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Axioms 2018, 7(3), 68; doi:10.3390/axioms7030068

Article
Characterizations of the Total Space (Indefinite Trans-Sasakian Manifolds) Admitting a Semi-Symmetric Metric Connection
1,†
and
2,†,*
1
Department of Mathematics, Dongguk University, Gyeongju 38066, Korea
2
Department of Mathematics Education and RINS, Gyeongsang National University, Jinju 52828, Korea
*
Correspondence: [email protected]; Tel.: +82-55-772-2251
These authors contributed equally to this work.
Received: 19 August 2018 / Accepted: 7 September 2018 / Published: 10 September 2018

Abstract

:
We investigate recurrent, Lie-recurrent, and Hopf lightlike hypersurfaces of an indefinite trans-Sasakian manifold with a semi-symmetric metric connection. In these hypersurfaces, we obtain several new results. Moreover, we characterize that the total space (an indefinite generalized Sasakian space form) with a semi-symmetric metric connection is an indefinite Kenmotsu space form under various lightlike hypersurfaces.
Keywords:
lightlike hypersurfaces; indefinite trans-Sasakian; Lie-recurrent; Hopf; semi-symmetric metric connection

1. Introduction

A semi-symmetric connection ¯ on a semi-Riemannian manifold ( M ¯ , g ¯ ) was introduced by Friedmann-Schouten [1] in 1924, whose torsion tensor T ¯ satisfies
T ¯ ( X ¯ , Y ¯ ) = θ ( Y ¯ ) X ¯ θ ( X ¯ ) Y ¯ ,
where θ is a 1-form associated with a vector field ζ by θ ( X ¯ ) = g ¯ ( X ¯ , ζ ) . In particular, if it is a metric connection (i.e., ¯ g ¯ = 0 ), then ¯ is said to be a semi-symmetric metric connection. This notion on a Riemannian manifold was introduced by Yano [2]. He proved that a Riemannian manifold admits a semi-symmetric metric connection whose curvature tensor vanishes if and only if a Riemannian manifold is conformally flat.
In a semi-Riemannian manifold, Duggal and Sharma [3] studied some properties of the Ricci tensor, affine conformal motions, geodesics, and group manifolds admitting a semi-symmetric metric connection. They also showed the geometric results had physical meanings.
In the following, we denote by X ¯ , Y ¯ , and Z ¯ the smooth vector fields on M ¯ .
Remark 1.
Let ˜ be the Levi-Civita connection of the semi-Riemannian manifold ( M ¯ , g ¯ ) with respect to the metric g ¯ . A linear connection ¯ on M ¯ is a semi-symmetric metric connection if and only if
¯ X ¯ Y ¯ = ˜ X ¯ Y ¯ + θ ( Y ¯ ) X ¯ g ¯ ( X ¯ , Y ¯ ) ζ .
On the other hand, Bejancu and Duggal [4] showed the existence of almost contact metric manifolds and established examples of Sasakian manifolds in semi-Riemannian manifolds. They also classified real hypersurfaces of indefinite complex space forms with parallel structure vector field, and then proved that Sasakian real hypersurfaces of a semi-Euclidean space are either open sets of the pseudo-sphere or of the pseudo-hyperbolic. In trans-Sasakian manifolds, which generalizes Sasakian manifolds and Kenmotsu manifolds, Prasad et al. [5] studied some special types of trans-Sasakian manifolds. De and Sarkar [6] studied the notion of ( ϵ ) -Kenmotsu manifolds. Shukla and Singh [7] extended the study to ( ϵ ) -trans-Sasakian manifolds with indefinite metric. Siddiqi et al. [8] also studied some properties of indefinite trans-Sasakian manifolds, which is closely related to this topic.
The object of study in this paper is recurrent, Lie-recurrent, and Hopf lightlike hypersurfaces of an indefinite trans-Sasakian manifold ( M ¯ , J , ζ , θ , g ¯ ) with a semi-symmetric metric connection ¯ . We provide several results on such a lightlike hypersurface. In the last section, we characterize that an indefinite generalized Sasakian space form with a semi-symmetric metric connection is an indefinite Kenmotsu space form under various lightlike hypersurfaces.

2. Lightlike Hypersurfaces

An odd-dimensional pseudo-Riemannian manifold ( M ¯ , g ¯ ) is called an indefinite almost contact metric manifold if there exists an indefinite almost contact metric structure { J , ζ , θ , g ¯ } with a ( 1 , 1 ) -type tensor field J, a vector field ζ , and a 1-form θ such that
J 2 X ¯ = X ¯ + θ ( X ¯ ) ζ , g ¯ ( J X ¯ , J Y ¯ ) = g ¯ ( X ¯ , Y ¯ ) ϵ θ ( X ¯ ) θ ( Y ¯ ) , θ ( ζ ) = ϵ ,
where ϵ = 1 or 1 if ζ is spacelike or timelike, respectively.
From (3), we derive
J ζ = 0 , θ J = 0 , θ ( X ¯ ) = ϵ g ¯ ( X ¯ , ζ ) , g ¯ ( J X ¯ , Y ¯ ) = g ¯ ( X ¯ , J Y ¯ ) .
Without loss of generality, we assume that the structure vector field ζ is spacelike (i.e., ϵ = 1 ) in the entire discussion of this article.
Definition 1.
An indefinite almost contact metric manifold M ¯ , J , ζ , θ , g ¯ is called an indefinite trans-Sasakian manifold [9] if, for the Levi-Civita connection ˜ with respect to g ¯ , there exist two smooth functions α and β such that
( ˜ X ¯ J ) Y ¯ = α { g ¯ ( X ¯ , Y ¯ ) ζ θ ( Y ¯ ) X ¯ } + β { g ¯ ( J X ¯ , Y ¯ ) ζ θ ( Y ¯ ) J X ¯ } .
Here, { J , ζ , θ , g ¯ } is called an indefinite trans-Sasakian structure of type ( α , β ) .
Note that Sasakian α = 1 , β = 0 , Kenmotsu α = 0 , β = ϵ and cosymplectic α = β = 0 manifolds are important kinds of trans-Sasakian manifolds.
Let ¯ be a semi-symmetric metric connection on an indefinite trans-Sasakian manifold M ¯ = ( M ¯ , J , ζ , θ , g ¯ ) . By using (2), (3) and the fact that J ζ = 0 and θ J = 0 , we see that
( ¯ X ¯ J ) Y ¯ = α { g ¯ ( X ¯ , Y ¯ ) ζ θ ( Y ¯ ) X ¯ } + ( β + 1 ) { g ¯ ( J X ¯ , Y ¯ ) ζ θ ( Y ¯ ) J X ¯ } .
Setting Y ¯ = ζ in (4), J ζ = 0 , and θ ( ¯ X ¯ ζ ) = 0 imply that
¯ X ¯ ζ = α J X ¯ + ( β + 1 ) { X ¯ θ ( X ¯ ) ζ } .
From the covariant derivative of θ ( Y ¯ ) = g ¯ ( Y ¯ , ζ ) in terms of X ¯ with (1), (3), and (5), we have
d θ ( X ¯ , Y ¯ ) = α g ¯ ( X ¯ , J Y ¯ ) .
Let ( M , g ) be a hypersurface of M ¯ . Denote by T M and T M the tangent and normal bundles of M, respectively. Then, there exists a screen distribution S ( T M ) on M [10] such that
T M = T M o r t h S ( T M ) ,
where o r t h denotes the orthogonal direct sum. Throughout this article, we assume that F ( M ) is the algebra of smooth functions on M and Γ ( E ) is the F ( M ) -module of smooth sections of a vector bundle E over M. Also, we denote the i-th equation of (3) by (3) i . These notations may be used in several terms throughout this paper.
For a null section ξ Γ ( T M | U ) on a coordinate neighborhood U M , there exists a unique null transversal vector field N of a unique transversal vector bundle t r ( T M ) in S ( T M ) [10] satisfying
g ¯ ( ξ , N ) = 1 , g ¯ ( N , N ) = g ¯ ( N , X ) = 0 , X Γ ( S ( T M ) ) .
Then, we have the decomposition of the tangent bundle T M ¯ of M ¯ as follows:
T M ¯ = T M t r ( T M ) = { T M t r ( T M ) } o r t h S ( T M ) .
Let P : T M S ( T M ) be the projection morphism. Then, we have the local Gauss–Weingarten formulas of M and S ( T M ) as follows:
¯ X Y = X Y + B ( X , Y ) N ,
¯ X N = A N X + τ ( X ) N ,
X P Y = X * P Y + C ( X , P Y ) ξ ,
X ξ = A ξ * X τ ( X ) ξ ,
respectively, where * is the induced linear connection on T M S ( T M ) , resp . , B C is the local second fundamental form on T M S ( T M ) , resp . , A N A ξ * is the shape operator on T M S ( T M ) , resp . , and τ is a 1-form on T M . Then, it is well known that ∇ is a semi-symmetric non-metric connection and
( X g ) ( Y , Z ) = B ( X , Y ) η ( Z ) + B ( X , Z ) η ( Y ) ,
T ( X , Y ) = θ ( Y ) X θ ( X ) Y .
B is symmetric on T M , where T is the torsion tensor with respect to the induced connection ∇ on M and η ( ) = g ¯ ( , N ) is a 1-form on T M .
B ( X , Y ) = g ¯ ( ¯ X Y , ξ ) implies that B is independent of the choice of the screen distribution S ( T M ) , and we have
B ( X , ξ ) = 0 .
Moreover, two local second fundamental forms B and C for T M and S ( T M ) give the relations with their shape operators, respectively, as follows:
B ( X , Y ) = g ( A ξ * X , Y ) , g ¯ ( A ξ * X , N ) = 0 ,
C ( X , P Y ) = g ( A N X , P Y ) , g ¯ ( A N X , N ) = 0 .
From (13), A ξ * is a S ( T M ) -valued real self-adjoint operator and satisfies
A ξ * ξ = 0 .

3. Semi-Symmetric Metric Connections

Let M be a lightlike hypersurface of an indefinite almost contact metric manifold M ¯ , and denote by J ( T M ) and J ( t r ( T M ) ) sub-bundles of S ( T M ) , of rank 1 [11], respectively. Now we assume that the structure vector field ζ is tangent to M. Cǎlin [12] proved that if ζ Γ ( T M ) , then ζ Γ S ( T M ) . Then, there exist two non-degenerate almost complex distributions D o i . e . , J ( D o ) = D o and D i . e . , J ( D ) = D with respect to J such that
S ( T M ) = J ( T M ) J ( t r ( T M ) ) o r t h D o , D = T M o r t h J ( T M ) o r t h D o .
From these two distributions, we have a decomposition of T M as follows:
T M = D J ( t r ( T M ) ) .
Consider two null vector fields U and V and their 1-forms u and v such that
U = J N , V = J ξ , u ( X ) = g ( X , V ) , v ( X ) = g ( X , U ) .
Denote by S : T M D the projection morphism of T M on D. X Γ ( T M ) is expressed as X = S X + u ( X ) U . Then, it is obtained
J X = F X + u ( X ) N ,
where F is the structure tensor field of type (1, 1) globally defined on M by F X = J S X .
Applying J to (18) with (17) and (18), we have
F 2 X = X + u ( X ) U + θ ( X ) ζ .
Here, the vector field U is called the structure vector field of M.
Replacing Y by ζ in (6) with (5) and (18), one gets
X ζ = α F X + ( β + 1 ) { X θ ( X ) ζ } ,
B ( X , ζ ) = α u ( X ) .
From the covariant derivative of g ¯ ( ζ , N ) = 0 in terms of X with (5), (7), and (14), it is obtained that
C ( X , ζ ) = α v ( X ) + ( β + 1 ) η ( X ) .
Applying ¯ X to (17) and (18) and using (4), (6), and (7), we get
B ( X , U ) = C ( X , V ) ,
X U = F ( A N X ) + τ ( X ) U { α η ( X ) + ( β + 1 ) v ( X ) } ζ ,
X V = F ( A ξ * X ) τ ( X ) V ( β + 1 ) u ( X ) ζ ,
( X F ) ( Y ) = u ( Y ) A N X B ( X , Y ) U + α { g ( X , Y ) ζ θ ( Y ) X } + ( β + 1 ) { g ¯ ( J X , Y ) ζ θ ( Y ) F X } ,
( X u ) ( Y ) = u ( Y ) τ ( X ) B ( X , F Y ) ( β + 1 ) θ ( Y ) u ( X ) ,
( X v ) ( Y ) = v ( Y ) τ ( X ) g ( A N X , F Y ) { α η ( X ) + ( β + 1 ) v ( X ) } θ ( Y ) .
Theorem 1.
Let M be a lightlike hypersurface of an indefinite trans-Sasakian manifold M ¯ with a semi-symmetric metric connection. If either U = 0 or V = 0 , then τ = 0 and M ¯ is an indefinite Kenmotsu manifold. That is, α = 0 and β = 1 .
Proof. 
(1) If U = 0 , then, taking the scalar product with ζ and V to (24) by turns, it is obtained
α = 0 , β = 1 , τ = 0 .
As α = 0 and β = 1 , M ¯ is an indefinite Kenmotsu manifold. Applying F to (24): F ( A N X ) = 0 and using (19) and (22), it is obtained that
A N X = u ( A N X ) U .
(2) If V = 0 , then, taking the scalar product with ζ and U to (25) by turns, we have β = 1 and τ = 0 . Applying F to (25): F ( A ξ * X ) = 0 and using (19) and (21), one gets
A ξ * X = α u ( X ) ζ + u ( A ξ * X ) U .
Taking the scalar product with U to the above equation, we have
B ( X , U ) = 0 .
Replacing X by ζ in (30) and using (21), we have α = 0 . Hence, M ¯ is an indefinite Kenmotsu manifold. ☐

4. Recurrent, Lie-Recurrent, and Hopf Hypersurfaces

Definition 2.
The structure tensor field F of M is said to be recurrent [13] if there exists a 1-form ϖ on M such that
( X F ) Y = ϖ ( X ) F Y .
A lightlike hypersurface M of an indefinite trans-Sasakian manifold M ¯ is said to be recurrent if its structure tensor field F is recurrent.
Theorem 2.
Let M be a recurrent lightlike hypersurface of an indefinite trans-Sasakian manifold M ¯ with a semi-symmetric metric connection. Then
(1) 
α = 0 and β = 1 (i.e., M ¯ is an indefinite Kenmotsu manifold),
(2) 
F is parallel in terms of the induced connection ∇ on M,
(3) 
D and J ( t r ( T M ) ) are parallel distributions on M, and
(4) 
M is locally a product manifold C U × M , where C U is a null curve tangent to J ( t r ( T M ) ) and M is a leaf of the distribution D.
Proof. 
(1) From (26), we have
ϖ ( X ) F Y = u ( Y ) A N X B ( X , Y ) U + α { g ( X , Y ) ζ θ ( Y ) X } + ( β + 1 ) { g ¯ ( J X , Y ) ζ θ ( Y ) F X } .
Setting Y = ζ in (31) with (3) and (21), it is obtained that
α { X + u ( X ) U + θ ( X ) ζ } ( β + 1 ) F X = 0 .
Taking X = ξ to this equation and using the fact that F ξ = V , we have
α ξ + ( β + 1 ) V = 0 .
Taking the scalar product with N and U to the above equation by turns, we get
α = 0 , β = 1 .
Therefore, M ¯ is an indefinite Kenmotsu manifold.
(2) Taking Y by ξ to (31) and using (12), we get ϖ ( X ) V = 0 . It follows that ϖ = 0 . Thus, F is parallel with respect to the connection ∇.
(3) Taking the scalar product with V to (31), it is obtained that
B ( X , Y ) = u ( Y ) u ( A N X ) .
Setting Y = V and Y = F Z o , Z o Γ ( D o ) to the above equation by turns with the fact that u ( F Z o ) = 0 as F Z o = J Z o Γ ( D o ) , we have
B ( X , V ) = 0 , B ( X , F Z o ) = 0 .
Generally, from (6), (9), (13), and (25), we derive
g ( X ξ , V ) = B ( X , V ) , g ( X V , V ) = 0 , g ( X Z o , V ) = B ( X , F Z o ) , Z o Γ ( D o ) .
From these equations and (33), we see that
X Y Γ ( D ) , X Γ ( T M ) , Y Γ ( D ) ,
and hence D is a parallel distribution on M.
On the other hand, setting Y = U in (31) with (32), we have
A N X = B ( X , U ) U .
Using F U = 0 in (34), it is obtained that
F ( A N X ) = 0 .
Using this result and (32), Equation (24) is reduced to
X U = τ ( X ) U .
It follows that
X U Γ ( J ( t r ( T M ) ) ) , X Γ ( T M ) ,
and hence J ( t r ( T M ) ) is parallel on M.
(4) From (16), D and J ( t r ( T M ) ) are parallel. By the decomposition theorem [14], M is locally a product manifold C U × M , where C U is a null curve tangent to J ( t r ( T M ) ) and M is a leaf of D. ☐
Definition 3.
The structure tensor field F of M is said to be Lie-recurrent [13] if
( L X F ) Y = ϑ ( X ) F Y ,
for some 1-form ϑ on M, where L X denotes the Lie derivative on M with respect to X. That is,
( L X F ) Y = [ X , F Y ] F [ X , Y ] .
F is said to be Lie-parallel if L X F = 0 . A lightlike hypersurface M of an indefinite trans-Sasakian manifold M ¯ is said to be Lie-recurrent if its structure tensor field F is Lie-recurrent.
Theorem 3.
Let M be a Lie-recurrent lightlike hypersurface of an indefinite trans-Sasakian manifold M ¯ with a semi-symmetric metric connection. Then, the following statements are satisfied:
(1) 
F is Lie-parallel,
(2) 
α = 0 and M ¯ is an indefinite β-Kenmotsu manifold,
(3) 
τ = β θ on T M , and
(4) 
A ξ * U = 0 and A ξ * V = 0 .
Proof. 
(1) From (11) and θ ( F Y ) = 0 , it is obtained that
ϑ ( X ) F Y = ( X F ) Y F Y X + F Y X + θ ( Y ) F X .
(26) implies that
ϑ ( X ) F Y = F Y X + F Y X + u ( Y ) A N X B ( X , Y ) U + α { g ( X , Y ) ζ θ ( Y ) X } + ( β + 1 ) g ¯ ( J X , Y ) ζ β θ ( Y ) F X .
Taking Y = ξ in (36) with (12), we have
ϑ ( X ) V = V X + F ξ X + ( β + 1 ) u ( X ) ζ .
Taking the scalar product with both V and ζ in (37) by turns, we get
u ( V X ) = 0 , θ ( V X ) = ( β + 1 ) u ( X ) .
Replacing Y by V in (36) and using θ ( V ) = 0 , we have
ϑ ( X ) ξ = ξ X + F V X B ( X , V ) U + α u ( X ) ζ .
Applying F to the above equation with (19) and (38), it is obtained that
ϑ ( X ) V = V X + F ξ X + ( β + 1 ) u ( X ) ζ .
Comparing the above equation with (37), we get ϑ = 0 . Therefore, F is Lie-parallel.
(2) Replacing X by U in (36) and using (14), (17), (19), (22)–(24), and F U = 0 and F ζ = 0 , it is obtained that
u ( Y ) A N U F ( A N F Y ) A N Y τ ( F Y ) U + { α v ( Y ) + ( β + 1 ) η ( Y ) } ζ α θ ( Y ) U = 0 .
Taking the scalar product with ζ into (39) and using (22), it is obtained that α v ( Y ) = 0 , and hence, α = 0 . That is, M ¯ is an indefinite β -Kenmotsu manifold.
(3) Taking the scalar product with N to (36) and using (14) 2 , we have
g ¯ ( F Y X , N ) + g ¯ ( Y X , U ) = β θ ( Y ) v ( X ) ,
because α = 0 . Replacing X by ξ in (40) and using (9) and (13), we get
B ( X , U ) = τ ( F X ) .
Taking X = U to (41) and using (23) and F U = 0 , we have
C ( U , V ) = B ( U , U ) = 0 .
Taking the scalar product with V in (39) and using (14), (23), (42), and α = 0 , it is obtained that
B ( X , U ) = τ ( F X ) .
Comparing the above equation with (41), it is obtained that τ ( F X ) = 0 .
Replacing X by V in (40) and using (25), we have
B ( F Y , U ) + β θ ( Y ) = τ ( Y ) .
Taking Y = U and Y = ζ and using F U = F ζ = 0 , it is obtained that
τ ( U ) = 0 , τ ( ζ ) = β .
Replacing X by F Y to τ ( F X ) = 0 and using (19) and (43), it is obtained that τ ( X ) = β θ ( X ) . Thus, we have (3).
(4) As τ ( F X ) = 0 , from (13) and (41), we have g ( A ξ * U , X ) = 0 . The non-degeneracy of S ( T M ) implies A ξ * U = 0 . Replacing X by ξ to (37) and using (15) and τ ( F X ) = 0 , it is obtained that A ξ * V = 0 . ☐
Definition 4.
The structure vector field U is said to be principal [13] (with respect to the shape operator A ξ * ) if there exists a smooth function κ such that
A ξ * U = κ U .
A lightlike hypersurface M of an indefinite almost contact manifold is called a Hopf lightlike hypersurface if its structure vector field U is principal.
Taking the scalar product with X in (44) and using (13), we get
B ( X , U ) = κ v ( X ) , C ( X , V ) = κ v ( X ) .
Theorem 4.
Let M be a Hopf-lightlike hypersurface of an indefinite trans-Sasakian manifold with a semi-symmetric metric connection. Then, α = 0 .
Proof. 
Replacing X by ζ in (45) 1 and using (21), we get α = 0 . ☐

5. Indefinite Generalized Sasakian Space Forms

For the curvature tensors R ¯ , R , and R * of the semi-symmetric metric connection ¯ on M ¯ , and the induced linear connections ∇ and * on M and S ( T M ) , respectively, two Gauss equations for M and S ( T M ) follow as
R ¯ ( X , Y ) Z = R ( X , Y ) Z + B ( X , Z ) A N Y B ( Y , Z ) A N X + { ( X B ) ( Y , Z ) ( Y B ) ( X , Z ) + τ ( X ) B ( Y , Z ) τ ( Y ) B ( X , Z ) + B ( T ( X , Y ) , Z ) } N ,
R ( X , Y ) P Z = R * ( X , Y ) P Z + C ( X , P Z ) A ξ * Y C ( Y , P Z ) A ξ * X + { ( X C ) ( Y , P Z ) ( Y C ) ( X , P Z ) τ ( X ) C ( Y , P Z ) + τ ( Y ) C ( X , P Z ) + C ( T ( X , Y ) , P Z ) } ξ ,
respectively.
Definition 5.
An indefinite generalized Sasakian space form M ¯ ( f 1 , f 2 , f 3 ) [15] is an indefinite trans-Sasakian manifold ( M ¯ , J , ζ , θ , g ¯ ) with
R ˜ ( X , Y ) Z = f 1 { g ¯ ( Y ¯ , Z ¯ ) X ¯ g ¯ ( X ¯ , Z ¯ ) Y ¯ } + f 2 { g ¯ ( X ¯ , J Z ¯ ) J Y ¯ g ¯ ( Y ¯ , J Z ¯ ) J X ¯ + 2 g ¯ ( X ¯ , J Y ¯ ) J Z ¯ } + f 3 { θ ( X ¯ ) θ ( Z ¯ ) Y ¯ θ ( Y ¯ ) θ ( Z ¯ ) X ¯ + g ¯ ( X ¯ , Z ¯ ) θ ( Y ¯ ) ζ g ¯ ( Y ¯ , Z ¯ ) θ ( X ¯ ) ζ }
for some three smooth functions f 1 , f 2 and f 3 on M ¯ , where R ˜ denote the curvature tensor of the Levi-Civita connection ˜ on M ¯ .
Note that Sasakian f 1 = c + 3 4 , f 2 = f 3 = c 1 4 , Kenmotsu f 1 = c 3 4 , f 2 = f 3 = c + 1 4 , and cosymplectic f 1 = f 2 = f 3 = c 4 space forms are important kinds of generalized Sasakian space forms, where c is a constant J-sectional curvature of each space form.
By directed calculations from (1) and (2), we see that
R ¯ ( X ¯ , Y ¯ ) Z ¯ = R ˜ ( X ¯ , Y ¯ ) Z ¯ + g ¯ ( X ¯ , Z ¯ ) ¯ Y ¯ ζ g ¯ ( Y ¯ , Z ¯ ) ¯ X ¯ ζ + { ( ¯ X ¯ θ ) ( Z ¯ ) g ¯ ( X ¯ , Z ¯ ) } Y ¯ { ( ¯ Y ¯ θ ) ( Z ¯ ) g ¯ ( Y ¯ , Z ¯ ) } X ¯ .
Taking the scalar product with ξ and N in (49) by turns and substituting (46) and (48) to the resulting equations and using (5) and (47), we get
( X B ) ( Y , Z ) ( Y B ) ( X , Z ) + { τ ( X ) θ ( X ) } B ( Y , Z ) { τ ( Y ) θ ( Y ) } B ( X , Z ) + α { u ( Y ) g ( X , Z ) u ( X ) g ( Y , Z ) } = f 2 { u ( Y ) g ¯ ( X , J Z ) u ( X ) g ¯ ( Y , J Z ) + 2 u ( Z ) g ¯ ( X , J Y ) } ,
( X C ) ( Y , P Z ) ( Y C ) ( X , P Z ) { τ ( X ) + θ ( X ) } C ( Y , P Z ) + { τ ( Y ) + θ ( Y ) } C ( X , P Z ) { ( ¯ X θ ) ( P Z ) + β g ( X , P Z ) } η ( Y ) + { ( ¯ Y θ ) ( P Z ) + β g ( Y , P Z ) } η ( X ) + α { v ( Y ) g ( X , P Z ) v ( X ) g ( Y , P Z ) } = f 1 { g ( Y , P Z ) η ( X ) g ( X , P Z ) η ( Y ) } + f 2 { v ( Y ) g ¯ ( X , J P Z ) v ( X ) g ¯ ( Y , J P Z ) + 2 v ( P Z ) g ¯ ( X , J Y ) } + f 3 { θ ( X ) η ( Y ) θ ( Y ) η ( X ) } θ ( P Z ) .
Theorem 5.
Let M be a lightlike hypersurface of an indefinite generalized Sasakian space form M ¯ ( f 1 , f 2 , f 3 ) with a semi-symmetric metric connection. Then, α , β , f 1 , f 2 , and f 3 satisfy that α is a constant on M, α β = 0 , and
f 1 f 2 = α 2 β 2 , f 1 f 3 = α 2 β 2 ζ β .
Proof. 
From the covariant derivative of θ ( V ) = 0 with respect to X and (6) and (25), it is obtained that
( ¯ X θ ) ( V ) = ( β + 1 ) u ( X ) .
Applying X to (23): B ( Y , U ) = C ( Y , V ) and using (21)–(25), we get
( X B ) ( Y , U ) = ( X C ) ( Y , V ) 2 τ ( X ) C ( Y , V ) α ( β + 1 ) { u ( Y ) v ( X ) u ( X ) v ( Y ) } α 2 u ( Y ) η ( X ) ( β + 1 ) 2 u ( X ) η ( Y ) g ( A ξ * X , F ( A N Y ) ) g ( A ξ * Y , F ( A N X ) ) .
Substituting this equation and (23) into (50) with Z = U , we have
( X C ) ( Y , V ) ( Y C ) ( X , V ) { τ ( X ) + θ ( X ) } C ( Y , V ) + { τ ( Y ) + θ ( Y ) } C ( X , V ) α ( 2 β + 1 ) { u ( Y ) v ( X ) u ( X ) v ( Y ) } { α 2 ( β + 1 ) 2 } { u ( Y ) η ( X ) u ( X ) η ( Y ) } = f 2 { u ( Y ) η ( X ) u ( X ) η ( Y ) + 2 g ¯ ( X , J Y ) } .
Comparing the above equation with (51) such that P Z = V and using (52), it is obtained that
{ f 1 f 2 α 2 + β 2 } { u ( Y ) η ( X ) u ( X ) η ( Y ) } = 2 α β { u ( Y ) v ( X ) u ( X ) v ( Y ) } .
Taking Y = U , X = ξ and Y = U , X = V to the above equation by turns, it is obtained that
f 1 f 2 = α 2 β 2 , α β = 0 .
From the covariant derivative of θ ( ζ ) = 1 with respect to X, (5) implies
( ¯ X θ ) ( ζ ) = 0 .
From the covariant derivative of η ( Y ) = g ¯ ( Y , N ) with respect to X, (7) implies
( X η ) ( Y ) = g ( A N X , Y ) + τ ( X ) η ( Y ) .
Applying Y to (22) and using (20), (22), (28), and (55), we get
( X C ) ( Y , ζ ) = ( X α ) v ( Y ) + ( X β ) η ( Y ) α { v ( Y ) τ ( X ) g ( A N X , F Y ) g ( A N Y , F X ) α θ ( Y ) η ( X ) + θ ( X ) v ( Y ) θ ( Y ) v ( X ) } + ( β + 1 ) { τ ( X ) η ( Y ) g ( A N X , Y ) g ( A N Y , X ) + ( β + 1 ) θ ( X ) η ( Y ) } .
Substituting this and (22) into (51) with P Z = ζ and using (54), we get
( X α ) v ( Y ) + ( Y α ) v ( X ) + ( X β ) η ( Y ) ( Y β ) η ( X ) = ( f 1 f 3 α 2 + β 2 ) { θ ( Y ) η ( X ) θ ( X ) η ( Y ) } .
Taking Y = ζ , X = ξ and Y = U , X = V to this by turns, it is obtained that
f 1 f 3 = α 2 β 2 ζ β , U α = 0 .
Applying Y to (21) and using (20), (21), and (27), we have
( X B ) ( Y , ζ ) = ( X α ) u ( Y ) ( β + 1 ) B ( X , Y ) + α { u ( Y ) τ ( X ) + θ ( Y ) u ( X ) θ ( X ) u ( Y ) + B ( X , F Y ) + B ( Y , F X ) } .
Substituting this equation and (21) into (50) with Z = ζ , it is obtained that
( X α ) u ( Y ) = ( Y α ) u ( X ) .
Taking Y = U , we get X α = 0 . It follows that α is a constant on M. ☐
Definition 6.
(a) A screen distribution S ( T M ) is said to be totally umbilical [10] in M if
C ( X , P Y ) = γ g ( X , Y )
for some smooth function γ on a neighborhood U . In particular, case S ( T M ) is totally geodesic in M if γ = 0 .
(b) A lightlike hypersurface M is said to be screen conformal [11] if
C ( X , P Y ) = φ B ( X , Y )
for some non-vanishing smooth function φ on a neighborhood U .
Theorem 6.
Let M be a lightlike hypersurface of an indefinite generalized Sasakian space form M ¯ ( f 1 , f 2 , f 3 ) with a semi-symmetric metric connection. If one of the following five conditions is satisfied,
(1) 
M is recurrent,
(2) 
S ( T M ) is totally umbilical,
(3) 
M is screen conformal,
(4) 
U = 0 , and
(5) 
V = 0 ,
then M ¯ ( f 1 , f 2 , f 3 ) is an indefinite Kenmotsu space form such that
α = 0 , β = 1 ; f 1 = 1 , f 2 = f 3 = 0 .
Proof. 
Applying ¯ X to θ ( U ) = 0 and using (6) and (24), it is obtained
( ¯ X θ ) ( U ) = α η ( X ) + ( β + 1 ) v ( X ) .
(a) Theorem 2 implies that α = 0 and β = 1 . By directed calculation from (35), it is obtained that
R ( X , Y ) U = 2 d τ ( X , Y ) U .
On the other hand, since α = 0 and β = 1 , we have ¯ X ζ = 0 by (5) and f 1 + 1 = f 2 = f 3 by Theorem 5. Comparing the tangential components of the right and left terms of (49) and using (46) and (48), it is obtained that
R ( X , Y ) Z = B ( Y , Z ) A N X B ( X , Z ) A N Y + ( ¯ X θ ) ( Z ) Y ( ¯ Y θ ) ( Z ) X + ( f 1 + 1 ) { g ( Y , Z ) X g ( X , Z ) Y } + f 2 { g ¯ ( X , J Z ) F Y g ¯ ( Y , J Z ) F X + 2 g ¯ ( X , J Y ) F Z } + f 3 { θ ( X ) θ ( Z ) Y θ ( Y ) θ ( Z ) X + g ¯ ( X , Z ) θ ( Y ) ζ g ¯ ( Y , Z ) θ ( X ) ζ } .
Setting Z = U in the above equation and using (57) and (58), we get
2 d τ ( X , Y ) U = B ( Y , U ) A N X B ( X , U ) A N Y + ( f 1 + 1 ) { v ( Y ) X v ( X ) Y } + f 2 { η ( X ) F Y η ( Y ) F X } + f 3 { v ( X ) θ ( Y ) v ( Y ) θ ( X ) } ζ .
Taking the scalar product with N to the above equation and using (14) 2 , we get
2 f 2 { v ( Y ) u ( X ) v ( X ) u ( Y ) } .
It follows that f 2 = 0 . Thus, f 1 + 1 = f 2 = f 3 = 0 .
(b) Since S ( T M ) is totally umbilical, (22) is reduced to
γ θ ( X ) = α v ( X ) + ( β + 1 ) η ( X ) .
Taking X = ζ , X = V , and X = ξ to this equation by turns, we get γ = 0 , α = 0 , and β = 1 , respectively. As γ = 0 , S ( T M ) is totally geodesic in M. As α = 0 and β = 1 , M ¯ is an indefinite Kenmotsu manifold and f 1 + 1 = f 2 = f 3 by Theorem 5.
Taking P Z = V in (51) and using (52) and the result: C = 0 , we have
f 2 { u ( Y ) η ( X ) u ( X ) η ( Y ) + 2 g ¯ ( X , J Y ) } = 0 .
Taking X = ξ and Y = U , we get f 2 = 0 . Thus, f 1 = 1 and f 2 = f 3 = 0 , and M ¯ ( f 1 , f 2 , f 3 ) is an indefinite Kenmotsu space form with c = 1 .
(c) Taking P Y = ζ in (56) and using (21) and (22), we get
α v ( X ) ( β + 1 ) η ( X ) = α φ u ( X ) .
Taking X = V and X = ξ by turns, we have α = 0 and β = 1 , respectively. Thus, M ¯ is an indefinite Kenmotsu manifold and we get f 1 + 1 = f 2 = f 3 .
Applying X to C ( Y , P Z ) = φ B ( Y , P Z ) , we have
( X C ) ( Y , P Z ) = ( X φ ) B ( Y , P Z ) + φ ( X B ) ( Y , P Z ) .
Substituting this equation into (51) and using (50), we have
{ X φ 2 φ τ ( X ) } B ( Y , P Z ) { Y φ 2 φ τ ( Y ) } B ( X , P Z ) { ( ¯ X θ ) ( P Z ) g ( X , P Z ) } η ( Y ) + { ( ¯ Y θ ) ( P Z ) g ( Y , P Z ) } η ( X ) = f 1 { g ( Y , P Z ) η ( X ) g ( X , P Z ) η ( Y ) } + f 2 { [ v ( Y ) φ u ( Y ) ] g ¯ ( X , J P Z ) [ v ( X ) φ u ( X ) ] g ¯ ( Y , J P Z ) + 2 [ v ( P Z ) φ u ( P Z ) ] g ¯ ( X , J Y ) } + f 3 { θ ( X ) η ( Y ) θ ( Y ) η ( X ) } θ ( P Z ) .
Replacing Y by ξ in the above equation, it is obtained that
{ ξ φ 2 φ τ ( ξ ) } B ( X , P Z ) + ( ¯ X θ ) ( P Z ) g ( X , P Z ) ( ¯ ξ θ ) ( P Z ) η ( X ) = f 1 g ( X , P Z ) + f 2 { v ( X ) φ u ( X ) } u ( P Z ) + 2 f 2 { v ( P Z ) φ u ( P Z ) } u ( X ) f 3 θ ( X ) θ ( P Z ) .
Taking X = V , P Z = U and then X = U , P Z = V to the above equation by turns and using (52), (57), and the fact that f 1 + 1 = f 2 , we have
{ ξ φ 2 φ τ ( ξ ) } B ( V , U ) = 2 f 2 , { ξ φ 2 φ τ ( ξ ) } B ( U , V ) = 3 f 2 ,
respectively. From the last two equations, it is obtained that f 2 = 0 . Therefore, f 1 = 1 and f 2 = f 3 = 0 . Consequently, we see that M ¯ ( f 1 , f 2 , f 3 ) is an indefinite Kenmotsu space form such that c = 1 .
(d) Theorem 1 implies τ = 0 , α = 0 , β = 1 , and (29). Thus, f 1 + 1 = f 2 = f 3 by Theorem 5.
Taking the scalar product with U in (29), it is obtained that
C ( X , U ) = 0 .
Applying X to C ( Y , U ) = 0 and using X U = 0 , we have
( X C ) ( Y , U ) = 0 .
Substituting the last two equations into (51) with P Z = U and using (57) and the fact that f 1 + 1 = f 2 , we have
2 f 2 { v ( Y ) η ( X ) v ( X ) η ( Y ) } = 0 .
Taking X = V and Y = ξ , we get f 2 = 0 . Thus f 1 + 1 = f 2 = f 3 = 0 and M ¯ ( f 1 , f 2 , f 3 ) is an indefinite Kenmotsu space form such that c = 1 .
(e) Theorem 1 implies τ = 0 , α = 0 , β = 1 and (30). Thus f 1 + 1 = f 2 = f 3 by Theorem 5.
From (23) and (30), we get
C ( X , V ) = 0 .
Applying X to C ( Y , V ) = 0 and using the fact that X V = 0 , we have
( X C ) ( Y , V ) = 0 .
Substituting these into (51) with P Z = V and using (52), we get
f 2 { u ( Y ) η ( X ) u ( X ) η ( Y ) + 2 g ¯ ( X , J Y ) } = 0 .
Taking U = U and X = ξ , we have f 2 = 0 . Thus, f 1 + 1 = f 2 = f 3 = 0 and M ¯ ( f 1 , f 2 , f 3 ) is an indefinite Kenmotsu space form with c = 1 . ☐
Theorem 7.
Let M be a lightlike hypersurface of an indefinite generalized Sasakian space form M ¯ ( f 1 , f 2 , f 3 ) with a semi-symmetric non-metric connection. If M is a Lie-recurrent or Hopf lightlike hypersurface, then M ¯ is an indefinite β-Kenmotsu space form with
f 1 = β 2 , f 2 = 0 , f 3 = ζ β .
Proof. 
(a) Theorem 3 implies α = 0 and
B ( X , U ) = 0 .
Applying X to B ( Y , U ) = 0 and using (21) and (24), we have
( X B ) ( Y , U ) = B ( Y , F ( A N X ) ) .
Setting Z = U in the last two equations into (50), we have
B ( X , F ( A N Y ) ) B ( Y , F ( A N X ) ) = f 2 { u ( Y ) η ( X ) u ( X ) η ( Y ) + 2 g ¯ ( X , J Y ) } .
Taking X = ξ and Y = U to the above equation and using (12) and (59), it is obtained that f 2 = 0 .
Therefore, Theorem 5 implies
f 1 = β 2 , f 2 = 0 , f 3 = ζ β .
(b) Applying Y to (45) 1 and using (21), (24), and (28), it is obtained that
( X B ) ( Y , U ) = ( X κ ) v ( Y ) B ( Y , F ( A N X ) ) κ { ( β + 1 ) θ ( Y ) v ( X ) + g ( A N X , F Y ) } ,
because α = 0 . Substituting this equation and (45) 1 into (50), we have
( X κ ) v ( Y ) ( Y κ ) v ( X ) + B ( X , F ( A N Y ) ) B ( Y , F ( A N X ) ) + κ { β [ θ ( X ) v ( Y ) θ ( Y ) v ( X ) ] + τ ( X ) v ( Y ) τ ( Y ) v ( X ) + g ( A N Y , F X ) g ( A N X , F Y ) } = f 2 { u ( Y ) η ( X ) u ( X ) η ( Y ) + 2 g ¯ ( X , J Y ) } .
Taking Y = U and X = ξ to the above equation and using (3), (18), (12), (14) 1 , 2 , and (45) 1 , 2 , we get f 2 = 0 . Thus, by Theorem 5 we have
f 1 = β 2 , f 2 = 0 , f 3 = ζ β .
This completes the proof of the theorem. ☐

6. Conclusions

In the submanifold theory, some properties of a base space (a submanifold) is investigated from the total space. In our case, we characterize that the total space (an indefinite generalized Sasakian space form) with a semi-symmetric metric connection is an indefinite Kenmotsu space form under various lightlike hypersurfaces, such as recurrent, Lie-recurrent, and Hopf lightlike hypersurfaces of an indefinite trans-Sasakian manifold with a semi-symmetric metric connection. The structure of a lightlike hypersurface in a semi-Riemannian manifold is not same as the one of a lightlike submanifold (half lightlike submanifolds, generic lightlike, and several CR-type lightlike, etc.) in a semi-Riemannian manifold. Our paper helps in solving more general cases in semi-Riemannian manifolds with a semi-symmetric metric connection.

Author Contributions

D.H.J. gave the idea to investigate various hypersurfaces of an indefinite tran-Sasakian manifold with a semi-symmetric metric connection. J.W.L. improve D.H.J.’s idea to characterize the total space as well as J.W.L. checked and polished the draft.

Funding

This research received no external funding.

Conflicts of Interest

The authors declare no conflict of interest.

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