Article
Kyungpook Mathematical Journal 2020; 60(2): 223-238
Published online June 30, 2020
Copyright © Kyungpook Mathematical Journal.
Lightlike Hypersurfaces of an Indefinite Nearly Trans-Sasakian Manifold with an (ℓ,m)-type Connection
Chul Woo Lee, Jae Won Lee∗
Department of Mathematics, Kyungpook National University, Daegu 41566, Korea
e-mail : mathisu@knu.ac.kr
Department of Mathematics Education and RINS, Gyeongsang National University, Jinju 52828, Korea
e-mail : leejaew@gnu.ac.kr
Received: February 14, 2020; Revised: March 31, 2020; Accepted: April 27, 2020
Abstract
We study a lightlike hypersurface
Keywords: (ℓ,
1. Introduction
A linear connection ∇̄ on a semi-Riemannian manifold (
for any vector fields
The notion of (
Remark 1.1.([8])
Denote by ∇̃ a unique Levi-Civita connection of a semi-Riemannian manifold (
The subject of study in this paper is lightlike hypersurfaces of an indefinite nearly trans-Sasakian manifold
Călin [3] proved that
2. On (ℓ, m )-type Connections
A hypersurface
where ⊕, there exists a unique null section
In this case, the tangent bundle
We call
In the following, we denote by
where ∇ and ∇
An odd dimensional semi-Riemannian manifold (
It is known [5, 6] that, for any lightlike hypersurface
In this case, the decomposition form of
Consider two lightlike vector fields
Denote by
where
Using (
where
From the fact that
The local second fundamental forms are related to their shape operators by
As
Applying ∇
Applying ∇
Applying ∇
3. Recurrents and Lie Recurrents
Definition 3.1.([7])
The structure tensor field
Theorem 3.2
Comparing (
Also, comparing (
Taking the scalar product with
Applying
Definition 3.3.([7])
The structure tensor field
where ℒ
The structure tensor field
Theorem 3.4
As the induced connection ∇ from ∇̄ is torsion-free, from (
Comparing (
Also, comparing (
Taking the scalar product with
Applying
4. Indefinite Nearly Trans-Sasakian Manifolds
Definition 4.1.([9])
An indefinite almost contact metric manifold
where ∇̃ is the Levi-Civita connection of
Note that the indefinite nearly Sasakian manifolds, indefinite nearly Kanmotsu manifolds and indefinite nearly cosymplectic manifolds are important examples of indefinite nearly trans-Sasakian manifold such that
Replacing the Levi-Civita connection ∇̃ by the (
Applying ∇̄
Definition 4.2.([4])
A lightlike hypersurface
-
totally umbilical if there is a smooth functionρ on a coordinate neighborhoodin
M such thator equivalently In case
ρ = 0 on, we say that
M istotally geodesic . -
screen totally umbilical if there exist a smooth functionγ on a coordinate neighborhoodsuch that
A N X =γPX or equivalentlyIn case
γ = 0 on, we say that
M isscreen totally geodesic .
Theorem 4.3
-
If M is totally umbilical, then M is totally geodesic and m = 0. -
If M is screen totally umbilical, then M is screen totally geodesic.
(1) If
(2) If
Applying ∇̄
Substituting (
Lemma 4.4
Applying ∇
From these two equations, we obtain
Comparing this result with (
Taking the scalar product with
By direct calculation from
Taking the scalar product with
Substituting (
By directed calculation from
Taking the scalar product with
Taking
Taking the scalar product with
Substituting (
Taking
due to (
Comparing this equation with (
Lemma 4.5
-
(∇
X F )Y + (∇Y F )X = 0, -
F is parallel with respect to the induced connection ∇on M, that is , ∇X F = 0, -
F is recurrent ,
(1) Assume that (∇
Taking
Taking the scalar product with
Taking
Replacing
Using (
Taking
Replacing
Taking
(2) If
(3) If
Theorem 4.6
If
Taking the scalar product with
due to (
Taking the scalar product with
Comparing this with (
Applying ∇
Taking the scalar product with
Replacing
As
Substituting the last two results into (
5. Indefinite Nearly Generalized Sasakian Space Forms
Denote by
Definition 5.1
An indefinite nearly trans-Sasakian manifold
where
The notion of (Riemannian) generalized Sasakian space form was introduced by Alegre
respectively, where
By direct calculations from (
Comparing the tangential, transversal and radical components of the left-right terms of (
due to the following equations:
Using the Gauss-Weingarten formulae for
Replacing
Comparing the radical components of the last two equations, we obtain
Applying ∇̄
Theorem 5.2
-
(∇
X F )Y + (∇Y F )X = 0, -
F is parallel with respect to the induced connection ∇,that is , ∇X F = 0, -
F is recurrent ,
If one of the items (1)~(3) is satisfied, then
Applying ∇
Substituting the last two equations into (
Taking
By using (
From the last three equations, we get
Definition 5.3
A lightlike hypersurface
Theorem 5.4
Taking the scalar product with
Applying ∇
Applying ∇
due to (
If
Theorem 5.5
If
Taking
Theorem 5.6
If
Applying ∇̄
due to
From these equations, we have our theorem.
References
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