On Co-screen Conformality of 1-lightlike Submanifolds in a Lorentzian Manifold

In this paper, the co-screen conformal 1-lightlike submanifolds of a Lorentzian manifold are introduced as a generalization of co-screen locally half-lightlike submanifolds in (Wang, Wang & Liu, 2013; Wang & Liu, 2013) and two examples are given which one is co-screen locally conformal and the other is not. Some results are obtained on these submanifolds which the co-screen distribution is conformal Killing on the ambient manifold. The induced Ricci tensor of co-screen conformal 1-lightlike submanifolds is investigated.


Introduction
Inspired by Einstein's theory of general relativity, the Kaluza-Klein's theory and the string theory, many physicists consider that the universe, we live in, can be a 4-dimensional submanifold embedded in high dimensional spacetime manifold and many mathematicians study not only submanifolds of Riemannian manifolds but also study semi-Riemannian manifolds.One can consider that semi-Riemannian submanifolds are two types which one is non-degenerate submanifolds and the other is lightlike submanifolds.In (Duggal & Bejancu, 1996;Duggal & Jin, 2007;Duggal & Sahin, 2010), Duggal and his colleagues published books related with geometry of lightlike submanifolds and they presented general theory of lightlike submanifolds.Since then large numbers of papers have been published on lightlike submanifolds of semi-Riemannian manifolds.
Unfortunately, due to degenerate metric on lightlike submanifolds and the screen distribution is not canonical, induced nations of the submanifold (e.g sectional curvature, Ricci curvature, shape operator etc.) depend on choosing screen distribution that creates a problem.Therefore, it is necessary to find some classes of lightlike submanifold, whose geometry is essentially the same as that of their chosen screen distribution.Therefore, many mathematicians have been presented variety of methods to overcome this problem and have identified some special submanifolds.For example, the authors are used specific suitable methods for this problem in (Akivis & Goldberg, 1998;Akivis & Goldberg, 1999;Akivis & Goldberg, 2000;Bolós, 2005;Bonnor, 1992;Leistner, 2006).Furthermore, Kupeli (Kupeli, 1996) has shown that any screen distribution of a lightlike submanifold is isometric to the factor bundle on the tangent space the submanifold.In (Atindogbe & Duggal, 2004), Atindogbe and Duggal introduced screen locally conformal lightlike submanifold as a special lightlike submanifold of a semi-Riemannian manifold whose screen distribution is integrable and induced notions of the submanifold are independent of the screen distribution as follows: A lightlike hypersurface (M, g, S (T M)) of a semi-Riemannian manifold is called screen locally conformal if there is the following relation between the shape operator A N and the local shape operator A * ξ of the submanifold where φ is a non-vanishing smooth function on a neighborhood in M.
Recently, another special lightlike submanifolds whose screen distribution is integrable and the induced notions of the submanifold are independent of the screen distribution is defined in (Wang, Wang & Liu, 2013;Wang & Liu, 2013) and given by where φ c is a non-vanishing smooth function on a neighborhood in M, u is is a unit vector field of screen transversal bundle of the submanifold, A u , A * ξ are the shape operator and the local shape operator on M, respectively.

Preliminaries
Let M be a semi-Riemannian manifold equipped with semi-Riemannian metric g of index q.The manifold ( M, g) is called a Lorentzian manifold if q = 1.
Let (M, g) be an (n + 1)-dimensional lightlike submanifold of an (n + m + 2)-dimensional Lorentzian manifold.The radical space Rad T p M on the tangent space at a point p ∈ M is one-dimensional subspace, defined by (3) The submanifold (M, g) is called a lightlike hypersurface if m is equal to zero.The complementary vector bundle S (T M) of Rad T M in T M is called the screen bundle of M (Duggal & Bejancu, 2007).For these submanifolds, any screen bundle is non-degenerate and where ⊕ orth denotes the orthogonal direct sum.Consider a complementary vector bundle S (T M ⊥ ) of Rad T M in T M ⊥ .Then, we have the following orthogonal direct sum Here, S (T M ⊥ ) is the non-degenerate distribution with respect to g.It is said to be co-screen distribution on M (Jin, 2009c;Jin, 2010b).
Let tr(T M) and ltr(T M) be complementary but not orthogonal vector bundles to T M in T M| M and Rad T M in tr(T M), respectively.In this situation, we have Let (M, g) be an (n + 1)-dimensional lightlike submanifold of an (n + m + 2) dimensional ( M, g) and U be a local coordinate neighborhood of M.Then, there exists a quasi-orthonormal frame of M along M, on U: {ξ, e 1 , e 2 , . . ., e n , N, u 1 , u 2 , . . ., u m }, where We note that the quasi-orthonormal basis of M satisfies that for all u ∈ Γ(tr(T M)| U ).
Let ∇ be the Levi-Civita connection of M and P be the projection morphism of Γ(T M) to Γ(S (T M)).The Gauss and Weingarten formulas are given by for any X, Y ∈ Γ(T M), where ∇ and ∇ * are the induced linear connections on T M and S (T M), respectively; B and D α are coefficients of the lightlike second fundamental form and coefficients of the screen second fundamental form of T M, respectively, C is coefficients of the local second fundamental form on S (T M), A N , A u β are the shape operators on M, A * ξ is the shape operator on S (T M) and ε, ε α , ρ, ρ α are 1-forms on M. Let us define a local 1-form η by Consider ( 9), ( 14) and ∇ is a metric connection on M it is known that the induced connection ∇ is not a metric connection (Duggal & Bejancu, 2007).
The second fundamental form h and the local second fundamental form h * are given by, respectively, where for any X, Y ∈ T M. Here, we note that h ℓ is the light part of the second fundamental form and h s is the nondegenerate part of the second fundamental form.
It is known that B = 0 on Rad T M and it is independent of the screen distribution S (T M) for any α ∈ {1, . . .m} and the following relations satisfy that for any X, Y ∈ T M and α ∈ {1, . . ., m}.
The submanifold (M, g, S (T M)) is called totally umbilical if there exist smooth functions λ ∈ F(tr(T M)) and for all X, Y ∈ Γ(T M).
The mean curvature vector on T M and on Γ(S (T M)) are given by Let Π = sp{e i , e j } be 2-dimensional non-degenerate plane of the tangent space T p M at p ∈ M.Then, the number i , e j ) g p (e i , e i )g p (e j , e j ) − g p (e i , e j ) 2 ( 23) is called the sectional curvature of the section Π at p ∈ M. Since the operator C isn't symmetric the sectional curvature function doesn't need to be symmetric on any lightlike submanifold of a semi-Riemannian manifold (Duggal & Sahin, 2010).
Let ξ be a null vector of T p M. A plane Π of T p M is called a null plane if it contains ξ and e i such that g(ξ, e i ) = 0 and g(e i , e i ) 0. The null sectional curvature of Π be given in (Beem, Ehrlich, & Easley, 1996) as follows: We note that the null sectional curvature measures differences in length of two spacelike geodesic constructed from the degenerate plane section Π and it is independent of the choice of the spacelike vector e i but it depends quadratically on the null vector ξ (Albujer & Haesen, 2010).
Let (M, g, S (T M)) be an (n + 1) dimensional 1-lightlike submanifold of an m-dimensional Lorentzian manifold ( M, g) and {e 1 , .. . . ., e n } be an orthonormal basis of Γ(S (T M)).The induced Ricci type tensor R (0,2) of M is defined by It is known that R (0,2) is not symmetric and has no geometric meaning.The tensor R (0,2) is called the Ricci curvature if it is symmetric (Duggal & Sahin, 2010).

Co-screen Conformal 1-lightlike Submanifolds
We begin this section with the canonical theorems for 1-lightlike submanifolds of a Lorentzian manifold.
Let (M, g) be an (n + 1)-dimensional 1-lightlike submanifold of an (m + n + 2)-dimensional Lorentzian manifold ( M, g).Let us consider two quasi-orthonormal frame {ξ, N, e i , u α } and {ξ, N ′ , e ′ i , u ′ α } induced on U.In this case, the followings can be written and where e b a , u β α , N 1 , f a and Q α are differentiable functions on U for a ∈ {1, . . ., n} and α ∈ {1, . . ., m}.Since g(N, N) = 0, we have Let h ℓ and h ′ℓ are light parts of the second fundamental forms and B and B ′ are coefficients of the light parts of the second fundamental forms on screen distributions S (T M) and S (T M) ′ , respectively.Taking ξ = θ ξ and thus N = 1 θ N, where θ is some function, we obtain which implies that h ℓ is independent of the screen distribution S (T M).
Let h s and h ′s are non-degenerate parts of the second fundamental forms and D and D ′ are coefficients of the non-degenerate parts of the second fundamental forms on screen distributions S (T M) and S (T M) ′ , respectively.Using (26), we have which implies that non-degenerate part of the second fundamental form h s depends on the screen distribution.
Using similar method, one can easily get where e = ∑ n a=1 f a e a is called the characteristic vector field.Let us consider the first derivative of a screen distribution S (T M) given by where [, ] denotes the Lie-bracket.Then, we have the following: Theorem 1 Let (M, g, S (T M)) be an (n + 1)-dimensional 1-lightlike submanifold of a Lorentzian manifold.If the first derivative S defined by (32) coincides with S (T M), then S (T M) is a canonical screen of M, up to an orthogonal transformation with a canonical lightlike transversal vector bundle and the screen second fundamental form h * is independent of a screen distribution.
The proof is same as that of Theorem 2.1 in (Duggal, 2007), so we omit it here.Now, we recall a class of 1-lightlike submanifolds of a Lorentzian manifold which admits an integrable canonical screen distribution as follows.
Definition 2 (Atindogbe & Duggal, 2004) where U is a local coordinate neighborhood of M and φ is a smooth function on a neighborhood U in M. If φ is non-zero constant then the submanifold is called screen homothetic.Now, we state the following theorem:

g). Suppose that S (T M) is integrable and S (T M) is totally umbilical immersed in M and it is parallel along integral curves of the radical distribution. Then, M is screen locally conformal if and only if
µ * µ 1 0.
Proof.Let us denote M ′ as a leaf of S (T M).Then, we have for all X, Y ∈ T M ′ .The mean curvature vector of M ′ is a vector field of the rank (m + 2).From ( 22) and since M ′ is totally umbilical, it is clear that In other words, for any α ∈ {1, . . ., m}.Also, it is known that the mean curvature vector H * on Γ(S (T M)) satisfies that Since S (T M) is parallel along integral curves of the radical distribution, we have A N ξ = 0 so that for all X ∈ T M which implies that M is screen locally conformal.The proof of the converse part is straightforward.
Using same proof way of Theorem 2.3 in (Duggal & Bejancu, 1996), we immediately have the following theorem: Theorem 4 Let (M, g, S (T M)) be a 1-lightlike submanifold of a Lorentzian manifold.Then, the following assertions are equivalent: 3) the shape operator A N on M is symmetric.
From the above theorem, it is clear that screen conformal 1-lightlike submanifolds have the important features that their screen distributions are always integrable and the sectional curvature function is always symmetric and it has significant geometric meanings as in Riemannian manifolds.
We give now the following definition that shows that there is another type lightlike submanifold which its screen distributions are always integrable and the sectional curvature function defined on it is always symmetric.
Definition 5 Let M be an (n + 1)-dimensional lightlike submanifold of a Lorentzian manifold.The submanifold M is called co-screen locally conformal on a coordinate neighborhood U if there exists a non-zero smooth function φ c such that for any null transversal vector field N ∈ Γ(ltr(T M)) and α ∈ {1, . . .m}.
We now state the following theorem to characterize the co-screen conformal 1-lightlike submanifolds.
Theorem 6 Let (M, g, S (T M)) be a 1-lightlike submanifold of a Lorentzian manifold, then M is co-screen conformal if and only if where φ c is a non-zero smooth function on M.
Theorem 7 The conditions given in Theorem 4 are always satisfied for any co-screen locally conformal 1-lightlike submanifold of a Lorentzian manifold.
Example 8 Consider in R 7 1 with signature (−, +, +, +, +, +, +) a submanifold M given by the equations Then, we have and Thus, Rad T M = Span{ξ} is a distribution on M and S (T Also, the lightlike transversal bundle ltr(T M) is spanned by By direct calculations, we get the manifold (M, g, S (T M)) isn't co-screen conformal since φ c can't be vanishing function.
Let M be a submanifold of R 8 1 given by where all of u 1 , u 2 , u 3 , u 4 are non-vanishing coordinate functions.Then, we have By straightforward computations, it can be obtained that the submanifold is co-screen conformal with φ c is arbitrary.
Now, we give the following: Theorem 10 Let (M, g, S (T M)) be a totally geodesic, totally umbilical or minimal screen locally conformal 1-lightlike submanifold of a Lorentzian manifold ( M, g).Any leaf M ′ of S (T M) immersed in M as a (m + 2)dimensional non-degenerate submanifold if and only if the following assertions must be occurred: Proof.Let M is a co-screen locally conformal irrotational 1-lightlike submanifold.Suppose that X, Y be tangent vector fields of the leaf M ′ of a screen distribution and h ′ is its second fundamental form in M. If we put ( 16), ( 17), ( 18) and ( 39) in (34) we get which implies that h ′ (X, Y) = B(X, Y)(φξ + N + u α ), for any α = {1, . . ., m}, ( and so where ( ) is a unit spacelike vector field on M ′ .Since M is irrotational D α (X, ξ) = 0 for all α ∈ {1, . . ., m} and B(X, ξ) = 0 for all X ∈ Γ(T M ′ ).Therefore, the leaf M ′ of S (T M) immersed in M as a (m + 2)-dimensional non-degenerate submanifold.The proof of the converse part is straightforward.
Corollary 11 If (M, g, S (T M)) is a lightlike hypersurface of a Lorentzian manifold ( M, g).Then, any leaf M ′ of S (T M) immersed in M as a 2-dimensional non-degenerate submanifold.

Remark 12
The above corollary is also valid for lightlike hypersurfaces of a semi-Riemannian manifold that proved in ( Atindogbe, Ezin & Tossa, 2006).
Theorem 13 Let (M, g, S (T M)) be a co-screen conformal 1-lightlike submanifold of a Lorentzian manifold.The co-screen distribution S (T M ⊥ ) is a conformal Killing on M if and only if S (T M) is totally umbilical.
Proof.Let L denote the Lie derivative on M. If S (T M ⊥ ) is conformal Killing on M, then for any u α ∈ Γ(S (T M ⊥ )) and X, Y ∈ Γ(T M).Here, δ is a smooth function.Putting ( 17) in ( 48), we get for any X, Y ∈ Γ(T M).Since M is co-screen conformal, we obtain which shows that S (T M) is totally umbilical.Now, we assume that S (T M) is totally umbilical.Then, where λ ′ is a smooth function.Putting λ ′ = − δ 2φ , it is clear that S (T M ⊥ ) is a conformal Killing on M. Corollary 14 Let (M, g, S (T M)) be co-screen conformal 1-lightlike submanifold of a Lorentzian manifold.The co-screen distribution S (T M ⊥ ) is a Killing distribution on M if and only if S (T M) is totally geodesic and M is minimal.
Using similar proof method of Theorem 3.10 in (Wang & Liu, 2013) we have also the following: Theorem 15 Let (M, g, S (T M)) be a co-screen conformal 1-lightlike submanifold of a Lorentzian manifold.Then, 1) Any leaf of (S T M) is totally geodesic on M.
2) The submanifold M is a lightlike product manifold of M ′ and F where M ′ is a leaf of S (T M) and F is a null curve of M.

Ricci Curvature on Co-screen Conformal 1-lightlike Submanifolds
In this section, we study on the sectional curvature, the null sectional curvature and the induced Ricci curvature on co-screen conformal 1-lightlike submanifolds of a Lorentzian manifold.We begin this section herewith the following lemma.
Lemma 16 Let (M, g, S (T M)) be a co-screen conformal 1-lightlike submanifold of a Lorentzian manifold.Let us denote the Riemannian curvature tensors R and R of the submanifold M and the ambient manifold M, respectively.
Since K i j are symmetric for all i, j ∈ {1, . . ., n} in co-screen conformal 1-lightlike submanifolds of a Lorentzian manifold, it is clear that the screen Ricci curvature is well defined.
We note that the screen Ricci curvature vanishes identically if n = 1, it is equal to the sectional curvature if n = 2.
Theorem 20 Let (M, g, S (T M)) be a 4-dimensional co-screen conformal 1-lightlike submanifold of a Lorentzian manifold.The screen Ricci curvature Ric S (T M) is constant at every unit vector on Γ(S (T M)) if and only if the following conditions are occurred.Theorem 21 Let (M, g, S (T M)) be an (n + 1)-dimensional co-screen conformal 1-lightlike submanifold of a semi-Euclidean space with the screen Ricci curvature on the submanifold vanishes identically.Then, at least one of the following situations are occurred: a) µ 1 C(X, X) = ∑ n i=1 B(e i , X)C(e i , X) for all X ∈ Γ(S (T M)).b) φ c = ∓1.
Proof.If we take trace in (56), we have ) n ∑ i=1 [B(e i , X)C(e i , X)] − µ 1 C(X, X) = 0, which implies that B(e i , X)C(e i , X), This is proof of the theorem.
Theorem 22 Let (M, g, S (T M)) be an (n + 1)-dimensional flat co-screen conformal 1-lightlike submanifold with B = 0 of a semi-Euclidean space.Then, the local shape operator of M takes the form as follows: for a is a real number.