Lightlike hypersurfaces in spaces with concircular fields

Lightlike hypersurfaces in semi-Riemannian manifolds admitting concircular vector fields are investigated. We prove that such hypersurfaces are generally products of lightlike curves and warped product manifolds. In special cases, we show that these hypersurfaces are totally geodesic or totally screen geodesic provided such concircular fields belong to their normal or transversal bundles. A number of examples are furnished, where possible, to illustrate the main concepts.

further, motivated other scholars to take an active role in the study of lightlike geometry. For instance see [1,[11][12][13], and many other references cited therein.
The paper is arranged as follows: in Sect. 2, we quote some basic notions on lightlike hypersurfaces needed in the rest of the paper. In Sect. 3, lightlike hypersurfaces in semi-Riemannian manifolds with concircular vector fields are introduced. It is shown that such hypersurfaces are totally geodesic (resp., totally screen geodesic) providedŪ belongs to their normal bundle (resp., transversal bundle). In Sect. 4, we study the geometry of hypersurfaces in space forms with concircular vector fields, by showing that the ambient manifolds are always flat wheneverŪ belongs to the normal or the transversal bundle of the hypersurface. Finally, as an application, in Sect. 5, we characterise lightlike hypersurfaces, whose screen distributions admit a Ricci soliton induced from the screen component ofŪ .

Preliminaries
Let (M,ḡ) be an (n + 2)-dimensional semi-Riemannian manifold, whereḡ is a metric tensor of index 0 < ν < n + 2. Suppose that (M, g) is a hypersurface ofM, such that g =ḡ |M . If g degenerates on M, then there exists a nonzero vector field ξ on M such that g(ξ, υ) = 0, for any υ tangent to M. Moreover, the radical space (also known as the null space) [16, p. 53] of T p M, at each point p ∈ M, is a subspace Rad T p M defined by Rad T p M = {ξ ∈ T p M : g p (ξ, υ) = 0, ∀ υ ∈ T p M}, whose dimension is called the nullity degree of g and M is called a lightlike hypersurface ofM. It follows from the above discussion that the normal space T p M ⊥ of M is also lightlike and satisfies Rad T p M = T p M ∩ T p M ⊥ . Since M is a hypersurface ofM, it follows that dim T p M ⊥ = 1, which implies that dim Rad T p M = 1 and Rad T p M = T p M ⊥ . We often call Rad T M a radical distribution of M. It is now clear that for a lightlike hypersurface M, T M and T M ⊥ have a nontrivial intersection and their sum is not the whole of tangent bundle space TM. This anomaly implies that a vector of T pM cannot be decomposed uniquely into a component tangent to T p M and a component of T p M ⊥ . Moreover, the standard text-book definitions of the second fundamental form and the Gauss-Weingarten formulas are not applicable in the lightlike case.
As a remedy to this anomaly, Duggal-Bejancu [8] (also see Duggal Bejancu [9]) introduced an approach to lightlike geometry, which we have followed in the present paper. More precisely, their approach consists of fixing, on the lightlike hypersurface, a geometric data formed by a lightlike section and a screen distribution. By a screen distribution of M, we mean a complementary subbundle of T M ⊥ in T M, which is a rank n nondegenerate distribution over M. A downside to this approach is that there are infinitely many possibilities of choices for such a screen distribution provided the hypersurface M is paracompact, but each of them is canonically isomorphic to the factor vector bundle T M/T M ⊥ studied in detail by Kupeli [14]. Furthermore, we denote by S(T M) the screen distribution over M. With such notation, we have the decomposition T M = S(T M) ⊥ T M ⊥ , where ⊥ denotes the orthogonal direct sum. It is known [8] (also see [9]) that for a lightlike hypersurface equipped with a screen distribution, there exists a unique rank one vector subbundle ltr(T M) of TM over M, called a lightlike transversal vector bundle, such that for any nonzero section ξ of T M ⊥ on a coordinate neighbourhood U ⊂ M, there exists a unique section N of ltr(T M) on U satisfyingḡ(ξ, N ) = 1, andḡ(N , N ) =ḡ(N , Z ) = 0, for any Z tangent to S(T M). It then follows that where ⊕ denote the direct (nonorthogonal) sum. Throughout this paper, we denote by ( ) the set of smooth sections of a vector bundle over M, and F(M) the set of smooth functions on M.
Let M be a lightlike hypersurface ofM, and let∇ denote the Levi-Civita connection onM. Then, the Gauss-Weingarten equations of M are given bȳ where, ∇ X Y and A N X belongs to (T M), while B(X, Y ) and τ (X ) belongs to F(M). Note that, B is a symmetric tensor field of type (0, 2), called the local second fundamental form of M, τ is a differential 1-form, A N is a tensor field of type (1, 1), called the shape operator of M and ∇ is a torsion-free linear connection on M. Let P be the projection morphism of T M onto S(T M). Then, we have Here, ∇ * X PY belongs to S(T M) and A * ξ X belongs to S(T M), while C(X, PY ) belongs to F(M). Unlike B, C is generally not symmetric and called the local second fundamental of S(T M). In addition, A * ξ is a (1,1) tensor, called the screen shape operator. On the other hand, ∇ * is the Levi-Civita connection on S(T M). The local second fundamental forms B and C are not related by means of g. In fact, by (2.2)-(2.5), we have for any X and Y tangent to M. It follows from the second equality in (2.7) that A N is screen-valued. In addition, fromḡ(∇ X ξ, ξ ) = 0, (2.2) leads to for any X tangent to M. As A * ξ is screen-valued, (2.6) and (2.8) implies that The induced linear connection ∇ is not a metric connection on M. In fact, let θ be a 1-form given by θ(X ) = g(X, N ), then for any X , Y and Z tangent to M. Let (M, g) be a lightlike hypersurface of a semi-Riemannian manifoldM. Duggal and Bejancu [8, p. 94] have showed that the curvature tensor,R, ofM satisfies the following relations: (2.12) where ∇ B and ∇C are given by for any X , Y and Z tangent to M.
where B is the local second fundamental form defined around p and k ∈ R. One says that M is totally umbilic if any point of M is umbilic. It is easy to see that M is totally umbilic if and only if, locally, on each U there exists a smooth function φ such that B(X, Y ) = φg(X, Y ), for any X and Y tangent to M. In case φ = 0 (resp., φ = 0) on U one says M is totally geodesic (resp., properly totally umbilic).

Definition 2.2
Let (M, g) be a lightlike hypersurface of a semi-Riemannian manifold (M,ḡ). S(T M) is totally umbilic [8] if on any coordinate neighbourhood U there exists a smooth function ψ such that C(X, PY ) = ψg(X, PY ), for any X and Y tangent to M. In case ψ = 0 (resp., ψ = 0) on U one says S(T M) is totally geodesic (resp., properly totally umbilic). We call M a totally screen umbilic lightlike hypersurface if its screen distribution is totally umbilic.

Definition 2.3
A lightlike hypersurface (M, g) of a semi-Riemannian manifold is called screen locally conformal [9] if the shape operators A N and A * ξ of M and S(T M), respectively, are related by A N = ϕ A * ξ , where ϕ is a non-vanishing smooth function on a neighbourhood U of M. We say that M is screen homothetic if ϕ is nonzero constant.
We windup this section by giving an example of a lightlike hypersurface satisfying all the above three definitions above.

Concircular vector fields and hypersurfaces
Let (M,ḡ) be a semi-Riemannian manifold. A nonzero vector fieldŪ tangent toM is called a concircular [6] (also see [4] and [5]) vector field if, for anyX tangent toM, one has where ρ :M → R is a smooth function.Ū is called a concurrent vector field if ρ = 1. A direct calculation, while using (3.1), leads tō for anyX andȲ tangent toM.
for eachX tangent toM. It follows thatŪ is concircular with ρ = 1. In particular,Ū is a concurrent vector field onŪ .
where π I and π F are the natural projections of I × F onto I and F, respectively, and ε the constant sectional curvature of F. The timelike vector fieldŪ = f ∂ t satisfy∇XŪ = f X , for anyX tangent toM. Therefore, U is concircular with ρ = f .

Corollary 3.8
For any X tangent to M, the following holds: (3) IfŪ is tangent to S(T M), then the following hold:
Proof Suppose thatŪ belongs to and T M ⊥ , then the first relation in (3.14) gives A * ξ = 0, and hence, by Definition 2.1, M is totally geodesic. Now, from (2.10) we see that ∇g = 0, showing that ∇ is a metric connection. Next, ifŪ belongs to ltr(T M), the first relation in (3.16) gives A N = 0. That is, by Definition 2.2, M is totally screen geodesic. Moreover, from (2.4), we see that S(T M) is parallel with respect to ∇, and hence an integrable distribution on M. It follows that M is locally a product manifold ξ × M * , where ξ is a lightlike curve tangent to T M ⊥ and M * is a leaf of S(T M). Finally, whenŪ belongs to S(T M) ⊥ , we take X = ξ in the first relation of (3.21) and using (2.9), to have β A N ξ = 0. It follows that either β = 0 or A N ξ = 0. In the first case, we note thatŪ belongs to T M ⊥ and M lies in the first category of the theorem. On the other hand, the second case implies that β = 0 and so A N + α β A * ξ = ρ β P, which in turn means that A N is symmetric on S(T M) since A * ξ is symmetric. This means that S(T M) is an integrable distribution over M. As before, M is locally a product manifold ξ × M * , where ξ is a lightlike curve tangent to T M ⊥ and M * is a leaf of S(T M). Now, in view of the second relations in (3.14), (3.16) and (3.21), we note that dτ = 0, and hence, by [9, Theorem 2.4.1, p. 68], M admits an induced Ricci tensor.

Theorem 3.11 There does not exist any proper totally umbilic or proper totally screen umbilic lightlike hypersurface such thatŪ is tangent to S(T M).
Proof Suppose that M is totally umbilic andŪ belongs to S(T M), then Corollary 3.8, relation (3.19) and Definition 2.1 implies that B(X, U * ) = φg(X, U * ) = 0, for any X tangent to S(T M). Since S(T M) is nondegenerate, we have φ = 0, that is M is totally geodesic. In a similar way, if S(T M) is totally umbilic, we have C(P X, U * ) = ψg(P X, U * ) = 0, which gives ψ = 0. Next, a direct calculation shows that g(U * , ∇ * X Y ) = 0, for any X , Y tangent to M * and orthogonal to U * . Thus, if D is the complementary distribution of the line bundle U * spanned by U * in M * , then D is parallel and therefore, an integrable distribution over S(T M). Hence, by de-Ram's decomposition theorem in [7], each leaf M * of S(T M) is locally a product manifold l U * × M , where l U * is a curve of U * and M is an (n − 1)-dimensional leaf of D.

Theorem 3.12 Any totally screen umbilic or screen conformal lightlike hypersurface such thatŪ is tangent to S(T M) is locally a triple product
We can generalise Theorem 3.12 to the following result.
for any E tangent to S(T M) and f = (1/2)g(U * , U * ), which turns out to be a nonzero smooth function on the leaves M * . Now, the rest of the proof will follow similar steps as in the proof of [5,Theorem 1]. In that line, let E 0 be a unit vector field parallel to U * , i.e. there exist a nonzero smooth function γ , on M * , such that We may now extend E 0 to a quasi-orthonormal basis It follows from (3.22) and (3.24) that the gradient ∇ * ρ of ρ is collinear with U * . Next, using (3.22) and (3.23), we derive It then follows from (3.25) that From the second relation in (3.26), we note that the integral curves of E 0 are geodesics in the leaves M * . It follows that the line bundle E 0 spanned by E 0 is a totally geodesic foliation. Next, consider the distribution D spanned by E i , i = 1, . . . , n − 1. Taking E = E i in (3.22), and then consider (3.23), we have We see from (3.27) that D is integrable, whose leaves are totally umbilic in M * , with umbilicity factor − ρ γ . From (3.24) and the first relation of (3.27) the mean curvature vector − ρ γ E 0 of leaves of D, as hypersurfaces in M * , is parallel in E 0 (the normal space of leaves of D in M * ). Therefore, D is a spherical distribution. By (3.28) By a direct calculation, while considering (3.22), we have In view of (3.29) and (3.23), we calculate for any E orthogonal to E 0 . Thus, by (3.28), (3.30) and the first relation in (3.26), we have (3.31) Therefore, in view of (3.31), we may take β(t) = γ (t) and hence M * is a warped product I × γ (t) M . Moreover, by a direct calculation, we have ∇ * E γ ∂ t = γ (t)E for any E tangent to I × γ (t) M .

Space forms and concircular vector fields
Let (M,ḡ) be a semi-Riemannian manifold. We say thatM is a semi-Riemannian space form [16, p. 80] if the sectional curvatureκ ofM is constant, sayκ = c. We denote such a space form byM(c). Moreover,R satisfies the relationR
Finally, suppose thatŪ belongs to S(T M) ⊥ such that β = 0. We know from Theorem 3.9 that S(T M) is integrable. One may be interested in the geometry of a leaf M * of S(T M), as an isometric immersion in M = R n+2 k . In this regard, we may consider the vector fieldŪ = αξ +β N , which is everywhere normal to M * . We define theŪ -shape operator AŪ of M * as a submanifold of R n+2 k . As ξ and N are linearly independent, we express AŪ , using (2.3) and (2.5), in terms of A * ξ and A N as AŪ X = α A * ξ X + β A N X, (4.11) for any X tangent to M * . Thus, using (3.21) and (4.11), we have It follows, from (4.12), that M * is a partially umbilic immersion in R n+2 k (see [15] for details). Moreover, by a direct calculation, while considering ∇ * t X ξ = −τ (X )ξ and ∇ t X N = τ (X )N , we have for all X tangent to M * , where ∇ ⊥ is the normal connection of M * . Therefore, using (4.12), (4.13) and Section 0.9 of [15, p. 37], we have the following result: if αβ < 0.

U * as a Ricci soliton potential on leaves of S(T M)
Let (M, g) be a lightlike hypersurface of a semi-Riemannian manifold (M,ḡ), admitting a concircular vector fieldŪ . In this section, we consider a screen integrable lightlike hypersurface M such that (M * , g * , U * , λ) is a Ricci soliton. Here, M * is a leaf of S(T M) and g * = g |S(T M . In this case, we have where £ U * is the Lie derivative operator, Ric * is the Ricci tensor of M * and λ is a real constant.
for any X, Y tangent to M * .
Proof As the connection ∇ * on leaves is a metric connection, one has for each X, Y tangent to M * . It then follows from (3.5) and (5.3) that Then, (5.2) follows easily from (5.4) and (5.1).

Theorem 5.3 LetŪ be a concircular vector field on a Lorentzian manifold (M,ḡ)
, and let (M n+1 , g), where n ≥ 3, be a lightlike hypersurface ofM. Suppose that (M * , g * , U * , λ) is a Ricci soliton on M such that α B + βC = g * , for some smooth function . Then, the following are all true: (1) ρ + is constant function, and given by ρ + = λ; (2) M = ξ × I × (λt+c 0 ) M , where ξ is a lightlike curve tangent to T M ⊥ , I is an open interval with arc length t, c 0 is a constant and M is an Einstein (n − 1)-dimensional manifold whose Ricci tensor satisfies Ric = (n − 1)λ 2 g , g being the metric of M .
Proof As S(T M) is integrable, we already know that M is locally a product manifold ξ × M * , where ξ is a lightlike curve tangent to T M ⊥ and M * is a Riemannian leaf of S(T M). Moreover, from (5.2) and the assumption α B + βC = g * , we see that Ric * (X, Y ) = (λ − ρ − )g * (X, Y ), which shows that M * is Einstein manifold. Now, since M * is Einstein and n ≥ 3, it follows that M * has constant scalar curvature [6]. Thus, λ − ρ − is a constant. Hence, ρ + is also constant. Moreover, by (3.5), we have ∇ * X U * = (ρ + )X, (5.5) for each X tangent to M * . By a direct calculation, we have

Concluding remarks
Concircular vector fields on (semi-)Riemannian manifolds have attracted the attention of many researchers, due to their numerous applications. For instance, in general relativity, it has been noted that trajectories of timelike concircular fields in the de-Sitter model determine the world lines of receding or colliding galaxies satisfying the Weyl hypothesis (see [18] for details). Furthermore, concircular vector fields are useful in the characterisation of (Ricci) solitons [6], as well as providing a simpler way of characterising generalised Robertson-Walker spacetimes [5]. Using the above literature, we have proved that lightlike hypersurfaces admitting concircular vector fields are generally totally geodesic or products of lightlike curves and warped product manifolds. In addition, following Theorem 5.3, we mention that this paper opens a way to further research in lightlike hypersurfaces whose screen distributions admit (Ricci) solitons.