Concurrent Vector Fields on Lightlike Hypersurfaces

Concurrent vector fields lying on lightlike hypersurfaces of a Lorentzian manifold are investigated. Obtained results dealing with concurrent vector fields are discussed for totally umbilical lightlike hypersurfaces and totally geodesic lightlike hypersurfaces. Furthermore, Ricci soliton lightlike hypersurfaces admitting concurrent vector fields are studied and some characterizations for this frame of hypersurfaces are obtained.


Introduction
In 1943, K. Yano [1] proved that there exists a smooth vector field v, so-called concurrent on a Riemannian manifold (M, g) which satisfies the following condition: for every vector field X tangent to M; where ∇ is the Levi-Civita connection with respect to Riemannian metric g on M.
Beside these facts, the notion of a Ricci soliton is initially observed by Hamilton's Ricci flow and Ricci solitons drew attention after G. Perelman [8] applied Ricci solitons to solve the Poincaré conjecture.
A Riemannian manifold (M, g) with a metric tensor g is called a Ricci soliton if there exists a smooth vector field v tangent to M satisfying the following equation: where L v g is the Lie derivative of g with respect to v, Ric denotes the Ricci tensor and λ is a constant. A Ricci soliton is called shrinking if λ > 0, steady if λ = 0, expanding if λ < 0. Some interesting applications and characterizations dealing the Ricci soliton equation given in (2) for Riemannian manifolds, semi-Riemannian manifolds and their submanifolds have been obtained in [9][10][11][12][13][14][15][16] recently.
The main purpose of this paper is to investigate concurrent vector fields on lightlike hypersurfaces and Ricci solitons lightlike hypersurfaces of a Lorentzian manifold. However, there are some difficulties to deal with while examining concurrent vector fields and Ricci solitons for these kinds of submanifolds. The first problem is that since the induced metric is degenerate and hence not invertible for a lightlike hypersurface, some significant differential operators such as the gradient, divergence, Laplacian operators with respect to the degenerate metric cannot be defined. To get rid of this problem, we consider the associated metric defined with the help of a rigging vector field. The second main problem is that the Ricci tensor of any lightlike hypersurface is not symmetric. In this case, the Ricci soliton equation loses its geometric and physical meanings. To get rid of this problem, we investigate this equation on lightlike hypersurfaces with the genus zero screen distribution whose Ricci tensor is symmetric.

Preliminaries
In this section, we shall recall some basic definitions and theorems related to lightlike hypersurfaces of a Lorentzian manifold by following [17][18][19].
Let ( M, g) be a Lorentzian manifold with the Lorentzian metic g of constant index 1 and (M, g) be an (n + 1)−dimensional lightlike hypersurface, n ≥ 2, of ( M, g), where g is the induced degenerate metric on M. Then the intersection of tangent bundle TM and normal bundle TM ⊥ is a one-dimensional subbundle such that this bundle is called the radical distribution of M and it is denoted by Rad TM. Therefore, we write the radical distribution at any point p ∈ M by the following equation: For any lightlike hypersurface (M, g), there exists the complementary non-degenerate (Riemannian) vector bundle of Rad TM in TM, called the screen distribution S(TM) of M such that we have where ⊕ orth denotes the orthogonal direct sum. For any ξ in Rad TM, there exists a unique section N of the lightlike transversal bundle tr(TM) such that we have Therefore, the tangent bundle T M of M is decomposed as follows: where ⊕ denotes the direct sum which is not orthogonal. From (5) and (6), one can consider a basis {e 1 , .. . . . , e n , ξ, N} on T M such that {e 1 , .. . . . , e n } is an orthonormal basis of Γ(S(TM)). The basis {e 1 , .. . . . , e n , ξ, N} is called a quasi-orthonomal basis on T M.
Suppose that P to be the projection morphism of TM onto S(TM) and ∇ to be the Levi-Civita connection of M. The Gauss and Weingarten formulas for the hypersurface are given by for any X, Y ∈ Γ(TM), where ∇ and ∇ * are the induced linear connection on TM and S(TM), respectively [17,19]. It is well known that there exist the following equalities involving B and C and their shape operators A * ξ and A N , respectively: C(X, PY) = g(A N X, PY).
Note that S(TM) is not unique [20] and the second fundamental form B is independent of the choice of a screen distribution and satisfies the condition B(X, ξ) = 0 for any X ∈ Γ(TM). It is known that the induced connection ∇ given in (7) is not metric connection and there exists the following relation for any X, Y, Z ∈ Γ(TM): The Lie derivative of g with respect to the its Levi-Civita connection ∇ is defined by for any X, Y, Z ∈ Γ(TM). For any lightlike hypersurface (M, g, S(TM)) of ( M, g), we have from (13) and (14) that or equivalently we can write the Equation (15) that for any X, Y, Z ∈ Γ(TM).
where λ is a constant. Furthermore, M is called totally umbilical in M if every points of M is umbilical [21]. A lightlike hypersurface (M, g, S(TM)) is called screen locally conformal if the shape operators A N and A * ξ are related by Here, ϕ is a non-vanishing smooth function on a neighborhood U on M. We note that M is called screen homothetic if ϕ is a constant [22].
Let us denote the Riemann curvature tensors of M and M by R and R, respectively. The Gauss-Codazzi type equations are given as follows: for any X, Y, Z, U ∈ Γ(TM).
Let Π = Span{X, Y} be a 2-dimensional non-degenerate plane in T p M at a point p ∈ M. The sectional curvature of Π is given by We note that since C is not symmetric, it is clear from (18) and (22) that the sectional curvature map does not need to be symmetric on any lightlike hypersurface. Theorem 1 ([17]). Let (M, g, S(TM)) be a lightlike hypersurface of a semi-Riemannian manifold ( M, g). Then the following assertions are equivalent: As a result of Theorem 1, we see that the sectional curvature map is symmetric on every lightlike hypersurface whose screen distribution is integrable.
Let {e 1 , . . . . . . , e n , ξ, N} be a quasi orthonormal basis on T M, where {e 1 , . . . . . . , e n } be an orthonormal basis of Γ(S(TM)). The induced Ricci type tensor R (0,2) of M is defined by for any X, Y ∈ TM. We note that the induced Ricci type tensor R (0,2) is not symmetric for any lightlike hypersurface.
Considering the Equation (20) and Theorem 1, we obtain the following corollary immediately: Corollary 1. The Ricci tensor R (0,2) is symmetric on lightlike hypersurface whose screen distribution is integrable. Let the Ricci tensor R (0,2) be symmetric on lightlike hypersurface (M, g, S(TM)). The manifold M is called as an Einstein lightlike hypersurface [24] if, for any X, Y ∈ Γ(TM), the following relation satisfies: where γ is a constant.

Concurrent Vector Fields
For any lightlike hypersurface (M, g, S(TM)), some significant differential operators such as the gradient, divergence, Laplacian operators could be defined by the help of a rigging vector field and its associated metric (see [25][26][27][28][29]). Therefore, we shall initially recall some basic facts related to rigging vector fields and their some basic properties before studying concurrent vector fields on lightlike hypersurfaces. Definition 1. Let (M, g, S(TM)) be a lightlike hypersurface of a Lorentzian manifold ( M, g) and ζ be a vector field defined in some open set containing M. Suppose that ζ p / ∈ T p M for any p ∈ M. If there exists a 1-form η satisfying η(X) = g(X, ζ) for any X ∈ Γ(TM), then ζ is called a rigging vector field for M. Now, let N ∈ tr(TM) be a rigging vector field for M and η be a 1-form defined by η(X) = g(X, N) for any X ∈ Γ(TM). In this case, one can define a (0, 2) type tensor g as follows: for any X, Y ∈ Γ(TM). We note that the associated metric g is non-degenerate. From (3), (5) and (25) we have and Let f : U ⊂ M → R be a smooth function and (x 1 , x 2 , . . . , x n ) be a coordinate system on U. Then the gradient of f with respect to g is defined by Here, [g ij ] denotes the inverse of g coincided with g and [g ij ] is defined to be pseudoinverse of g [25]. Now, let v be a concurrent vector field on Γ(T M). Then, we can write v as the tangential and transversal components by where v T ∈ Γ(TM) and v N ∈ tr(TM). From (25) and (29), we have For any X ∈ Γ(TM), we write Now, we suppose that v = v T , that is, v lies in Γ(TM). In this case, we can write where v s ∈ Γ(S(TM)) and η(v) = a.
Taking into consideration the above facts, we get the following lemma: Lemma 1. Let (M, g, S(TM)) be a lightlike hypersurface of a Lorentzian manifold ( M, g). If v is a concurrent vector field on Γ(TM), then we have Proof. From (30)-(32), we have Therefore, we get By a straightforward computation, we have which completes the proof of lemma.

Lemma 2.
Let v be a concurrent vector field on Γ(TM). Then we have for any X ∈ Γ(TM).
where v is a concurrent vector field on Γ(TM).
Proof. Using the fact the second fundamental form B vanishes on Rad(TM) and from (9), we write From (4) and (39), the proof of lemma is completed.
Theorem 2. Let (M, g, S(TM)) be a screen conformal lightlike hypersurface and v be a concurrent vector field lying on Γ(TM). Then at least one of the following statements occurs.
Proof. From the Gauss and Weingarten formulas and Lemma 3, we have and Since v is the concurrent vector field, we have from Lemma 2 that If we write v = v s + aξ in (43), we see that .
From (44), we get Since A * ξ in Γ(S(TM)), we see from (45) that Using Lemma 4 and (45), we get Using the fact that τ(v) = −1, we have which shows that Therefore, we get at least which implies the proof of theorem.
From Theorem 2, we get the following corollary immediately: Corollary 2. Let (M, g, S(TM)) be a screen conformal lightlike hypersurface and v is a concurrent vector field such that v / ∈ Γ(S(TM)), that is a = 0. Then τ(ξ) = 0. Now, we shall recall the following proposition of K. L. Duggal and B. Sahin [19]: Under the hypothesis of the above proposition, one can obtain the following corollary: In the light of Corollaries 2 and 3, we get the following corollary immediately: Corollary 4. Let (M, g, S(TM)) be a screen homothetic lightlike hypersurface of ( M(c), g) and v be a concurrent vector field such that v / ∈ Γ(S(TM)). Then M is the semi-Euclidean space. If the concurrent vector field v lies on Γ(S(TM)) then M is totally umbilical. Proof. Under the assumption, we have from Theorem 2 that τ(ξ) = 0. Then we obviously have Furthermore, using the fact that B(ξ, Y) = g(A * ξ ξ, Y) we obtain that A * ξ ξ = 0. From (51), we have Therefore, we get Considering the Equation (43), we have Using the fact that M is totally umbilical, we have Thus, we obtain from the Equation (40) that From (20) and (56), we have Finally we see that λ = 0 by considering Proposition 1 and the equation (57). This fact shows us that M is totally geodesic. Remark 1. From Theorem 2, we see that τ(ξ) does not have to be equal to zero. Therefore, Theorem 3 is not correct when the concurrent vector field v lies on Γ(S(TM)). In a similar manner, considering (53), we see that Theorem 3 is not correct when v lies on Rad(TM). Now we shall investigate concurrent vector fields on the Levi-Civita connection with respect to the associated metric g.
Let ∇ be the Riemannian connection of M with respect to the associated metric g given in the Equation (25) and ∇ 1 be the induced Riemann connection from ∇ onto TM. Then we have the following:

Theorem 4.
Let v be a concurrent vector field with respect to ∇. Then v is also concurrent with respect to ∇ 1 . (25), for any vector field X ∈ Γ(TM), we write

Proof. From
Beside this fact, we have From (58) and (59), we obtain Using (60), we get which show that v is also concurrent with respect to ∇ 1 .
We note that the converge part of Theorem 4 is not correct. Taking into consideration (61), we obtain the following corollaries.

Corollary 5.
Let v be a concurrent vector field with respect to ∇ 1 . Then v is also concurrent with respect to ∇ if and only if the following relation holds for all X ∈ Γ(TM): Corollary 6. Let (M, g, S(TM)) be screen conformal with respect to ∇ and v be a concurrent vector field with respect to ∇ 1 . Then v is also concurrent with respect to ∇ if and only if the following relation holds: Corollary 7. Let (M, g, S(TM)) be totally geodesic screen conformal with respect to ∇ and v be a concurrent vector field with respect to ∇ 1 . Then v is also concurrent with respect to ∇ if and only if the following relation holds: Corollary 8. Let (M, g, S(TM)) be totally umbilical screen conformal with respect to ∇. Suppose v is a concurrent vector field with respect to ∇ 1 . Then v is also concurrent with respect to ∇ if and only if the following relation holds:

Ricci Solitons on Lightlike Hypersurfaces
Let (M, g, S(TM)) be a lightlike hypersurface of a Lorenzian manifold ( M, g). Suppose that v is a concurrent vector field on Γ(T M). Then we can write the vector v as the tangential and transversal components by where v T ∈ Γ(TM) and f = g(v, ξ). In this case we have the followings: and Proof.
Since v is a concurrent vector field with respect to ∇, we have for any X ∈ Γ(TM). Therefore, we get  Proof. From (68), if (M, g, S(TM)) is totally geodesic, then we have Using the above equation, we get for any X, Y ∈ Γ(TM). Considering Corollary 1 and Proposition 2, we get which indicates that f is constant.

Lemma 6.
For any lightlike hypersurface (M, g, S(TM)) of a Lorentzian manifold ( M, g) we have or equivalent to (73) we have for any X, Y ∈ Γ(TM).
If v lies on Γ(TM), that is, v = v T (or f = 0), then we have the following special result: Lemma 7. Let (M, g, S(TM)) be a lightlike hypersurface of a Lorentzian manifold and v be a concurrent vector field on Γ(TM). Then, for any X, Y ∈ Γ(TM), we have Definition 2. A lightlike hypersurface (M, g, S(TM)) is called a Ricci soliton if the following relation satisfies for any X, Y ∈ Γ(TM): where λ is a constant. A Ricci soliton lightlike hypersurface is called shrinking if λ > 0, steady if λ = 0, expanding if λ < 0.
Using the equation (7.1.2) in [18], we have the local field of frames {∂u 0 , N, δ u 1 , δ u 2 } of R 4 1 such that the matrix of metric on R 4 1 satisfies Then, the Ricci tensor is symmetric and If we consider the position vector v = tξ of (M, g, S(TM), we see that v is concurrent for M. From the above equations and (81), we get (M, g, S(TM)) is a expanding Ricci soliton with λ = − 1 2 , where v is the potential concurrent vector.
Now we shall give the following lemma for later use: Lemma 8. Let (M, g, S(TM)) be an (n + 1)-dimensional lightlike hypersurface of the semi-Euclidean space IR n+2 q . Then for any X, Y ∈ Γ(TM).
Proof. Let {e 1 , . . . , e n , ξ} be a basis of M such that Span = {e 1 , . . . , e n } is an orthonormal basis on Γ(S(TM)). Using the fact that M = IR n+2 q is the semi-Euclidean space and from (18), we have for any X, Y, Z ∈ Γ(TM). Furthermore, we get which completes the proof of the lemma.
Using (86) in (80) and putting X = e i , the proof of proposition is straightforward.
Corollary 11. Let (M, g, S(TM)) be a lightlike hypersurface and v be the concurrent vector field on Γ(TM). The manifold (M, g, S(TM)) is a Ricci soliton if and only if it is an Einstein lightlike hypersurface.