Abstract

The object of the present paper is to study invariant pseudo parallel submanifolds of a LP-Sasakian manifold and obtain the conditions under which the submanifolds are pseudoparallel, 2-pseudoparallel, generalized pseudoparallel and 2-generalized pseduoparallel. Finally, a non-trivial example is used to demonstrate that the method presented in this paper is effective.

1991 Mathematics Subject Classification. 53C15; 53C40, 53D10.

1. Introduction

Invariant submanifolds of a (para) contact metric manifold have been a major area of research for a long time. It helps us to understand several important topics of applied mathematics. For example, in studying non-linear autonomous systems the idea of invariant submanifolds plays an important role ¹.

A submanifold of a Riemannian manifold is said to be totally geodesic if every geodesic in that submanifold is also geodesic in the ambient manifold. In ², Kon proved that invariant submanifolds of a Sasakian manifold are totally geodesic if the second fundamental form of the immersion is covariantly constant. On the other hand, any submanifold of a Kenmotsu manifold is totally geodesic if and only if the second fundamental form of the immersion is covariantly constant ³.

Furthermore, in ⁴, authors proved that some equivalent conditions of an invariant submanifold of trans-Sasakian to be totally geodesic. Recently, in ⁵, authors considered invariant submanifolds of (κ,µ)-contact metric manifold and obtained some conditions for an invariant submanifold to be totally geodesic.

In the present paper, we also introduce some new equivalent conditions for an invariant submanifold of a LP-Sasakian manifold to be pseudoparallel.

2. Preliminaries

On the analogy of Sasakian manifolds, Matsumoto introduced the notion of LP-Sasakian manifolds ⁶. After then, some properties of the their submanifolds and LP-Sasakian manifolds are also studied many geometers ^{4, 6, 8, 9, 13}.

An n-dimensional smooth manifold is said to be an Lorentzian para(briefly LP) Sasakian manifold if it admits a (1,1)-type tensor field a unit timelike vector field ξ, a 1-form η and a Lorentzian metric tensor g which satisfy;

(1)

(2)

(3)

for all (), where and denote the Levi-Civita connection and set of the differentiable vector fields on respectively.

Moreover, we can easily to see that in an LP-Sasakian manifold,

(4)

(5)

(6)

(7)

for all where and S are the Riemannian curvature tensor and Ricci tensor of respectively.

Now, let be an immersed submanifold of an LP-Sasakian manifold Then the Gauss and Weingarten formulae are, respectively, given by

(8)

And

(9)

for all and where is the induced Levi-Civita connection on is the normal connection on the normal bundle σ is the second fundamental form of M and A_V is the shape operator with respect to the normal vector field V.

Moreover the shape operator A_V and second fundamental form σ are related by

(10)

for all and

The covariant derivatives of σ and A_V are defined

(11)

And

(12)

for all They are also related

If then the submanifold said to have parallel second fundamental form.

By we denote the Riemannian curvature tensor of submanifold we have the following Gauss equation;

(13)

for all where if then submanifold is called curvature-invariant submanifold.

For a -type tensor field and a (0, 2)-type tensor field on a Riemannian manifold -tensor field is defined by

(14)

for all where

(15)

In ¹⁰, J. Deprez defined the semiparallel immersion in the following way;

(16)

for all where denote the Riemannian curvature tensor of the normal bundle If then is said to be semiparallel.

In ¹¹, Authors called pseudoparallel submanifold the satisfying the curvature condition

(17)

On the other hand, C. Murathan, K. Arslan and R. Ezentas defined and studied submanifolds satisfying the condition ¹²

(18)

This kind of submanifolds are called generalized Ricci-pseudoparallel, where (18) is defined by

Furthermore, the submanifolds satisfying the condition

(19)

are called 2-pseudoparallel ¹³. Particularly, then submanifold is said to be 2-semiparallel.

Also, the submanifolds satisfying the condition

(20)

are called 2-generalized Ricci pseudoparallel.

Riemannian curvature tensor satisfies

(21)

Particularly, if = 0, it is called semisymmetric manifold.

3. Some classes of Invariant Submanifolds of LP-Sasakian Manifolds

Now, let be an immersed submanifold of an LP-Sasakian manifold manifold

If for each point at then is said to be invariant submanifold. We note that all of the properties of an invariant submanifold inherit the ambient manifold.

From (14), we have the following proposition for later use.

Proposition 3.1. Let be an invariant submanifold of an LP-Sasakan manifold Then the following equalities hold on

(22)

(23)

(24)

(25)

(26)

(27)

(28)

for all

Next, we will give the main results of this paper.

Theorem 3.2. If an LP-Sasakian manifold is a pseudosymmetric, then it is an η-Einstein provided that L 1 and n 3.

Proof. If is a pseudosymmetric, (21) implies that

for all It follows that

(29)

The relation (29) yields for

It follows that

(30)

for all Putting in (30) for orthonormal basis of Then by a straightforward calculations, we obtain

which proves our assertion.

From the Theorem 3.2 and (21), we have following proposition.

Proposition 3.3. LP-Sasakian manifold is semisymmetric if and only if it has 1-constant sectional curvature.

Theorem 3.4. Let M be an invariant pseudoparallel submanifold of an LP-Sasakain manifold Then M is either totally geodesic or

Proof. Let us assume that M is an invariant pseudoparallel submanifold. Then from (14) and (17), we have

for all This implies that

(31)

The relation (31) yields for

This proves our assertion.

Theorem 3.5. Let M be an invariant 2-pseudoparallel submanifold of an LP-Sasakain manifold Then M is either totally geodesic or

Proof. If is a 2-pseudoparallel, then (14) and (19) imply that

for all It follows that

(32)

Putting in (32) and taking into account of (15), we have

(33)

Now, Let’s calculate each term. Also taking into account that (11) and (28), we obtain

(34)

Making use of (5), we reach at

(35)

(36)

(37)

(38)

(39)

Finally,

(40)

If (34), (35), (36), (37), (38), (39) and (40) statements are substituted in (33), we obtain

(41)

The relation (41) yields for

(42)

On the other hand, by means of (11) and (28), we conclude

(43)

From (42) and (43), we get

which proves our assertion.

Theorem 3.6. Let M be an invariant Ricci generalized pseudoparallel submanifold of an LP-Sasakian manifold Then M is either totally geodesic or

Proof. We assume that is an invariant Ricci-generalized pseudoparallel. Then from (18), we have

for all (14) and (15) lead to

(44)

Replacing by in (44), we have

This completes the proof.

Theorem 3.7. Let M be an invariant 2-Ricci generalized pseudoparallel submanifold of an LP-Sasakian manifold Then M is either totally geodesic or

Proof. Let M be an invariant 2-Ricci generalized pseudoparallel submanifold of an LP-Sasakian manifold Then (20) implies that

for all By virtue of (14) and (15), it follows that

This yields for

(45)

Now, let’s examine these situation separately. Also, by using of (11), (22) and (28), we obtain

(46)

(47)

(48)

(49)

(50)

(51)

Finally,

(52)

Consequently, substituting (46), (47), (48), (49), (50), (51) and (52) into (45), we reach at

(53)

Replacing by in (53), we conclude that

By means of (43), we have

which proves our assertion.

Example 3.8. Let us the 5-dimensional manifold

where (x_i, t) denote the cartesian coordinates in R⁵ for Then the vector fields

are linearly independent at each point of By g, we denote the semi-Riemannian metric tensor such that

Let η be the 1-form defined by for all X Now, we define the tensor field (1,1)-type ϕ such that

Then we can easily to see that

and

for all Thus defines an almost Lorentzian paracontact metric manifold. By we denote the Levi-Civita connection on Then by direct calculations, we have

thus one can easily verified

this tell us that is a LP-Sasakian manifold.

Now, let us a submanifolds of is defined by immersion as follows;

Then the tangent space of is spanned by the vector fields

Moreover, we can easily observe that ϕU=U and ϕV = V that is, M is a 3-dimensional invariant submanifold of an LP-Sasakian manifold Furthermore, we can easily verify that

This tell us that is pseudoparallel, Ricci generalized pseudoparallel submanifold because of it is a totally geodesic submanifold of

References