₂,𝑊₃,𝑊₄} of Type 2 as follows: 𝑡𝜉=sinh𝑡,²4𝑡+2,𝑡,²4,√,13,cosh𝑡𝑁=2𝑡sinh𝑡,−²4𝑡−2,−𝑡,−²4,−√,𝐿3,cosh𝑡₁=1√2𝑡cosh𝑡,2𝑡,1,2,𝐿,0,sinh𝑡₂=1√2𝑡−cosh𝑡,2𝑡,1,2,𝑊,0,−sinh𝑡₃𝑡=−0,²4+32𝑡,𝑡,²4−12,√,𝑊3,0₄=√0,32√,0,32,1,0. (26) It is easy to obtain the following Frenet equations for the above frame 𝐹₂: ∇_𝜉𝜉=√2𝐿₁,ℎ=0,∇_𝜉1𝑁=−√2𝐿₂,∇_𝜉𝐿₁1=−√21𝜉+√2𝐿₁,∇_𝜉𝐿₂=−√12𝑁+√2𝑊₃,∇_𝜉𝑊₃1=−√2𝐿₁−1√2𝐿₂,∇_𝜉𝑊₄=0. (27)

Remark 12. Note that there are 𝑚 and 𝑚−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⊂𝐶. Also, see [25] on null curves of 𝑅₂^𝑚+2.
Frenet Frames of Type 𝑞(≥3). Using the above procedure, we first construct Frenet frames of null curves 𝐶 in 𝑀₃^𝑚+2. Their screen distribution 𝑆(𝑇𝐶^⟂) is of index 2. Therefore, we have 2 timelike vector fields in {𝑊₁,…,𝑊_𝑚}. To understand this, take a case when {𝑊₁,𝑊₂} are timelike. The construction of Type 1 and Type 2 frames is exactly the same as that in the case 𝑞=2 so we give details for Type 3. Transform the Frenet frame 𝐹₁ of Type 1 into another frame which consists of two null vector fields 𝜉 and 𝑁 and additional four null vector fields 𝐿₁, 𝐿₂, 𝐿₃, and 𝐿₄ such that 𝐿₁=𝑊₁+𝑊₂√2,𝐿₂=𝑊₂−𝑊₁√2,𝑔𝐿₁,𝐿₂𝐿=1,₃=𝑊₃+𝑊₄√2,𝐿₄=𝑊₄−𝑊₃√2,𝑔𝐿₃,𝐿₄=1. (28) The remaining (𝑚−4) vector fields of subset {𝑊_𝛼} of 𝐹₁ are all spacelike. In this case, we have a Frenet frame of the form 𝐹₃=𝜉,𝑁,𝐿₁,𝐿₂,𝐿₃,𝐿₄,𝑊₅,…,𝑊_𝑚. (29) Denote 𝐹₃ frame by Type 3 which is preferable choice as any of its vector fields will not change its causal character on 𝒰⊂𝐶. In this way one can use all possible choices of two timelike vector fields from 𝐹₁ and construct corresponding forms of Frenet frames of Type 3.

The above procedure can be easily generalized to show that the null curves of 𝑀_𝑞^𝑚+2 have 𝐹_𝑞 Frenet frames of Type 1, Type 2,…, Type 𝑞. Also, there are a variety of each of such type and their corresponding Frenet equations.

Precisely, if 𝑞=1, then 𝑀₁^𝑚+2 has Type 1 of Frenet frames; if 𝑞=2, then 𝑀₂^𝑚+2 has two types of Frenet frames, labeled Type 1 and Type 2, up to the signs of 𝑊_𝑖, and if 𝑞=𝑛, then 𝑀_𝑛^𝑚+2 have 𝑛-types, labeled Type 1, Type 2,…, Type 𝑛, up to the signs of 𝑊_𝑖. However, for each 𝑞=𝑛>1, only one frame of Type 𝑛 will be a preferable frame as it is invariant with respect to the change of its causal character on 𝒰⊂𝐶.

Example 13. Let 𝐶 be a null curve in 𝐑⁶₃ given by 𝐶∶(cos𝑡,sin𝑡,sinh𝑡,cosh𝑡,𝑡,𝑡) (30) with Type 3 Frenet frame 𝐹₃={𝜉,𝑁,𝐿₁,𝐿₂,𝐿₃,𝐿₄} given as follows: (),1𝜉=−sin𝑡,cos𝑡,cosh𝑡,sinh𝑡,1,1𝑁=2(),𝐿sin𝑡,−cos𝑡,−cosh𝑡,−sinh𝑡,1,−1₁=(),𝐿−cos𝑡,−sin𝑡,sinh𝑡,cosh𝑡,0,0₂=12(),𝐿cos𝑡,sin𝑡,sinh𝑡,cosh𝑡,0,0₃=(),𝐿−sin𝑡,cos𝑡,0,0,0,1₄=(0,0,cosh𝑡,sinh𝑡,0,1). (31) Then one can calculate its following Frenet equations: ∇_𝜉𝜉=𝐿₁,∇_𝜉1𝑁=−2𝐿₁,∇_𝜉𝐿₁=−𝐿₃+𝐿₄,∇_𝜉𝐿₂=121𝜉−𝑁−2𝐿₃+𝐿₄,∇_𝜉𝐿₃=12𝐿₁−𝐿₂,∇_𝜉𝐿₄=12𝐿₁. (32) Related to the focus of this paper, we now discuss the issue of unique existence of null curves in a semi-Euclidean space (𝑅_𝑞^𝑚+2,𝑔).

Fundamental Theorems of Unique Null Curves in 𝑅_𝑞^𝑚+2. Suppose 𝐶 is a null curve in (𝑅_𝑞^𝑚+2,𝑔) locally given by 𝑥^𝐴=𝑥^𝐴(𝑡),𝑡∈𝐈⊂𝑅,𝐴∈{0,1,…,(𝑚+1)} (33) with a semi-Euclidean metric 𝑔𝑥,𝑦=−^𝑞−1_𝑖=0𝑥^𝑖𝑦^𝑖+^𝑚+1_𝛼=𝑞𝑥^𝛼𝑦^𝛼. (34) Let 𝐹₁={𝜉,𝑁,𝑊₁,…,𝑊_𝑚} be its general Frenet frame of Type 1. Take {𝑊₁,…,𝑊_𝑞−1} timelike and the rest (𝑚−𝑞−1) vector fields {𝑊_𝑞,…,𝑊_𝑚} spacelike. There is one fundamental theorem for each of 𝑞-types. We give details for the last Type 𝑞 whose Frenet frame is given by 𝐹_𝑞=𝜉=𝐿₁,𝑁=𝐿^∗₁,𝐿₂𝐿,…,_𝑞,𝐿^∗₂,…,𝐿^∗_𝑞,𝑊_𝑞+1,…,𝑊_𝑚, (35) where its vector fields are defined by 𝐿₁=1√21,0,0,0,…,0,0,…,0,0,√2,𝐿^∗₁=−1√21,0,0,0,…,0,0,…,0,0,√2,𝐿₂=10,√21,0,0,…,0,0,…,0,√2,𝐿,0_∗2=10,−√21,0,0,…,0,0,…,0,√2,𝐿,0₃=10,0,√21,0,…,0,0,…,0,√2,𝐿,0,0_∗3=10,0,−√21,0,…,0,0,…,0,√2,⋮𝐿,0,0_𝑞=10,…,0,√2,1√2,𝐿,0,…,0_∗𝑞=10,…,0,−√2,1√2,0,…,0, (36) where {𝐿₁,…,𝐿_𝑞,𝐿^∗₁,…,𝐿^∗_𝑞} are null vector fields such that 𝑔𝐿_𝑖,𝐿^∗_𝑗=𝛿_𝑖𝑗,𝑔𝐿_𝑖,𝐿_𝑗𝑔𝐿=0,_∗𝑖,𝐿^∗_𝑗=0. (37) In this case, we find ^𝑞−1_𝑖=1𝐿^𝐴_𝑖𝐿_𝑖^∗𝐵+𝐿^𝐵_𝑖𝐿_𝑖^∗𝐴=ℎ^𝐴𝐵, (38) for any 𝐴,𝐵∈{0,…,(𝑚+1)}, where we put ℎ^𝐴𝐵=;(;−1,𝐴=𝐵∈0,…,𝑞−11,𝐴=𝐵∈𝑞,…,𝑚+1)0,𝐴≠𝐵. (39) Now we state the following fundamental existence and uniqueness theorem for null curves of 𝑅_𝑞^𝑚+2 (which also includes Theorem 1 for the case 𝑞=1).

Theorem 14 (see [7]). Let 𝜅_𝑖,𝜏_𝑖,𝜇_𝑖,𝜈_𝑖,𝜂_𝑖,…∶[−𝜀,𝜀]→𝑅 be everywhere continuous functions, 𝑥_𝑜=(𝑥^𝑖_𝑜) a fixed point of 𝑅_𝑞^𝑚+2, and {𝐿_𝑖,𝐿^∗_𝑖,𝑊_𝛼}, 1≤𝑖≤𝑞; 𝑞+1≤𝛼≤𝑚 the quasiorthonormal basis of a Frenet frame 𝐹_𝑞 as displayed above. Then there exists a unique null curve 𝐶∶[−𝜀,𝜀]→𝑅_𝑞^𝑚+2 such that 𝑥^𝑖=𝑥^𝑖(𝑡), 𝐶(0)=𝑥_𝑜, and {𝜅_𝑖,𝜏_𝑖,𝜇_𝑖,𝜈_𝑖,𝜂_𝑖,…} are curvature and torsion functions with respect to this Frenet frame 𝐹_𝑞 of Type 𝑞 satisfying 𝑑𝜉=𝑑𝑝=𝐿₁,𝑁(0)=𝐿^∗₁,𝑊_𝛼(0)=𝑊_𝛼,𝑞+1≤𝛼≤𝑚. (40)

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, Chapters 2–4] for proofs of the fundamental theorems, the geometry of all possible types of null curves in 𝑀_𝑞^𝑚+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 𝐶 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_𝑞^𝑚+2, 𝑔)? 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 𝑞, where 𝑞>2.

3. Unique or Canonical Theorems in Lightlike Hypersurfaces

Let (𝑀,𝑔) be a hypersurface of a prope (𝑚+2)-dimensional semi-Riemannian manifold (𝑀,𝑔) of constant index 𝑞∈{1,…,𝑚+1}. Suppose 𝑔 is degenerate on 𝑀. Then, there exists a vector field 𝜉≠0 on 𝑀 such that 𝑔(𝜉,𝑋)=0, for all 𝑋∈Γ(𝑇𝑀). The radical subspace Rad𝑇_𝑥𝑀 of 𝑇_𝑥𝑀, at each point 𝑥∈𝑀, is defined by Rad𝑇_𝑥𝑀=𝜉∈𝑇_𝑥𝑀∶𝑔_𝑥(𝜉,𝑋)=0=𝑇_𝑥𝑀^⟂,∀𝑋∈𝑇_𝑥𝑀, (41)dim(Rad𝑇_𝑥𝑀)=1 and (𝑀,𝑔) is called a lightlike hypersurface of (𝑀,𝑔). We call Rad𝑇𝑀 a radical distribution of 𝑀. Since 𝑇𝑀⊃𝑇𝑀^⟂, contrary to the nondegenerate case, their sum is not the whole of tangent bundle space 𝑇𝑀. In other words, a vector of 𝑇_𝑥𝑀 cannot be decomposed uniquely into a component tangent to 𝑇_𝑥𝑀 and a component of 𝑇_𝑥𝑀^⟂. 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 [10] introduced a geometric technique by splitting the tangent bundle 𝑇𝑀 into two nonintersecting complementary (but not orthogonal) vector bundles (one null and one nonnull) as follows. Consider a complementary vector bundle 𝑆(𝑇𝑀) of 𝑇𝑀^⟂=Rad𝑇𝑀 in 𝑇𝑀. This means that 𝑇𝑀=Rad𝑇𝑀⊕_orth𝑆(𝑇𝑀), (42) where 𝑆(𝑇𝑀) is called a screen distribution on 𝑀 which is nondegenerate. Thus, along 𝑀 we have the following decomposition: 𝑇𝑀_∣𝑀=𝑆(𝑇𝑀)⟂𝑆(𝑇𝑀)^⟂,𝑆(𝑇𝑀)∩𝑆(𝑇𝑀)^⟂≠{0}; (43) that is, 𝑆(𝑇𝑀)^⟂ is orthogonal complement to 𝑆(𝑇𝑀) in 𝑇𝑀_∣𝑀 which is also nondegenerate, but it includes Rad𝑇𝑀 as its sub bundle. We need the following taken from [6, Chapter 4].

There exists a unique vector bundle tr(𝑇𝑀) of rank1 over 𝑀, such that for any nonzero section 𝜉 of Rad𝑇𝑀 on a coordinate neighborhood 𝒰⊂𝑀 we have a unique section 𝑁 of tr(𝑇𝑀) on 𝒰 satisfying 𝑔(𝑁,𝜉)=1,𝑔(𝑁,𝑁)=𝑔(𝑁,𝑊)=0,∀𝑊∈Γ𝑆𝑇(𝑀)∣_𝒰. (44) It follows that tr(𝑇𝑀) is lightlike such that tr(𝑇𝑀)_𝑢∩𝑇_𝑢𝑀={0} for any 𝑢∈𝑀. Moreover, we have the following decompositions: 𝑇𝑀_∣𝑀=𝑆(𝑇𝑀)⊕_orth𝑇𝑀^⟂⊕tr(𝑇𝑀)=𝑇𝑀⊕tr(𝑇𝑀). (45) Hence for any screen distribution 𝑆(𝑇𝑀) there is a unique tr(𝑇𝑀) which is complementary vector bundle to 𝑇𝑀 in 𝑇𝑀_|𝑀, called the lightlike transversal vector bundle of 𝑀 with respect to 𝑆(𝑇𝑀). Denote by (𝑀,𝑔,𝑆(𝑇𝑀)) a lightlike hypersurface of (𝑀,𝑔). The Gauss and Weingarten type equations are ∇_𝑋𝑌=∇_𝑋𝑌+𝐵(𝑋,𝑌)𝑁,∇_𝑋𝑁=−𝐴_𝑁𝑋+𝜏(𝑋)𝑁, (46) respectively, where 𝐵 is the local second fundamental form of 𝑀 and 𝐴_𝑁 is its shape operator. It is easy to see that 𝐵(𝑋,𝜉)=0, for all 𝑋∈Γ(𝑇𝑀_|𝒰). Therefore, 𝐵 is degenerate with respect to 𝑔. Moreover, the connection ∇ on 𝑀 is not a metric connection and satisfies ∇_𝑋𝑔(𝑌,𝑍)=𝐵(𝑋,𝑌)𝜂(𝑍)+𝐵(𝑋,𝑍)𝜂(𝑌). (47) In the lightlike case, we also have another second fundamental form and its corresponding shape operator which we now explain as follows.

Let 𝑃 denote the projection morphism of Γ(𝑇𝑀) on Γ(𝑆(𝑇𝑀)). We obtain ∇_𝑋𝑃𝑌=∇^∗_𝑋()∇𝑃𝑌+𝐶𝑋,𝑃𝑌𝜉,_𝑋𝜉=−𝐴^∗_𝜉𝑋−𝜏(𝑋)𝜉,∀𝑋,𝑌∈Γ(𝑇𝑀), (48) where 𝐶(𝑋,𝑃𝑌) is the screen fundamental form of 𝑆(𝑇𝑀). The two second fundamental forms of 𝑀 and 𝑆(𝑇𝑀) are related to their shape operators by 𝐵(𝑋,𝑌)=𝑔𝐴^∗_𝜉𝑋,𝑌,𝑔𝐴^∗_𝜉𝐶()𝐴𝑋,𝑁=0,𝑋,𝑃𝑌=𝑔_𝑁𝑋,𝑃𝑌,𝑔𝐴_𝑁𝑌,𝑁=0. (49)

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 𝑆(𝑇𝑀) which, in general, is not unique. This raises the question of finding those lightlike hypersurfaces which admit a unique or canonical 𝑆(𝑇𝑀), needed in the lightlike geometry. Although a positive answer to this question for an arbitrary 𝑀 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 [29]; Leistner [30]; Bolós [31] are samples of many more authors in the literature. Also, Kupeli [5] has shown that 𝑆(𝑇𝑀) is canonically isometric to the factor vector bundle 𝑇𝑀^∗=𝑇𝑀/𝑇𝑀^⟂ and used canonical projection 𝜋∶𝑇𝑀→𝑇𝑀^∗ 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 𝑆(𝑇𝑀) for lightlike hypersurfaces 𝑀 of semi-Euclidean spaces 𝑅_𝑞^𝑛+1. Then, in [10, Chapter 4] it was proved that such a canonical screen distribution is integrable on any lightlike hypersurface of 𝑅^𝑛₁ and on any lightlike cone ∧^𝑛_𝑞−1 of 𝑅_𝑞^𝑛+1. This information was used by Bejancu et al. [33] in showing some interested 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 𝑀. Therefore, in the same paper [28], they constructed invariant normalizations intrinsically connected with the geometry of 𝑀 and investigated induced linear connections by these normalizations, using relative and absolute invariant defined by the first and second fundamental forms of 𝑀.

Let 𝐹={𝜉,𝑁,𝑊_𝑎}, 𝑎∈{1,…,𝑚} be a quasiorthonormal basis of 𝑀 along 𝑀, where {𝜉}, {𝑁}, and {𝑊_𝑎} are null basis of Γ(Rad𝑇𝑀_|𝒰), Γ(tr(𝑇𝑀)_∣𝒰), and orthonormal basis of Γ(𝑆(𝑇𝑀)_∣𝒰), respectively. For the same 𝜉, consider other quasiorthonormal frames fields 𝐹^′={𝜉,𝑁^′,𝑊^′_𝑎} induced on 𝒰⊂𝑀 by {𝑆^′(𝑇𝑀), (tr)^′(𝑇𝑀)}. It is easy to obtain 𝑊^′_𝑎=^𝑚_𝑏=1𝑊^𝑏_𝑎𝑊_𝑏−𝜖_𝑏𝐟_𝑏𝜉,𝑁^′=𝑁+𝐟𝜉+^𝑚_𝑎=1𝐟_𝑎𝑊_𝑎, (50) where {𝜖_𝑎} are signatures of orthonormal basis {𝑊_𝑎} and 𝑊^𝑏_𝑎, 𝐟, and 𝐟_𝑎 are smooth functions on 𝒰 such that [𝑊^𝑏_𝑎] is 𝑚×𝑚 semiorthogonal matrices. Computing 𝑔(𝑁^′,𝑁^′)=0 and 𝑔(𝑊_𝑎,𝑊_𝑎)=1 we get 2𝐟+∑^𝑚_𝑎=1𝜖_𝑎(𝐟_𝑎)²=0. Using this in the second relation of the above two equations, we get 𝑊^′_𝑎=^𝑚_𝑏=1𝑊^𝑏_𝑎𝑊_𝑏−𝜖_𝑏𝐟_𝑏𝜉,𝑁^′1=𝑁−2^𝑚_𝑎=1𝜖_𝑎𝐟_𝑎²𝜉+^𝑚_𝑎=1𝐟_𝑎𝑊_𝑎. (51) The above two relations are used to investigate the transformation of the induced objects when the pair {𝑆(𝑇𝑀),tr(𝑇𝑀)} 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 (𝑀,𝑔,𝑆(𝑇𝑀)) of a semi-Riemannian manifold is called screen locally conformal if the shape operators 𝐴_𝑁 and 𝐴^∗_𝜉 of 𝑀 and 𝑆(𝑇𝑀), respectively, are related by 𝐴_𝑁=𝜑𝐴^∗_𝜉, (52) where 𝜑 is a nonvanishing smooth function on a neighborhood 𝒰 in 𝑀.

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 𝐵 and 𝐶 of a screen conformal lightlike hypersurface 𝑀 and its 𝑆(𝑇𝑀), respectively, are related by 𝐶(𝑋,𝑃𝑌)=𝜑𝐵(𝑋,𝑌),∀𝑋,𝑌∈Γ𝑇𝑀_|𝒰. (53) Denote by 𝒮¹ the first derivative of 𝑆(𝑇𝑀) given by 𝒮¹(𝑥)=span[𝑋,𝑌]|_𝑥,𝑋_𝑥,𝑌_𝑥∈𝑆𝑇_𝑥𝑀,∀𝑥∈𝑀. (54) Let 𝑆(𝑇𝑀) and 𝑆(𝑇𝑀)^′ be two screen distributions on 𝑀, 𝐵, and 𝐵^′ their second fundamental forms with respect to tr(𝑇𝑀) and tr(𝑇𝑀)^′, respectively, for the same 𝜉∈Γ(𝑇𝑀^⟂∣_𝒰). Denote by 𝜔 the dual 1-form of the vector field 𝑊=∑^𝑚_𝑎=1𝐟_𝑎𝑊_𝑎 with respect to 𝑔. Following is a unique existence theorem.

Theorem 16 (see [8], page 61). Let (𝑀,𝑔,𝑆(𝑇𝑀)) be a screen conformal lightlike hypersurface of a semi-Riemannian manifold (𝑀,𝑔), with 𝒮¹ the first derivative of 𝑆(𝑇𝑀) given by (54). Then, (1)a choice of the screen 𝑆(𝑇𝑀) of 𝑀 satisfying (52) is integrable;(2)the one form 𝜔 vanishes identically on 𝒮¹;(3)if 𝒮¹ coincides with 𝑆(𝑇𝑀), then 𝑀 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 𝐶 is independent of its choice.

Proof. It follows from the screen conformal condition (52) that the shape operator 𝐴^∗_𝜉 of 𝑆(𝑇𝑀) is symmetric with respect to 𝑔. Therefore, a result [6, page 89] says that a choice of screen distribution of a screen conformal lightlike hypersurface 𝑀 is integrable, which proves (1).
As 𝑆(𝑇𝑀) is integrable, 𝒮¹ is its subbundle. Assume 𝒮¹=𝑆(𝑇𝑀). Then, it is easy to show that 𝜔 vanishes on 𝑆(𝑇𝑀), which implies that the functions 𝐟_𝑎 of the transformation equations vanish. Thus, the transformation equation (51) becomes 𝑊^′_𝑎=∑^𝑚_𝑏=1𝑊^𝑏_𝑎𝑊_𝑏, (1≤𝑎≤𝑚) and 𝑁^′=𝑁, where (𝑊^𝑏_𝑎) is an orthogonal matrix of 𝑆(𝑇_𝑥𝑀) at any point 𝑥 of 𝑀, which proves the first part of (3). Then independence of 𝐶 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 𝑆(𝑇𝑀) of a lightlike hypersurface imply that 𝑆(𝑇𝑀) 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 𝑅_𝑞^𝑚+2; however, only the canonical screen distribution on any lightlike hypersurface of 𝑅₁^𝑚+2 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 𝒮¹≠𝑆(𝑇𝑀).
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 (𝑀,𝑔,𝑆(𝑇𝑀)) be a lightlike hypersurface of a semi-Riemannian manifold (𝑀_𝑞^𝑚+2,𝑔), with 𝐸 a complementary vector bundle of 𝑇𝑀^⟂ in 𝑆(𝑇𝑀)^⟂ such that 𝐸 admits a covariant constant timelike vector field. Then, with respect to a section 𝜉 of Rad𝑇𝑀, 𝑀 is screen conformal. Thus, 𝑀 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 [8], page 62). Consider a smooth function 𝐹∶Ω→𝑅, where Ω is an open set of 𝑅^𝑚+1. Then 𝑀=𝑥⁰,…,𝑥^𝑚+1∈𝑅_𝑞^𝑚+2∶𝑥⁰=𝐹𝑥¹,…,𝑥^𝑚+1 (55) is a Monge hypersurface. The natural parameterization on 𝑀 is 𝑥⁰=𝐹𝑣⁰,…,𝑣^𝑚;𝑥^𝛼+1=𝑣^𝛼,𝛼∈{0,…,𝑚}. (56) Hence, the natural frames field on 𝑀 is globally defined by 𝜕_𝑣𝛼=𝐹^′_𝑥𝛼+1𝜕_𝑥0+𝜕_𝑥𝛼+1,𝛼∈{0,…,𝑚}. (57) Then 𝜉=𝜕_𝑥0−^𝑞−1_𝑠=1𝐹^′_𝑥𝑠𝜕_𝑥𝑠+^𝑚+1_𝑎=𝑞𝐹^′_𝑥𝑎𝜕_𝑥𝑎 (58) spans 𝑇𝑀^⟂. Therefore, 𝑀 is lightlike (i.e., 𝑇𝑀^⟂=Rad𝑇𝑀), if and only if the global vector field 𝜉 is spanned by Rad𝑇𝑀 which means, if and only if, 𝐹 is a solution of the partial differential equation 1+^𝑞−1_𝑠=1𝐹^′_𝑥𝑠²=^𝑚+1_𝑎=𝑞𝐹^′_𝑥𝑎². (59) Along 𝑀 consider the constant timelike section 𝑉=𝜕_𝑥0 of Γ(𝑇𝑅_𝑞^𝑚+2). Then 𝑔(𝑉,𝜉)=−1 implies that 𝑉 is not tangent to 𝑀. Therefore, the vector bundle 𝐸=Span{𝑉,𝜉} is nondegenerate on 𝑀. The complementary orthogonal vector bundle 𝑆(𝑇𝑀) to 𝐸 in 𝑇𝑅_𝑞^𝑚+2 is a nondegenerate distribution on 𝑀 and is complementary to Rad𝑇𝑀. Thus 𝑆(𝑇𝑀) is a screen distribution on 𝑀. The transversal bundle 𝑡𝑟(𝑇𝑀) is spanned by 𝑁=−𝑉+(1/2)𝜉 and 𝜏(𝑋)=0 for any 𝑋∈Γ(𝑇𝑀). Indeed, 𝜏(𝑋)=𝑔(∇_𝑋𝑁,𝜉)=(1/2)𝑔(∇_𝑋𝜉,𝜉)=0. The Weingarten equations reduce to ∇_𝑋𝑁=−𝐴_𝑁𝑋 and ∇_𝑋𝜉=−𝐴^∗_𝜉𝑋, which implies 𝐴_𝑁1𝑋=2𝐴^∗_𝜉𝑋,∀𝑋∈Γ(𝑇𝑀). (60) Hence, any lightlike Monge hypersurface of 𝑅_𝑞^𝑚+2 is screen globally conformal with 𝜑(𝑥)=1/2. 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 (𝑀,𝑔) is a metric (Levi-Civita) connection if and only if the second fundamental form 𝐵 vanishes on 𝑀. 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 𝑀 with respect to an induced connection ∇ is a geodesic of 𝑀 with respect to ∇, we say that 𝑀 is a totally geodesic lightlike hypersurface of 𝑀. Also, note that a vector field 𝑋 on a lightlike manifold (𝑀,𝑔) is said to be a Killing vector field if £_𝑋𝑔=0. A distribution 𝐷 on 𝑀 is called a Killing distribution if each vector field of 𝐷 is Killing.

Now we quote the following theorem on the existence of a unique metric connection on 𝑀, which also shows, from the Gauss equation, that the definition of totally geodesic 𝑀 does not depend on the choice of a screen.

Theorem 20 (see [6]). Let (𝑀,𝑔,𝑆(𝑇𝑀)) be a lightlike hypersurface of a semi-Riemannian manifold (𝑀,𝑔). Then the following assertions are equivalent:(a)𝑀 is totally geodesic in 𝑀;(b)𝐵 vanishes identically on 𝑀;(c)𝐴^∗_𝜉 vanish on Γ(𝑇𝑀) for any 𝜉∈Γ(𝑅𝑎𝑑𝑇𝑀);(d)There exists a unique torsion-free metric connection ∇ on 𝑀;(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 𝑀, we proceed as follows. Consider a type (0,2) induced tensor on 𝑀 given by 𝑅(𝑋,𝑌)=trace{𝑍⟶𝑅(𝑋,𝑍)𝑌},∀𝑋,𝑌∈Γ(𝑇𝑀). (61) Let {𝜉;𝑊_𝑎} be an induced quasiorthonormal frame on 𝑀, where Rad𝑇𝑀=Span{𝜉} and 𝑆(𝑇𝑀)=Span{𝑊_𝑎} and let 𝐸={𝜉,𝑁,𝑊_𝑎} be the corresponding frames field on 𝑀. Then, we obtain 𝑅(𝑋,𝑌)=^𝑚_𝑎=1𝜖_𝑎𝑔𝑅𝑋,𝑊_𝑎𝑌,𝑊_𝑎+𝑔(𝑅(𝑋,𝜉)𝑌,𝑁), (62) where 𝜖_𝑎 denotes the causal character (±1) of respective vector field 𝑊_𝑎. Using Gauss-Codazzi equations, we obtain 𝑔𝑅𝑋,𝑊_𝑎𝑌,𝑊_𝑎=𝑔𝑅𝑋,𝑊_𝑎𝑌,𝑊_𝑎+𝐵(𝑋,𝑌)𝐶𝑊_𝑎,𝑊_𝑎−𝐵𝑊_𝑎,𝑌𝐶𝑋,𝑊_𝑎. (63) Substituting this into the previous equation and using the relations (49), we obtain 𝑅(𝑋,𝑌)=Ric(𝑋,𝑌)+𝐵(𝑋,𝑌)𝑡𝑟𝐴_𝑁−𝑔𝐴_𝑁𝑋,𝐴^∗_𝜉𝑌−𝑔(𝑅()),(),𝜉,𝑌𝑋,𝑁∀𝑋,𝑌∈Γ𝑇𝑀 (64) where Ric is the Ricci tensor of 𝑀. This shows that 𝑅(𝑋,𝑌) is not symmetric. Therefore, in general, it has no geometric or physical meaning similar to the symmetric Ricci tensor of 𝑀. Thus, this 𝑅(𝑋,𝑌) can be called an induced Ricci tensor of 𝑀 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 (𝑀,𝑔,𝑆(𝑇𝑀)) be a locally (or globally) screen conformal lightlike hypersurface of a semi-Riemannian manifold (𝑀(𝑐),𝑔) of constant sectional curvature 𝑐. Then, 𝑀 admits an induced symmetric Ricci tensor.

Proof. Using the curvature identity 𝑅(𝑋,𝑌)𝑍=𝑐{𝑔(𝑌,𝑍)𝑋−𝑔(𝑋,𝑍)𝑌 and the equation in (64), we obtain 𝑅(𝑋,𝑌)=Ric(𝑋,𝑌)+𝐵(𝑋,𝑌)tr𝐴_𝑁−𝑔𝐴_𝑁𝑋,𝐴^∗_𝜉𝑌−𝑐𝑔(𝑋,𝑌). (65) Then using the screen conformal relation (52) mentioned above it is easy to show that 𝑅(𝑋,𝑌) is symmetric and, therefore, it is an induced Ricci tensor of 𝑀.
In particular, if 𝑀 is totally geodesic in 𝑀, then using the curvature identity and proceeding similarly to what is mentioned above one can show that 𝑅(𝑋,𝑌)=Ric(𝑋,𝑌)−𝑐𝑔(𝑋,𝑌). (66) Since Ric and 𝑔 are symmetric we conclude that any totally geodesic lightlike hypersurface of 𝑀(𝑐) admits an induced symmetric Ricci tensor.

Finally, we quote a general result on the induced symmetric Ricci tensor.

Theorem 22 (see [6]). Let (𝑀,𝑔,𝑆(𝑇𝑀)) be a lightlike hypersurface of a semi-Riemannian manifold (𝑀,𝑔). Then the tensor 𝑅(𝑋,𝑌), defined in (61), of the induced connection ∇ is a symmetric Ricci tensor, if and only if each 1-form 𝜏 induced by 𝑆(𝑇𝑀) is closed; that is, 𝑑𝜏=0, on any 𝒰⊂𝑀.

Remark 23. The symmetry property of the Ricci tensor on a manifold 𝑀 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 (𝑀,∇) 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 𝜔 satisfying ∇𝜔=0.

Open Problem. Give an interpretation of Theorem 22 in terms of affine geometry.

3.3. Induced Scalar Curvature

To introduce a concept of induced scalar curvature for a lightlike hypersurface 𝑀 we observe that, in general, the nonuniqueness of screen distribution 𝑆(𝑇𝑀) and its nondegenerate causal structure rule out the possibility of a definition for an arbitrary 𝑀 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 𝑀 of a Lorentzian manifold (𝑀,𝑔) for which we know that any choice of its screen 𝑆(𝑇𝑀) is Riemannian. This case is also physically useful. To calculate an induced scalar function 𝑟 by setting 𝑋=𝑌=𝜉 and then 𝑋=𝑌=𝑊_𝑎 in (62) and using Gauss-Codazzi equations, we obtain 𝑟=𝑅(𝜉,𝜉)+^𝑚_𝑎=1𝑅𝑊_𝑎,𝑊_𝑎=^𝑚_𝑎=1^𝑚_𝑏=1𝑔𝑅𝑊_𝑎,𝑊_𝑏𝑊_𝑎,𝑊_𝑏+^𝑚_𝑎=1𝑔𝑅𝜉,𝑊_𝑎𝜉,𝑊_𝑎+𝑔𝑅𝑊_𝑎,𝜉𝑊_𝑎,𝑁. (67) In general, 𝑟 given by the above expression cannot be called a scalar curvature of 𝑀 since it has been calculated from a tensor quantity 𝑅(𝑋,𝑌). 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 𝜉. Thus to recover a scalar curvature, we recall the following conditions [36] on 𝑀.

A lightlike hypersurface 𝑀 (labeled by 𝑀^𝟎) of a Lorentzian manifold (𝑀,𝑔) is of genus zero with screen 𝑆(𝑇𝑀)^𝟎 if(a)𝑀 admits a canonical or unique screen distribution 𝑆(𝑇𝑀) that induces a canonical or unique lightlike transversal vector bundle 𝑁;(b)𝑀 admits an induced symmetric Ricci tensor, denoted by Ric.

Denote by 𝒞[𝑀]^𝟎=[(𝑀,𝑔,𝑆(𝑇𝑀))]^𝟎 a class of lightlike hypersurfaces which satisfy the above two conditions.

Definition 25. Let (𝑀⁰,𝑔⁰,𝜉⁰,𝑁⁰) belong to 𝒞[𝑀]^𝟎. Then, the scalar 𝑟, given by (67), is called its induced scalar curvature of genus zero.

It follows from (a) that 𝑆(𝑇𝑀)⁰ and 𝑁⁰ are either canonical or unique. For the stability of 𝑟 with respect to a choice of the second fundamental form 𝐵 and the 1-form 𝜏, it is easy to show that with canonical or unique 𝑁⁰, both 𝐵 and 𝜏 are independent of the choice of