We study characteristic r-lightlike submanifolds M tangent to the characteristic vector fields in an indefinite metric 𝒮-manifold, and we also discuss the existence of characteristic lightlike submanifolds of an indefinite 𝒮-space form under suitable hypotheses: (1) M is totally umbilical or (2) its screen distribution S(TM) is totally umbilical in M.

1. Introduction

In the theory of submanifolds of semi-Riemannian manifolds, it is interesting to study the geometry of lightlike submanifolds due to the fact that the intersection of normal vector bundle and the tangent bundle is nontrivial, making it interesting and remarkably different from the study of nondegenerate submanifolds. In particular, many authors study lightlike submanifolds on indefinite Sasakian manifolds (e.g., [1–4]).

Similar to Riemannian geometry, it is natural that indefinite 𝒮-manifolds are generalizations of indefinite Sasakian manifolds. Brunetti and Pastore analyzed some properties of indefinite 𝒮-manifolds and gave some characterizations in terms of the Levi-Civita connection and of the characteristic vector fields [5]. After then, they studied the geometry of lightlike hypersurfaces of indefinite 𝒮-manifold [6]. As Jin′s generalizations of lightlike submanifolds of the Sasakian manifolds with the general codimension [3, 4, 7], Lee and Jin recently extended lightlike hypersurfaces on indefinite 𝒮-manifold to lightlike submanifolds with codimension 2 on an indefinite 𝒮-manifold, called characteristic half lightlike submanifolds [8]. However, a general notion of characteristic lightlike submanifolds of an indefinite 𝒮-manifold have not been introduced as yet.

The objective of this paper is to study characteristic r-lightlike submanifolds M of an indefinite 𝒮-manifold subject to the conditions: (1) M is totally umbilcial, or (2) S(TM) is totally umbilcal in M. In Section 2, we begin with some fundamental formulae in the theory of r-lightlike submanifolds. In Section 3, for an indefinite metric g.f.f-manifold we consider a lightlike submanifold M tangent to the characteristic vector fields, we recall some basic information about indefinite 𝒮-manifolds and deal with the existence of irrotational characteristic submanifolds of an indefinite 𝒮-space form. Afterwards, we study characteristic r-lightlike submanifolds of in Sections 4 and 5.

2. Preliminaries

Let (M, g) be an m-dimensional lightlike submanifold of an (m + n)-dimensional semi-Riemannian manifold

. Then the radical distribution Rad(TM) = TM∩TM^⊥ is a vector subbundle of the tangent bundle TM and the normal bundle TM^⊥, of rank r (1 ≤ r ≤ min {m, n}). In general, there exist two complementary nondegenerate distributions S(TM) and S(TM^⊥) of Rad(TM) in TM and TM^⊥, respectively, called the screen and coscreen distributions on M, such that

(2.1)

where the symbol ⊕_orth denotes the orthogonal direct sum. We denote such a lightlike submanifold by (M, g, S(TM), S(TM^⊥)). Denote by F(M) the algebra of smooth functions on M and by Γ(E) the F(M) module of smooth sections of a vector bundle E over M. We use the same notation for any other vector bundle. We use the following range of indices:

(2.2)

Let tr (TM) and ltr(TM) be complementary (but not orthogonal) vector bundles to TM in

and TM^⊥ in S(TM) ^⊥, respectively, and let {N₁, …, N_r} be a lightlike basis of

consisting of smooth sections of

, where 𝒰 is a coordinate neighborhood of M, such that

(2.3)

where {ξ₁, …, ξ_r} is a lightlike basis of Γ(Rad(TM)). Then we have

(2.4)

We say that a lightlike submanifolds (M, g, S(TM), S(TM^⊥)) of

are characterized as follows:

(1)
r-lightlike if 1 ≤ r < min {m, n};
(2)
coisotropic if 1 ≤ r = n < m;
(3)
isotropic if 1 ≤ r = m < n;
(4)
totally lightlike if 1 ≤ r = m = n.

The above three classes (2)–(4) are particular cases of the class (1) as follows: S(TM^⊥) = {0}, S(TM) = {0}, and S(TM) = S(TM^⊥) = {0}, respectively. The geometry of r-lightlike submanifolds is more general form than that of the other three type submanifolds. For this reason, in this paper we consider only r-lightlike submanifolds M ≡ (M, g, S(TM), S(TM^⊥)), with the following local quasiorthonormal field of frames on

(2.5)

where the sets {F_r+1, …, F_m} and {W_r+1, …, W_n} are orthonormal basis of Γ(S(TM)) and Γ(S(TM^⊥)), respectively.

Let

be the Levi-Civita connection of

and P the projection morphism of Γ(TM) on Γ(S(TM)) with respect to (2.1). For an r-lightlike submanifold, the local Gauss-Weingartan formulas are given by

(2.6)

(2.7)

(2.8)

(2.9)

(2.10)

for any X, Y ∈ Γ(TM), where ∇ and ∇^* are induced linear connections on TM and S(TM), respectively, the bilinear forms

and

on M are called the local lightlike and screen second fundamental forms on TM, respectively,

are called the local radical second fundamental forms on S(TM).

, and

are linear operators on Γ(TM) and τ_ij, ρ_iα, ϕ_αi, and σ_αβ are 1-forms on TM. Since

is torsion-free, ∇ is also torsion-free and both

and

are symmetric. From the fact

, we know that

are independent of the choice of a screen distribution. We say that

(2.11)

is the second fundamental tensor of M.

The induced connection ∇ on TM is not metric and satisfies

(2.12)

for all X, Y ∈ Γ(TM), where η_is are the 1-forms such that

(2.13)

But the connection ∇^* on S(TM) is metric. The above three local second fundamental forms are related to their shape operators by

(2.14)

(2.15)

(2.16)

(2.17)

(2.18)

where X, Y ∈ Γ(TM) and ϵ_α is the sign of W_α but it is ±1 related to the causal character of W_α. From (2.18), we know that each

is shape operator related to the local second fundamental form

on S(TM). Replacing Y by ξ_j in (2.14), we have

(2.19)

for all X ∈ Γ(TM). It follows

(2.20)

Also, replacing X by ξ_j in (2.14) and using (2.20), we have

(2.21)

For an r-lightlike submanifold, replace Y by ξ_i in (2.16), we have

(2.22)

Note 1. Using (2.14) and the fact that are symmetric, we have

(2.23)

From this, (2.20) and (2.21), we show that

are self-adjoint on Γ(TM) with respect to g if and only if

for all X ∈ Γ(TM), i and j if and only if

for all i, j. We call self-adjoint

the lightlike shape operators of M. It follows from the above equivalence and (2.10) that the radical distribution Rad(TM) of a lightlike submanifold M, with the lightlike shape operators

, is always an integrable distribution.

3. Characteristic Lightlike Submanifolds

A manifold is called a globally framed f-manifold (or g.f.f-manifold) if it is endowed with a nonnull (1,1)-tensor field of constant rank, such that is parallelizable, that is, there exist global vector fields , α ∈ {1, …, k}, with their dual 1-forms , satisfying and .

The g.f.f-manifold

, α ∈ {1, …, k}, is said to be an indefinite metric g.f.f-manifold if

is a semi-Riemannian metric, with index ν, 0 < ν < 2n + k, satisfying the following compatibility condtion

(3.1)

for any X, Y

, being ϵ_α = ±1 according to whether

is spacelike or timelike. Then, for any α ∈ {1, …, k}, one has

. An indefinite metric g.f.f-manifold is called an indefinite 𝒮-manifold if it is normal and

, for any α ∈ {1, …, k}, where

for any X, Y

. The normality condition is expressed by the vanishing of the tensor field

being the Nijenhuis torsion of

. Furthermore, as proved in [5], the Levi-Civita connection of an indefinite 𝒮-manifold satisfies:

(3.2)

where

and

. We recall that

and

is an integrable flat distribution since

(more details in [5]).

Following the notations in [9], we adopt the curvature tensor R, and thus we have R(X, Y, Z) = ∇_X∇_YZ − ∇_Y∇_XZ − ∇_[X,Y]Z, and R(X, Y, Z, W) = g(R(Z, W, Y), X), for any X, Y, Z, W ∈ Γ(TM).

An indefinite 𝒮-manifold

is called an indefinite 𝒮-space form, denoted by

, if it has the constant

-sectional curvature c [5]. The curvature tensor

of this space form

is given by

(3.3)

for any vector fields

Note 2. Although S(TM) is not unique, it is canonically isomorphic to the factor vector bundle S(TM) ^* = TM/Rad(TM) considered by Kupeli [10]. Thus all screen distributions S(TM) are mutually isomorphic. For this reason, we newly define generic lightlike submanifolds of as follows.

Definition 3.1. Let M be a r-lightlike submanifold of such that all the characteristic vector fields are tangent to M. A screen distribution S(TM) is said to be characteristic if and .

Definition 3.2. A r-lightlike submanifold M of is said to be characteristic if and a characteristic screen distribution (S(TM)) is chosen.

Proposition 3.3 (see [6].)Let (M, g, S(TM)) be a lightlike hypersurface of an indefinite 𝒮-manifold such that the characteristic vector fields are tangent to M. Then there exists a screen distribution such that and , where E is a nonzero section of Rad(TM).

Proposition 3.4 (see [8].)Let (M, g, S(TM)) be a 1-lightlike submanifold of codimension 2 of an indefinite 𝒮-manifold . Then M is a characteristic lightlike submanifold of .

Definition 3.5. A lightlike submanifold M is said to be irrotational [10] if for any X ∈ Γ(TM) and ξ_i ∈ Γ(Rad(TM)) for all i.

Note 3. For an r-lightlike M, the above definition is equivalent to

(3.4)

The extrinsic geometry of lightlike hypersurfaces depends on a choice of screen distribution, or equivalently, normalization. Since the screen distribution is not uniquely determined, a well-defined concept of 𝒮-manifold is not possible for an arbitrary lightlike submanifold of a semi-Riemannian manifold, then one must look for a class of normalization for which the induced Riemannian curvature has the desired symmetries. Let (M, g) be a semi-Riemannian manifold, p ∈ M.

is said to be an algebraic curvature tensor [11] on T_pM if it satisfies the following symmetries:

(3.5)

Definition 3.6. A screen distribution S(TM) is said to be admissible if the associated induced Riemannian curvature is an algebraic curvature tensor.

Theorem 3.7. Let (M, g, S(TM)) be an irrotational generic characteristic lightlike submanifold of an indefinite 𝒮-space form with an admissible screen distribution S(TM). Then one has c = ϵ.

Proof. Denote by and R the curvature tensors of and ∇, respectively. Using the local Gauss-Weingarten formulas for M, we obtain

(3.6)

for all X, Y, Z ∈ Γ(TM). Replace Z by ξ_k in (3.6) and use (2.10), (2.15), (2.17), and (3.4), we have

(3.7)

Using (3.7), the fact R(X, Y)Z ∈ Γ(TM) for X, Y, Z ∈ Γ(TM), and a screen distribution S(TM) is admissible, we get

(3.8)

On the other hand, since

and

for any X ∈ Γ(TM),

is an indefinite 𝒮-space form implies the Riemannian curvature

in (3.3) is given by

(3.9)

for any X, Y, Z, ∈Γ(TM). So, replacing X, Y, Z by PX, ξ, PZ in (3.9), we find

(3.10)

Then, using (3.3), (3.8), and (3.9), we get

(3.11)

Choosing

, we obtain c = ϵ.

Corollary 3.8. There exist no irrotational characteristic r-lightlike submanifolds (M, g, S(TM)) of an indefinite 𝒮-space form with c ≠ ϵ such that the screen distribution S(TM) is admissible.

4. Totally Umbilical Characteristic Lightlike Submanifolds

Definition 4.1. An r-lightlike submanifold M of is said to be totally umbilical [1] if there is a smooth vector field ℋ ∈ Γ(tr (TM)) such that

(4.1)

for all X, Y ∈ Γ(TM). In case ℋ = 0, we say that M is totally geodesic.

It is easy to see that M is totally umbilical if and only if, on each coordinate neighborhood 𝒰, there exist smooth functions A_i and B_α such that

(4.2)

for any X, Y ∈ Γ(TM). From (4.2) we show that any totally umbilical r-lightlike submanifold of

is irrotational. Thus, by Theorem 3.7, we have the following.

Theorem 4.2. Let (M, g, S(TM)) be a totally umbilical characteristic r-lightlike submanifold of an indefinite 𝒮-space form . Then one has c = ϵ.

Theorem 4.3. Let (M, g, S(TM)) be a totally umbilical characteristic r-lightlike submanifold of an indefinite 𝒮-manifold . Then M is totally geodesic.

Proof. Apply to with X ∈ Γ(TM), for all i and α, and use (2.8), (2.10), (2.15), (2.17), (2.22), and (3.2), we have

(4.3)

Assume that M is totally umbilical. Then we have

(4.4)

Replace X by

and X by

by turns, we get A_i = 0 for all i and B_α = 0 for all α. Thus we show that

and M is totally geodesic.

Corollary 4.4 (see [1].)Let (M, g, S(TM)) be a totally umbilical characteristic r-lightlike submanifold of an indefinite 𝒮-manifold . Then there exists a unique torsion-free metric connection ∇ on M induced by the connection on .

Proof. From (4.2) and Theorem 4.3, we have for all X, Y ∈ Γ(TM) and i. Thus, using (2.12), we obtain our assertion.

5. Totally Umbilical Screen Distributions

Definition 5.1. A screen distribution S(TM) of M is said to be totally umbilical [1] in M if, for each locally second fundamental form , there exist smooth functions C_i on any coordinate neighborhood 𝒰 in M such that

(5.1)

In case C_i = 0 for all i, we say that S(TM) is totally geodesic in M.

Due to (2.18) and (5.1), we know that S(TM) is totally umbilical in M if and only if each shape operators

of S(TM) satisfies

(5.2)

for some smooth functions C_i on 𝒰⊆M.

In general, S(TM) is not necessarily integrable. The following result gives equivalent conditions for the integrability of a screen S(TM).

Theorem 5.2 (see [1].)Let M be an r-lightlike submanifold of a semi-Riemannian manifold . Then the following assertions are equivalent:

(1)
S(TM) is integrable,
(2)
is symmetric on Γ(S(TM)), for each i,
(3)
is self-adjoint on Γ(S(TM)) with respect to g, for each i.

We know that, from (5.2), each shape operator is self-adjoint on Γ(S(TM)) with respect to g, which further follows from that above theorem that any totally umbilical screen distribution S(TM) of M is integrable.

Theorem 5.3. Let (M, g, S(TM)) be a characteristic r-lightlike submanifold of an indefinite 𝒮-manifold . If S(TM) is totally umbilical in M, then S(TM) is totally geodesic in M.

Proof. Apply the operator to for some k, j such that k ≠ j, and use (2.7) and (2.18)

(5.3)

Assume that S(TM) is totally umbilical in M. Then we have

(5.4)

Replacing X by

in (5.4) and taking (k, j) = (1,2), (2,3), …, (r − 1, r) and (r, 1) by turns and use the above method, we have C_i = 0 for all i ∈ {1, …, r}. Thus we have our assertion.

Theorem 5.4. Let (M, g, S(TM)) be a characteristic r-lightlike submanifold of an indefinite 𝒮-manifold such that S(TM) is totally umbilical in M. Then M is not irrotational.

Proof. Apply the operator to for all i and j, and use (2.6), (2.10), (2.15), (2.18), and Theorem 5.3, we have

(5.5)

Since S(TM) is totally umbilical, by Theorem 5.3, we have that S(TM) is totally geodesic. Then, by (5.5), we have

(5.6)

Apply the operator

and use (2.6), (2.10), (2.15), we have

(5.7)

Replace X by

in this equation and (5.6), we have

(5.8)

It is a contradiction. Thus M is not irrotational.

Since any totally umbilical r-lightlike submanifold of is irrotational, by Theorem 5.4, we have the following result.

Corollary 5.5. There exist no totally umbilical characteristic r-lightlike submanifolds (M, g, S(TM)) of an indefinite 𝒮-manifold equipped with a totally umbilical screen distribution S(TM) in M.

References

1 Duggal K. L. and Bejancu A., Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications, 1996, Kluwer Academic, Dodrecht, The Netherlands, 1383318 (97e:53121).
10.1007/978-94-017-2089-2
Google Scholar
2 Duggal K. L. and Sahin B., Lightlike submanifolds of indefinite Sasakian manifolds, International Journal of Mathematics and Mathematical Sciences. (2007) 2007, 21, 2295734 (2007m:53062), ZBL1147.53044, 57585.
10.1155/2007/57585
Google Scholar
3 Jin D. H., Geometry of lightlike hypersurfaces of an indefinite Sasakian manifold, Indian Journal of Pure and Applied Mathematics. (2010) 41, no. 4, 569–581, https://doi.org/10.1007/s13226-010-0032-y, 2672691 (2011i:53077), ZBL1202.53024.
10.1007/s13226-010-0032-y
Web of Science® Google Scholar
4 Jin D. H., Geometry of half lightlike submanifolds of an indefinite Sasakian manifold, submitted to Bulletin of the Korean Mathematical Society.
Google Scholar
5 Brunetti L. and Pastore A. M., Curvature of a class of indefinite globally framed f-manifolds, Bulletin Mathématique de la Société des Sciences Mathématiques de Roumanie. (2008) 51(99), no. 3, 183–204, 2433497 (2009f:53110).
Web of Science® Google Scholar
6 Brunetti L. and Pastore A. M., Lightlike hypersurfaces in indefinite 𝒮-manifolds, Differential Geometry. (2010) 12, 18–40, 2606544 (2011d:53114).
Google Scholar
7 Jin D. H., Generic lightlike submanifolds of an indefinite Sasakian manifold.
Google Scholar
8 Lee J. W. and Jin D. H., Half lightlike submanifolds in indefinite 𝒮-manifolds, to appear in Taiwanese Journal of Mathematics.
Google Scholar
9 Kobayashi S. and Nomizu K., Foundations of Differential Geometry, 1963, 1-2, Interscience Publishers, New York, NY, USA, 0152974 (27#2945).
Google Scholar
10 Kupeli D. N., Singular Semi-Riemannian Geometry, 1996, Kluwer Academic, Dodrecht, The Netherlands, 1392222 (97f:53105).
10.1007/978-94-015-8761-7
Google Scholar
11 Atindogbe C. and Duggal K. L., Pseudo-Jacobi operators and Osserman lightlike hypersurfaces, Kodai Mathematical Journal. (2009) 32, no. 1, 91–108, https://doi.org/10.2996/kmj/1238594548, 2518556 (2010j:53139), ZBL1173.53013.
10.2996/kmj/1238594548
Web of Science® Google Scholar

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Characteristic Lightlike Submanifolds of an Indefinite 𝒮-Manifold

Abstract

1. Introduction

2. Preliminaries

3. Characteristic Lightlike Submanifolds

4. Totally Umbilical Characteristic Lightlike Submanifolds

5. Totally Umbilical Screen Distributions

References

References

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Characteristic Lightlike Submanifolds of an Indefinite 𝒮-Manifold

Abstract

1. Introduction

2. Preliminaries

3. Characteristic Lightlike Submanifolds

4. Totally Umbilical Characteristic Lightlike Submanifolds

5. Totally Umbilical Screen Distributions

References

References

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