Journal of Applied Mathematics
Volume 2015 (2015), Article ID 259146, 9 pages
http://dx.doi.org/10.1155/2015/259146

Research Article

Lightlike Hypersurfaces of Indefinite Generalized Sasakian Space Forms

Dae Ho Jin

Department of Mathematics, Dongguk University, Gyeongju 780-714, Republic of Korea

Received 19 January 2015; Accepted 16 March 2015

Academic Editor: Dimitris Fotakis

Copyright © 2015 Dae Ho Jin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We study lightlike hypersurfaces of an indefinite generalized Sasakian space form , with indefinite trans-Sasakian structure of type , subject to the condition that the structure vector field of is tangent to . First we study the general theory for lightlike hypersurfaces of indefinite trans-Sasakian manifold of type . Next we prove several characterization theorems for lightlike hypersurfaces of an indefinite generalized Sasakian space form.

1. Introduction

Oubiña [1] introduced the notion of indefinite trans-Sasakian manifold of type . Indefinite Sasakian, Kenmotsu, and cosymplectic manifolds are three important kinds of indefinite trans-Sasakian manifold such that

respectively. Alegre et al. [2] introduced indefinite generalized Sasakian space form . Indefinite Sasakian, Kenmotsu, and cosymplectic space forms are some kinds of indefinite generalized Sasakian space form such that

respectively, where denotes constant -sectional curvatures of each of them.

Recently author has been studying the geometry of lightlike hypersurfaces of indefinite Sasakian [3], Kenmotsu [4], and cosymplectic [5] manifolds. In this paper, we study lightlike hypersurfaces of an indefinite generalized Sasakian space form , with indefinite trans-Sasakian structure of type , subject to the condition that the structure vector field of is tangent to . First we study lightlike hypersurfaces of indefinite trans-Sasakian manifold of type . Next we prove two characterization theorems for lightlike hypersurfaces of an indefinite generalized Sasakian space form such that the following hold.

(i) Let be a lightlike hypersurface of an indefinite generalized Sasakian space form . Then is a constant, , and

(ii) Let be a screen conformal lightlike hypersurface of an indefinite generalized Sasakian space form . Then .

2. Preliminaries

An odd-dimensional semi-Riemannian manifold is called an indefinite trans-Sasakian manifold [1, 2] if there exist a -type tensor field , a vector field which is called the structure vector field, and a 1-form such that

for any vector fields and on , where or according to the fact that is spacelike or timelike, respectively. In this case, the set is called an indefinite trans-Sasakian structure of type .

In the entire discussion of this paper, we may assume that is unit spacelike; that is, , without loss of generality. From (4) and (6), we get

Let be a lightlike hypersurface, with a screen distribution , of an indefinite trans-Sasakian manifold . Denote by the algebra of smooth functions on and by the module of smooth sections of a vector bundle . Also donate by the th equation of several equations in (Equation number), for example, (7)₁ donates the first equation of the two equations in (7). We use same notations for any others.

We follow Duggal-Bejancu [6] for notations and structure equations used in this paper. It is well known that, for any null section of on a coordinate neighborhood , there exists a unique null section of a unique vector bundle of rank 1 in satisfying

In the following, let , and be the vector fields on , unless otherwise specified. Let be the Levi-Civita connection of and the projection morphism of on . Then the local Gauss and Weingarten formulas are given by

where and are the liner connections on and , respectively, and are the local second fundamental forms on and respectively, and are the shape operators on and , respectively, and is a 1-form on .

Since is torsion-free, is also torsion-free and is symmetric. From the fact that , we show that is independent of the choice of and satisfies

The induced connection of is not metric and satisfies

where is a 1-form such that

But the connection on is metric. The above two local second fundamental forms and are related to their shape operators by

Definition 1. A lightlike hypersurface of is said to be

(1)totally umbilical [6] if there is a smooth function on any coordinate neighborhood in such that , or equivalently,

 In case on , we say that is totally geodesic;(2)screen totally umbilical [6] if there exists a smooth function on such that , or equivalently,

 In case on , we say that is screen totally geodesic;(3)screen conformal [7] if there exists a nonvanishing smooth function on such that , or equivalently,

Denote by , , and the curvature tensors of the Levi-Civita connection of , the induced connection on , and the induced connection on , respectively. Using the Gauss-Weingarten formulas for and , we obtain the Gauss-Codazzi equations for and such that

3. Indefinite Trans-Sasakian Manifolds

Let be a lightlike hypersurface of a indefinite trans-Sasakian manifold such that is tangent to . Călin [8] proved that if is tangent to , then it belongs to which we assume in this paper. It is well known [3, 6] that, for any lightlike hypersurface of an indefinite almost contact metric manifold , and are subbundles of , of rank 1, and . Thus is a subbundle of of rank 2. First, we prove the following results.

Theorem 2. (1) Let be a totally umbilical lightlike hypersurface of an indefinite trans-Sasakian manifold . Then and is totally geodesic.
(2) Let be a screen conformal or screen totally umbilical lightlike hypersurface of an indefinite trans-Sasakian manifold . Then . In case is screen totally umbilical, is totally geodesic.

Proof. Applying to and , we have

(1) If is totally umbilical, then, from (18) and (25)₁, we have

Taking and by turns, we have and , respectively. As , is totally geodesic.
(2) If is screen conformal, then, from (20) and (25)1,2, we have

Taking and by turns, we have and , respectively.
If is screen totally umbilical, then, from (19) and (25)₂, we have

Taking , and to this equation by turns, we have , , and , respectively. As , is screen totally geodesic.

As is a subbundle of of rank 2, there exists a nondegenerate almost complex distribution with respect to ; that is, , such that

Consider the 2-lightlike almost complex distribution such that

and the local lightlike vector fields and and their 1-forms such that

Denote by the projection morphism of on . Any vector field of is expressed as . Applying to this, we have

where is a tensor field of type (1, 1) globally defined on by

Applying to the first two equations of (31) and (32) and using (9), (10), (12), (13), (6), (31), and (32), for any , we have

Theorem 3. Let be a lightlike hypersurface of an indefinite trans-Sasakian manifold . If   or is parallel with respect to , then and . If both and are parallel with respect to the induced connection , then is screen totally geodesic.

Proof. (1) If is parallel, then, from (32) and (35) we have

Taking the scalar product with and to (38) by turns and using (4), we have and , respectively. Taking and to the second result by turns, we have and , respectively.
(2) If is parallel with respect to , then, from (32) and (36), we have

Taking the scalar product with to (39) and using (4), we have . Taking the scalar product with to (39) and using (4) and , we get . Taking to this result, we have . From (25)₁ and (31)₃, we obtain

Applying to (39) and using (4) and the fact , we have

Taking the scalar product with to this equation, we get

Replacing by in (40) and using (42), we get

Thus . Then we have

(3) In case and are parallel with respect to , as is parallel, applying to (38) and using (4), (25)₂ and the fact , we obtain

As is parallel, from (34) and (42), we show that . Thus we obtain . Consequently is screen totally geodesic.

Theorem 4. Let be a lightlike hypersurface of an indefinite trans-Sasakian manifold . If is parallel with respect to the connection , then we have . Furthermore and are parallel distributions on and is locally a product manifold , where is a null curve tangent to and is a leaf of .

Proof. If is parallel with respect to , then, taking the scalar product with to (37) and using the facts and , we get

Taking and by turns, we get and . Taking and to the second equation, we have .
From (37) we have

Taking and in (47)₂ by turns, we have and . These results and (13) imply that

By using (4), (9), (12), (14), (32), and (36), we derive

for all and , or equivalently, we get

This result implies that is a parallel distribution on .
Taking the scalar product with to (47)₁, we get for all . Taking to this, we have

For all and , using (35) we derive

that is, for all .  Thus is also parallel. As , and and are parallel distributions, by the decomposition theorem of de Rham [9] we have , where is a null curve tangent to and is a leaf of .

Corollary 5. Let be a lightlike hypersurface of an indefinite trans-Sasakian manifold . If and are parallel with respect to , then is totally geodesic and screen totally geodesic.

Proof. As is parallel with respect to , we get the two equations of (47). As is also parallel with respect to , substituting (34) to (47)₂ and using (42), we have . Thus is totally geodesic. Replacing by to (47)₁, we obtain . Thus is also screen totally geodesic.

4. Indefinite Generalized Sasakian Space Form

An indefinite almost contact metric manifold is said to be an indefinite generalized Sasakian space form [2] and denote it by if there exist three smooth functions ,  , and on such that

for any vector fields , , and on .

Theorem 6. Let be a lightlike hypersurface of an indefinite generalized Sasakian space form . Then is a constant, , and

Proof. Comparing the tangential and transversal components of (21) and (53), and using (32), we get

Taking the scalar product with to (23), we have

Substituting (55) into the last equation and using (17)₂, we obtain

Applying to (34)₁: , we have

Using (25), (32), (34), (35), and (36), the above equation is reduced to

Substituting this equation and (34) into (56) such that , we get

Comparing this equation with (58) such that , we obtain

Taking and and and by turns, we have

Substituting (32) into (7) and using (9), we have

Applying to and using (9), (32), (34), and (35), we get

Applying to and using (4) and (6) we have

Using (31), the equation (25)₂ is reduced to

Applying to this equation and using (64), (65), and (66), we have

Substituting this and (67) into (58) such that , we get

where . Taking and and then taking and to this equation, we obtain

Applying to and using (9), (32), and (36), we get

Applying to (40) and using (40) and (64) and (71), we have

Substituting this into (56) such that and using the fact that , we have . Therefore the function is a constant.
From the facts that is a constant and , if , then we get .
Assume that . Then (64) is reduced to

By straightforward calculations form this equation, we obtain

Comparing this equation with (55) such that , we obtain

Taking the scalar product with to this equation, we get ; that is,

due to (32)₂. Taking and to this equation, we have .
As , (63) and (70) are reduced to and , respectively. From these two results, we get .

Corollary 7. There exist no indefinite generalized Sasakian space forms, endowed with -Kenmotsu structure, admitting a lightlike hypersurface.

Corollary 8. Let be a lightlike hypersurface of an indefinite Sasakian space form , endowed with -Sasakian structure. Then .

Theorem 9. Let is lightlike hypersurface of an indefinite generalized Sasakian space form . If is screen totally umbilical, then .

Proof. As is screen totally umbilical, by (2) of Theorem 2. Thus (58) is reduced to

for all . Replacing by to this equation, we obtain

Taking , ; , , and by turns, we have

From the first two equations we show that . As , is an indefinite cosymplectic manifold. Thus . This implies .

Theorem 10. Let be a screen conformal lightlike hypersurface of an indefinite generalized Sasakian space form . Then .

Proof. Substituting (55) into (57) and using (56), we have

Replacing by to the last equation, we obtain

Taking to this equation and using (40), we obtain . Also taking , , and , by turns, we have

respectively. Comparing these two equations, we obtain .
As is screen conformal, we obtain by Theorem 2. As , we show that is a cosymplectic manifold and . Therefore we get .
Let denote the induced Ricci type tensor of given by

for any , . Consider the induced quasi-orthonormal frame field on such that and . Put . Using this quasi-orthonormal frame field, we obtain

for any , , where is the causal character of . In general, the induced Ricci type tensor is not symmetric [6, 7]. A tensor field of lightlike submanifolds is called its induced Ricci tensor if it is symmetric. A symmetric tensor will be denoted by Ric. A lightlike manifold equipped with an induced Ricci tensor is called Ricci flat if its Ricci tensor vanishes. is called an Einstein manifold if the Ricci tensor of satisfies .
If is a screen conformal lightlike hypersurface of , then, using (55) and the fact that , we have

This implies that is a symmetric induced Ricci tensor Ric.

Theorem 11. Any screen conformal Einstein lightlike hypersurface of an indefinite generalized Sasakian space form is Ricci flat.

Proof. As is Einstein, from (85) and the fact

where is trace of . Define a nonnull vector field on by

Then belongs to . Using (20) and (34), satisfies

From this equation and (16), we show that

Taking to (86) and using (89), we get . Therefore, is Ricci flat.

5. Parallel Structure Fields

Definition 12. Let for any , where is the projection morphism of on . Then is a linear connection on . We say that is the transversal connection of . We define the curvature tensor of by

The transversal connection of is flat [3] if vanishes.

As , we show that the transversal connection of is flat if and only if the 1-form is closed; that is, , on any [3].

Denote and by the 1-forms such that

Theorem 13. Let be a lightlike hypersurface of an indefinite generalized Sasakian space form . If one of the following conditions,

(1) is parallel with respect to the connection ,(2) is parallel with respect to the connection ,(3) is parallel with respect to the connection ,

is satisfied, then is a flat manifold with indefinite cosymplectic structure and the lightlike transversal connection of is flat. In case (1), is also a flat manifold.

Proof. (1) Assume that is parallel with respect to . Then we get by Theorem 4. Thus by Theorem 6 and (37) is reduced to

Taking to (92) and using (31), we have

Taking the scalar product with to (92) and using (17) and (31), we have

As and belong to and is nondegenerate, we have

Taking the scalar product with to (93), we obtain

Applying to and using (37), (93) and , we get

Replacing by to (58) and using the last two equations, we have

Taking and to this equation, we get . Therefore, and is flat.
As , substituting (93) and (95) into (55), we get

Thus is flat. From (37), (93) and the fact that , we get

Substituting this equation into , we get . Thus the transversal connection of is flat.
(2) If is parallel with respect to , then, by Theorem 3. Thus by Theorem 6 and (35) is reduced to

Applying to (101) and using (4), (31), and (67), we have

Taking the scalar product with to (102), we get

Applying to this and using (35), (102) and the fact that , we get

Substituting the last two equation into (58) such that , we have

Taking and to this equation, we obtain . Therefore, and is flat. As , we obtain . Thus the transversal connection of is flat.
(3) If is parallel with respect to , then, by Theorem 3. Thus by Theorem 6 and (35) is reduced to

Applying to (106) and using (4) and (40), we have

Taking the scalar product with to this equation, we get

Applying to this equation and using (35), we have

Substituting the last two equations into (56), we obtain

Taking and to this equation and using (14) and (108), we obtain . Therefore, and is flat. As , we obtain . Thus the lightlike transversal connection of is flat.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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