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Original Article

Open Access Peer-reviewed

MEHMET ATÇEKEN^{ }

Received June 01, 2020; Revised July 02, 2020; Accepted July 10, 2020

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.

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.

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.

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** **pseudopar**allel **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** **pseudopar**allel **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** **th**e cartesian coordinates** **in** **R*^{5}* **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

[1] | Guojing. Z. and Jianguo, W. Invariant submanifolds and modes of non- linear auotonomous systems. Appl. Math. Mech. 1998, 19, 587-693. | ||

In article | View Article | ||

[2] | Kon, M. Invariant submanifolds of normal contact metric manifolds. Kodai Math. Sem. Rep. 1973, 27, 330-336. | ||

In article | View Article | ||

[3] | Kobayashi, M. Semi Invariant Submanifolds of a Certain class of Almost contact metric manifolds. Tensor(N.S),1986,43, 28-36. | ||

In article | |||

[4] | Sarkar, A. and Sen, M. On Invariant Submanifolds Trans-Sasakian Mani- folds. proc. Estonian Acad. Sci. 2012, 61, 29-37. | ||

In article | View Article | ||

[5] | Siddesha, M. S. and Bagewadi, C.S. On Some Classes an Invariant Sub- manifolds of (κ,µ)-Contact Manifold. J. of inforatics and mathematical Sciences. Vol.9,No.1, 13-26, 2017. | ||

In article | |||

[6] | K. Matsumoto. On Lorentzian Almost Paracontact Manifolds. Bull. Yam- agata Univ. Nat. Sci.12(1989), 151-156. | ||

In article | |||

[7] | A. A. Aqeel, U.C. De, G. C. Ghosh. On Lorentzian Para-Sasakian Mani- folds. Kuwait J. Sci. eng. 31(2), 2004, 1-13. | ||

In article | |||

[8] | I. Mihai, U. C. De, A. A. Shaikh. On Lorentzian para-Sasakian Manifolds. Korean J. Math. Sci. 6(1999), 1-13. | ||

In article | |||

[9] | C. Murathan, A. Yildiz, K. Arslan, U. C. De. On a Class of Lorentzian Para-Sasakian Manifolds. Proc. Estonian Acad. Sci. Phys. Math. 55(4), 2006, 210-219. | ||

In article | |||

[10] | J. Deprez. Semi-Parallel Surfaces in the Euclidean Space. J. of Geometry, 25(1985),192-200. | ||

In article | View Article | ||

[11] | A. C. Asperti, G. A. Lobos and F. Mercuri. Pseudo-Parallel immersions in Space Forms. Math. Contemp. 17(1999), 59-70. | ||

In article | |||

[12] | C. Murathan, K. Arslan and R. Ezenta. Ricci-Generalized Pseudoparallel Immersions. Diff. geom. and Its. Appl. 99-108. Matfyzpress Pragua. 2005. | ||

In article | |||

[13] | C. Özgür and C. Murathan. On Pseudoparallel Invariant Submanifolds of Contact Metric Manifolds. Bull. Transilv. Univ. Braov. Ser. B(N.S), 14(49). 2007. | ||

In article | |||

[14] | C. Özgür and C. Murathan. On Invariant Submanifolds of Lorentzian Para-Sasakian Manifolds. The Arabian J. of Sci. and Eng. Vol.34, Num.2A, 2009, 277-185. | ||

In article | |||

Published with license by Science and Education Publishing, Copyright © 2020 MEHMET ATÇEKEN

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

MEHMET ATÇEKEN. Some Classes of Invariant Submanifolds of LP-Sasakian Manifolds. *Turkish Journal of Analysis and Number Theory*. Vol. 8, No. 2, 2020, pp 28-33. http://pubs.sciepub.com/tjant/8/2/2

ATÇEKEN, MEHMET. "Some Classes of Invariant Submanifolds of LP-Sasakian Manifolds." *Turkish Journal of Analysis and Number Theory* 8.2 (2020): 28-33.

ATÇEKEN, M. (2020). Some Classes of Invariant Submanifolds of LP-Sasakian Manifolds. *Turkish Journal of Analysis and Number Theory*, *8*(2), 28-33.

ATÇEKEN, MEHMET. "Some Classes of Invariant Submanifolds of LP-Sasakian Manifolds." *Turkish Journal of Analysis and Number Theory* 8, no. 2 (2020): 28-33.

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[1] | Guojing. Z. and Jianguo, W. Invariant submanifolds and modes of non- linear auotonomous systems. Appl. Math. Mech. 1998, 19, 587-693. | ||

In article | View Article | ||

[2] | Kon, M. Invariant submanifolds of normal contact metric manifolds. Kodai Math. Sem. Rep. 1973, 27, 330-336. | ||

In article | View Article | ||

[3] | Kobayashi, M. Semi Invariant Submanifolds of a Certain class of Almost contact metric manifolds. Tensor(N.S),1986,43, 28-36. | ||

In article | |||

[4] | Sarkar, A. and Sen, M. On Invariant Submanifolds Trans-Sasakian Mani- folds. proc. Estonian Acad. Sci. 2012, 61, 29-37. | ||

In article | View Article | ||

[5] | Siddesha, M. S. and Bagewadi, C.S. On Some Classes an Invariant Sub- manifolds of (κ,µ)-Contact Manifold. J. of inforatics and mathematical Sciences. Vol.9,No.1, 13-26, 2017. | ||

In article | |||

[6] | K. Matsumoto. On Lorentzian Almost Paracontact Manifolds. Bull. Yam- agata Univ. Nat. Sci.12(1989), 151-156. | ||

In article | |||

[7] | A. A. Aqeel, U.C. De, G. C. Ghosh. On Lorentzian Para-Sasakian Mani- folds. Kuwait J. Sci. eng. 31(2), 2004, 1-13. | ||

In article | |||

[8] | I. Mihai, U. C. De, A. A. Shaikh. On Lorentzian para-Sasakian Manifolds. Korean J. Math. Sci. 6(1999), 1-13. | ||

In article | |||

[9] | C. Murathan, A. Yildiz, K. Arslan, U. C. De. On a Class of Lorentzian Para-Sasakian Manifolds. Proc. Estonian Acad. Sci. Phys. Math. 55(4), 2006, 210-219. | ||

In article | |||

[10] | J. Deprez. Semi-Parallel Surfaces in the Euclidean Space. J. of Geometry, 25(1985),192-200. | ||

In article | View Article | ||

[11] | A. C. Asperti, G. A. Lobos and F. Mercuri. Pseudo-Parallel immersions in Space Forms. Math. Contemp. 17(1999), 59-70. | ||

In article | |||

[12] | C. Murathan, K. Arslan and R. Ezenta. Ricci-Generalized Pseudoparallel Immersions. Diff. geom. and Its. Appl. 99-108. Matfyzpress Pragua. 2005. | ||

In article | |||

[13] | C. Özgür and C. Murathan. On Pseudoparallel Invariant Submanifolds of Contact Metric Manifolds. Bull. Transilv. Univ. Braov. Ser. B(N.S), 14(49). 2007. | ||

In article | |||

[14] | C. Özgür and C. Murathan. On Invariant Submanifolds of Lorentzian Para-Sasakian Manifolds. The Arabian J. of Sci. and Eng. Vol.34, Num.2A, 2009, 277-185. | ||

In article | |||