Higher order mean curvatures of SAC half-lightlike submanifolds of indefinite almost contact manifolds

We introduce higher order mean curvatures of screen almost conformal (SAC) half-lightlike submanifolds of indefinite contact manifolds, admitting a semi-symmetric non-metric connection, and use them to generalize some known results of [6]. Also, we derive a new set of integration formulae via the divergence of some special vector fields tangent to this submanifold. Several examples are also included to illustrate the main concepts.


INTRODUCTION
Null (or lightlike) subspaces exist naturally in semi-Riemannian spaces and they play a central role in general relativity. More precisely, in the study of black holes (small volumes of spacetime with infinite density). In fact, they are subspaces whose induced metrics are singular (or simply with vanishing determinants). Differential geometry of these subspaces was introduced by Duggal and Bejancu in their book [4], which was later updated by Dugal and Sahin to [5]. Their approach was later adopted by many other researchers, including but not limited to; [1], [6], [7], [9], [10], [11] and [14]. From the above pieces of work, we can see the theory of lightlike geometry rests on a number of operators, including, shape, Ricci, etc., together with functions constructed from them, like mean curvature, scalar curvature, etc. However, the most important of such functions are the ones derived from algebraic invariants of their respective operators. For instance, trace, determinant, and in a more general sense the r th symmetric functions, σ r . These functions play a central role in studying higher order mean curvatures in differential geometry of both Riemannian and semi-Riemannian manifolds. In fact, for any given point in a manifold, the r th symmetric function σ r coincides with the r th mean curvature S r . A great deal of work has been done for r = 1 (see [4], [5], [6], [9] and references therein). But the case r > 1 is strictly non-linear and complicated. The most efficient way of studying this case is the use of Newton transformations, T r , of a given operator A (or a system of operators) which, in some way, linearizes S r . That is to say (−1) r−1 rS r = tr(A • T r−1 ).
Let M m+1 be a half lightlike submanifold of an indefinite contact manifold M m+3 admitting a semi-symmetric non-metric connection. Then, M carries three shape operators A * E , A N and A W , where E, N and W are respectively vector fields in its radical distribution, lightlike transversal bundle and screen transversal bundle. When the structure vector field ξ is tangent to M but not necessarily in its screen distribution, then A * E is a self-adjoint operator on T M while A N and A W are generally not self-adjoint. If we suppose that M is a screen almost conformal (SAC) [12] half-lightlike submanifold , then the operator A N becomes self-adjoint on T M and therefore diagonalizable on T M and hence we can investigate its higher order mean curvatures. With such mean curvatures, one can also investigate integration geometry on such submanifolds. Integration geometry is fundamentally important as it provide obstructions to the existence of foliations whose leaves enjoy some special geometric properties-totally geodesic (or totally umbilical), minimal, constant mean curvature and many more. Also, it provides a way of minimizing volume (of submanifolds) as well as energy defined from smooth vector fields on manifolds (see [15] and references therein).
In this paper, we consider SAC half lightlike submanifold M of an indefinite contact manifold M , admitting a semi-symmetric non-metric connection. We derive equations relating the r th mean curvatures and Newton transformations of A N and A * E . We generalize some known results for r = 1 and also, derive new integration formulas by computing the divergence of some vector fields on in the tangent bundle of M . The rest of the paper is arranged as follows. Section 2 outlines the basic preliminary concepts needed in other parts of the paper. Section 3 introduces Newton transformations of A * E . In Section 4 we show that the r th mean curvatures and Newton transformations of A N and A * E are in partial variation (see Proposition 4.2 and Theorem 4.4). Also, we derive generalized differential equations for r th mean curvatures (Theorem 4.10). In Section 5 we present special integration formulas by computing the divergence of some vector fields (see Theorem 5.7 and its corollaries).

PRELIMINARIES
Let M be a (2n + 1)-dimensional manifold endowed with an almost contact structure (φ, ξ, η), i.e. φ is a tensor field of type (1, 1), ξ is a vector field, and η is a 1-form satisfying Then (φ, ξ, η, g) is called an indefinite almost contact metric structure on M if (φ, ξ, η) is an almost contact structure on M and g is a semi-Riemannian metric on M such that [3], for any vector field X, Y on M , It follows that, for any vector X on M , η(X) = g(ξ, X). We denote by Γ(Ξ) the set of smooth sections of the vector bundle Ξ. A connection ∇ on M is called a semi-symmetric non-metric connection [8,14] if ∇ and its corresponding torsion tensor T satisfy the following equations; and for all X, Y and Z vector fields on M . Let (M , g) be an (m + n)-dimensional semi-Riemannian manifold of constant index ν, 1 ≤ ν < m + n and M be a submanifold of M of codimension n. We assume that both m and n are ≥ 1. At a point p ∈ M , we define the orthogonal complement The submanifold M of M is said to be r-lightlike submanifold, if the mapping Rad T M : p ∈ M −→ Rad T p M, defines a smooth distribution on M of rank r > 0. We call Rad T M the radical distribution on M .
We say that M is a half-lightlike submanifold of M [5] if r = 1, n = 2 and there exist E, W ∈ Γ(T p M ⊥ ) such that (2.5) From (2.5), we observe that E ∈ Rad T p M and therefore, Thus, Rad T M is locally (or globally) spanned by E. Let S(T M ) be a screen distribution which is a semi-Riemannian complementary distribution of Rad T M in T M , that is, Note that the distribution S(T M ) is not unique, and is canonically isomorphic to the factor vector bundle T M/Rad T M [4]. Let P be the projection of T M on to S(T M ). Throughout this paper, we shall suppose that ξ is a unit space-lightlike vector field. Moreover, from (2.9) ξ is decomposed as follows ξ = ξ S + aE + bN + eW, (2.10) where ξ S denotes the projection of the tangential part of ξ on to S(T M ) and a = η(N ), b = η(E) and e = ǫη(W ), with ǫ = ±1, are smooth functions on M . The Gauss and Weingarten formulas are given by Notice that {∇ X Y, A V X} and {h(X, Y ), ∇ t X V } belongs to Γ(T M ) and Γ(tr(T M )) respectively. Further, ∇ and ∇ t are linear connections on M and tr T M , respectively. The second fundamental form h is a symmetric F(M )-bilinear form on Γ(T M ) with values in Γ(tr(T M )) and the shape operator A V is a linear endomorphism of Γ(T M ). Then, for all X, Y ∈ Γ(T M ), (2.11) and (2.12) gives 14) C is the local second fundamental form on S(T M ), {A N , A W } and A * E are the shape operators on T M and S(T M ) respectively, and τ , ρ, φ and δ are differential 1-forms on T M . Notice that ∇ * is a metric connection on S(T M ) while ∇ is generally not a metric connection. In fact, using (2.3) and (2.13), we deduce for all X, Y, Z ∈ Γ(T M ), where λ is a 1-form on T M given λ(·) = g(·, N ). It is well known [4,5] that B and D are independent of the choice of S(T M ) and they satisfy B(X, E) = 0, D(X, E) = φ(X), ∀ X ∈ Γ(T M ).
(2.19) The three local second fundamental forms B, D and C are related to their shape operators by the following equations (2.24) From the first equations of (2.19) and (2.20) we deduce that A * E is S(T M )-valued, self-adjoint and satisfies A * E E = 0. Let R and R denote the curvature tensors of M and M respectively. Then, using the Gauss-Weingarten equtions for M , we derive A half lightlike submanifold M of an indefinite almost contact manifold M , with ξ ∈ Γ(T M ), is called screen almost conformal (SAC) [12] if the shape operators A N and A * E of M and S(T M ), respectively, are linked to each other by where ϕ is a non vanishing smooth function on a coordinate neighborhood U of M . Furthermore, M is screen almost homothetic if ϕ is a non-vanishing constant function.
When ∇ is a metric connection, it is easy to show that g(A N X, N ) = 0 for any X ∈ Γ(T M ), for any half-lightlike submanifolds. Hence A N is screen-valued operator and thus, the screen almost conformallity condition (2.27) makes sense only if ξ ∈ Γ(S(T M )).
In the following example, we consider the connection in the ambient space to be Levi-Civita and construct a SAC half-lightlike submanifold of an indefinite Kenmotsu manifold.
Let ∇ be the Levi-Civita connection with respect to the semi-Riemannian metric g and consider the vector fields u 1 , · · · , u 9 , where for all 1 ≤ i ≤ 8 we have where functions f iα ′ and f iβ ′ are defined such that the action of the connection ∇ on the basis and ∇ u 9 u 9 = 0. From these constructions, (φ 0 , u 9 , η, g) defines n almost contact structure on R 9 2 . Therefore, (R 9 2 , φ 0 , u 9 , η, g) is an indefinite Kenmotsu manifold. Next, let us consider a submanifold M of R 9 2 above which is given by the following equation . By straightforward calculations, one can easily show that the vectors E = 1 2 (u 6 + u 2 ) + 1 2 u 1 and W = U 4 . Thus, M is a half-lightlike submanifold of (R 9 2 , φ 0 , u 9 , η, g). By straightforward calculations, we have From these connections, the 1-forms τ and ρ vanish on T M . Therefore, from (2.15) and When ∇ is semi-symmetric non-metric connection, one can easily verify that g(A N X, N ) = −η(N )λ(X), for any X ∈ Γ(T M ). See (2.23) for details. This shows that A N is generally not a screen-valued operator. Thus, the screen almost conformality condition (2.27) allows a ξ ∈ Γ(T M ) but not necessarily in S(T M ), given by ξ = ξ S + aE. Since on any SAC half-lightlike submanifold we have ξ ∈ Γ(T M ), then it is easy to see from (2.10) that b = 0 and e = 0. Hence, using (2.22) and (2.28) we deduce that M is SAC half-lightlike submanifold of M admitting a semi-symmetric non-metric connection if and only if Furthermore, if M is SAC half-lightlike submanifold of an almost contact manifold M admitting a semi-symmetric non-metric connection then from (2.29), the operator A N is self-adjoint on T M and thus diagonalizable on T M . Also, from (2.29) we have that any SAC half-lightlike submanifold of an almost contact manifold admitting a semi-symmetric non-metric connection, with the structure vector field ξ ∈ Γ(S(T M )) is screen conformal.

NEWTON TRANSFORMATIONS OF
Let M be an (n + 3)-dimensional almost contact metric manifold admitting a semi-symmetric non-metric connection and (M, g, S(T M )) be a codimension two SAC half-lightlike submanifold of M . Since A * E is a self-adjoint operator, from (2.20) and (2.29) we can see that A N is a self-adjoint linear operator on T M . Thus, A * E and A N are diagonalizable. Hence, A * E has (n + 1) real eigenvalues κ * 0 = 0, κ * 1 , · · · , κ * n (the principal curvatures) corresponding to a set of quasi-orthonormal frame field of eigenvector fields {Z 0 = E, Z 1 , · · · , Z n }. By the SAC condition (2.29) it is easy to see that −a, (ϕκ * 1 − a), · · · , (ϕκ * n − a) are eigenvalues of A N . Moreover the matrix of A N has the form . Associated to the shape operator A * E are (n + 1) algebraic invariants S * r = σ r (κ * 0 , κ * 1 , · · · , κ * n ), where σ r : M n+1 → R, for r = 0, 1, · · · , n + 1, are symmetric functions given by Let denote by I the identity map in Γ(T M ). Then, the characteristic polynomial of A * E is given by The normalized r-th mean curvature H * r of M is defined by In particular, when r = 1, then H * 1 = 1 n+1 tr(A * E ) which is called the mean curvature of the half-lightlike submanifold M . On the other hand, H * 2 relates directly with the (intrinsic) scalar curvature of M . Often times, H * r instead of S * r , is called the r-th mean curvature [1,2]. Moreover, the functions S * r (H * r respectively) are smooth on the whole M and, for any point p ∈ M , S * r coincides with the r-th mean curvature at p. Throughout this paper, we shall use S * r instead of H * r .
The Newton transformations T * r : Γ(T M ) → Γ(T M ), for r = 0, 1, · · · , n + 1, of a SAC half-lightlike submanifold M of an (n + 3)-dimensional almost contact metric manifold M with respect to A N are given by 2) or equivalently by the inductive formula Notice that, by Cayley-Hamiliton theorem, we have T * n+1 = 0. Moreover, T * r are also self-adjoint and commutes with A * E . It is important to note that the operators T * r depend on the choice of the transversal bundle tr(T M ) and the screen distribution S(T M ). Suppose a screen distribution S(T M ) changes to another screen S(T M ) ′ . The following are some of the local transformation equations due to this change (see [4, p. 87] for more details): for any X, Here c i and W j i are smooth functions on U and {ǫ 1 , · · · , ǫ n } is the signature of the basis {W 1 , · · · , W n }. Denote by ω is the dual 1-form of W , characteristic vector field of the screen change, with respect to the induced metric g = g| M , that is, Consider an orthogonal basis {Z i }, for i ∈ {1, · · · , n}, which diagonalizes A ′ * E and A * E . Let k ′ i and k i be the eigenvalues corresponding to eigenvector Z i . Then, which shows that the eigenvalues changes under the change of the screen distribution. Since the generalized expansion Θ r depends on on the eigenvalues k i , i.e. Θ r = (−1) r S * r = (−1) r σ r (k 1 , · · · , k n ), then a change of N will cause a change in it. Now, let {Θ, T * r } and {Θ ′ , T * ′ r } be two sets of the above objects under a change in N . Applying recurrence relation (3.3) and the fact that T r Z i = (−1) r S * i r Z i , we have Subtracting the second relation in (3.9) from the first and using relation (3.6) with X = Z i , we deduce that the operators T * r and T * ′ r are related by the following equation.
where θ r := (−1) r−1 (S * i ′ r−1 − S * i r−1 ). It is easy to see that the tensor T * r is unique if and only if the null hypersurface M is totally geodesic. For more details on Newton transformations and their properties, we refer the reader to [2], [15] and many more references therein.

FUNDAMENTAL SAC EQUATIONS OF A N
In this section, we derive SAC equations of A N from those of A * E . We use some of them to generalize some known results of [6]. Let T M = span{Z 0 = E, Z 1 , · · · , Z n } and S(T M ) = span{Z 1 , · · · , Z n }.  In the next proposition, we generalize Proposition 4.1.

Proposition 4.2.
Let M be a SAC half-lightlike submanifold of an almost contact manifold M admitting a semi-symmetric non-metric connection. Let S r and S * r be the r-th mean curvatures corresponding to the two shape operators A N and A * E respectively. Then, for all r ≥ 1 we have where for a given A * E , J * r are smooth functions in a and ϕ given by Proof. Let κ * 0 , · · · , κ * n be the eigenvalues (principal curvatures) of A * E and consider a linear factorization of a k th -degree monic polynomial in t below; where e s denotes the s th -degree symmetric function in variables X 1 , · · · , X k . Thus, if S r is the r th mean curvatures of A N , then we have from the definition of S r and (4.4) that which proves (4.2) and (4.3), hence the proof.
Notice that S * 0 = 1, S * 1 = 3, S * 2 = 3, etc. In a similar way, if S r is the r th mean curvature with respect to A N , then Finally, we notice that the mean curvatures S * r and S r are also conformally related, i.e., S r = ϕ r S * r and J * r (0, − 1 4 ) = 0. Theorem 4.4. Let M be a SAC half-lightlike submanifold of an almost contact manifold M admitting a semi-symmetric non-metric connection. Let T r and T * r be the r-th Newton transformations corresponding to the two shape operators A N and A * E respectively. Then, for all r ≥ 1 we have T r = ϕ r T * r + N * r (a, ϕ), (4.5) where N * r are operators depending on a, ϕ and A * E given by Proof. Using the fact that M is SAC half-lightlike submanifold, the definition of T r , (3.2) and Proposition 4.2 we get Applying the binomial theorem, the above equation leads to Expanding the two brackets in (4.7) gives which gives T r = ϕ r T * r + N * r (a, ϕ), which completes the proof. From now on, we shall write J * r instead of J * r (a, ϕ) and N * r instead of N * r (a, ϕ). Next, we use Proposition 4.2 above to state the following; Proposition 4.5. Let M be a SAC half-lightlike submanifold of an almost contact metric manifold M admitting a semi-symmetric non-metric connection. Let S * r and T * r denote the r-th mean curvature and Newton transformations with respect to A * E respectively. Then, for all r ≥ 1 we have tr(T r ) = ϕ r tr(T * r ) + (−1) r (n + 1 − r)J * r ; (4.8) for any X ∈ Γ(T M ).
Proof. The proof follows by straightforward calculations.
Notice that (4.12) implies that Proof. From (3.3), (3.18) and the fact that A N is self-adjoint, we derive for all X ∈ Γ(T M ).
Using the definition of covariant derivative we have for all X ∈ Γ(T M ). By virtue of (2.18) and the fact that A N is a self-adjoint operator, equation (4.15) reduces to Finally, substituting (4.17) in (4.14) and using Propositions 4.2, 4.5 and Theorem 4.4 we obtain the required equation (4.13).
A semi-Riemannian manifold M of constant curvature c is called a semi-Riemannian space form [4,5] and is denoted by M (c). Then, the curvature tensor R of M (c) is given by (4.18) Next, using Proposition 4.9 we have the following.
From the above theorem we have; ). Proof. The proof follows easily from Theorem 4.10 using the fact a = 0 when ξ ∈ Γ(S(T M )).

From Corollary 4.11 we have
which on simplifying gives Notice that (4.19) recovers Theorem 4.5 of [6], which says: for a conformal half-lightlike submanifold M (c) with mean curvature K and M ′ is a totally umbilical leaf in M (c), then, the submanifold is a semi-Euclidean space if and only if Ksatisfies E(K) − Kτ (E) − K 2 ϕ −1 = 0.

SPECIAL MINKOWSKI INTEGRATION FORMULAE
In this section, we present a new set of integration formulas on a special SAC half-lightlike submanifold (M, g, S(T M ), S(T M ⊥ )) of an indefinite nearly cosymplectic manifold (M , g), called SAC H-half lightlike submanifold, via the computation of div ∇ (T r HX ′ ) and div ∇ (T r HX ′ + T r E), where X ′ ∈ Γ(S(T M ) ⊥ ) and E ∈ Γ(Rad T M ). We shall suppose that M is closed and bounded (compact). An almost contact manifold M is said to be nearly cosymplectic if for any vector fields X, Y on M , where ∇ is the connection for the semi-Riemannian metric g.
Replacing Y by ξ in (5.1) we obtain where H is a (1,1) tensor given by H X = φ(∇ ξ φ)X. The linear operator H has the properties [13]:

5)
for all X ′ ∈ Γ(S(T M ) ⊥ ) and ∇ * is the metric connection on S(T M ).
Since M is null, the divergence div ∇ (Y ) of a vector Y ∈ Γ(T M ) with respect to the degenerate metric g on L is intrinsically defined by (see [5, p. 136], for more details and references therein) Also, g(∇ Z i T r E, Z i ) = (−1) r−1 (ϕ r S * r + J * r )g(A * E Z i , Z i ).