Characterizations of the Total Space (Indefinite Trans-Sasakian Manifolds) Admitting a Semi-Symmetric Metric Connection

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.

1. Introduction

A semi-symmetric connection $\overline{\nabla }$ on a semi-Riemannian manifold $(\overline{M},\overline{g})$ was introduced by Friedmann-Schouten [1] in 1924, whose torsion tensor $\overline{T}$ satisfies

$$\overline{T}(\overline{X},\overline{Y})=\theta \left(\overline{Y}\right)\overline{X}-\theta \left(\overline{X}\right)\overline{Y},$$

(1)

where $\theta$ is a 1-form associated with a vector field $\zeta$ by $\theta \left(\overline{X}\right)=\overline{g}(\overline{X},\zeta )$. In particular, if it is a metric connection (i.e., $\overline{\nabla }\overline{g}=0$), then $\overline{\nabla }$ 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 $\overline{X}$, $\overline{Y}$, and $\overline{Z}$ the smooth vector fields on $\overline{M}$.

Remark 1.

Let $\tilde{\nabla }$ be the Levi-Civita connection of the semi-Riemannian manifold $(\overline{M},\overline{g})$ with respect to the metric $\overline{g}$. A linear connection $\overline{\nabla }$ on $\overline{M}$ is a semi-symmetric metric connection if and only if

$${\overline{\nabla }}_{\overline{X}}\overline{Y}={\tilde{\nabla }}_{\overline{X}}\overline{Y}+\theta \left(\overline{Y}\right)\overline{X}-\overline{g}(\overline{X},\overline{Y})\zeta .$$

(2)

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 $\left(ϵ\right)$-Kenmotsu manifolds. Shukla and Singh [7] extended the study to $\left(ϵ\right)$-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 $(\overline{M},J,\zeta ,\theta ,\overline{g})$ with a semi-symmetric metric connection $\overline{\nabla }$. 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 $(\overline{M},\overline{g})$ is called an indefinite almost contact metric manifold if there exists an indefinite almost contact metric structure $\{J,\zeta ,\theta ,\overline{g}\}$ with a $(1,1)$-type tensor field J, a vector field $\zeta$, and a 1-form $\theta$ such that

$$J^2\overline{X}=-\overline{X}+\theta \left(\overline{X}\right)\zeta ,\overline{g}(J\overline{X},J\overline{Y})=\overline{g}(\overline{X},\overline{Y})-ϵ\theta \left(\overline{X}\right)\theta \left(\overline{Y}\right),\theta \left(\zeta \right)=ϵ,$$

(3)

where $ϵ=1$ or $-1$ if $\zeta$ is spacelike or timelike, respectively.

From (3), we derive

$$J\zeta =0,\theta \circ J=0,\theta \left(\overline{X}\right)=ϵ\overline{g}(\overline{X},\zeta ),\overline{g}(J\overline{X},\overline{Y})=-\overline{g}(\overline{X},J\overline{Y}).$$

Without loss of generality, we assume that the structure vector field $\zeta$ is spacelike (i.e., $ϵ=1$) in the entire discussion of this article.

Definition 1.

An indefinite almost contact metric manifold $\left(\overline{M},J,\zeta ,\theta ,\overline{g}\right)$ is called an indefinite trans-Sasakian manifold [9] if, for the Levi-Civita connection $\tilde{\nabla }$ with respect to $\overline{g}$, there exist two smooth functions α and β such that

$$\left({\tilde{\nabla }}_{\overline{X}}J\right)\overline{Y}=\alpha \{\overline{g}(\overline{X},\overline{Y})\zeta -\theta \left(\overline{Y}\right)\overline{X}\}+\beta \{\overline{g}(J\overline{X},\overline{Y})\zeta -\theta \left(\overline{Y}\right)J\overline{X}\}.$$

Here, $\{J,\zeta ,\theta ,\overline{g}\}$ is called an indefinite trans-Sasakian structure of type $(\alpha ,\beta )$.

Note that Sasakian$\left(\alpha =1,\beta =0\right)$, Kenmotsu$\left(\alpha =0,\beta =ϵ\right)$ and cosymplectic$\left(\alpha =\beta =0\right)$ manifolds are important kinds of trans-Sasakian manifolds.

Let $\overline{\nabla }$ be a semi-symmetric metric connection on an indefinite trans-Sasakian manifold $\overline{M}=(\overline{M},J,\zeta ,\theta ,\overline{g})$. By using (2), (3) and the fact that $J\zeta =0$ and $\theta \circ J=0$, we see that

$$\begin{array}{c}\hfill \left({\overline{\nabla }}_{\overline{X}}J\right)\overline{Y}=\alpha \{\overline{g}(\overline{X},\overline{Y})\zeta -\theta \left(\overline{Y}\right)\overline{X}\}+(\beta +1)\{\overline{g}(J\overline{X},\overline{Y})\zeta -\theta \left(\overline{Y}\right)J\overline{X}\}.\end{array}$$

(4)

Setting $\overline{Y}=\zeta$ in (4), $J\zeta =0$, and $\theta \left({\overline{\nabla }}_{\overline{X}}\zeta \right)=0$ imply that

$${\overline{\nabla }}_{\overline{X}}\zeta =-\alpha J\overline{X}+(\beta +1)\{\overline{X}-\theta \left(\overline{X}\right)\zeta \}.$$

(5)

From the covariant derivative of $\theta \left(\overline{Y}\right)=\overline{g}(\overline{Y},\zeta )$ in terms of $\overline{X}$ with (1), (3), and (5), we have

$$d\theta (\overline{X},\overline{Y})=\alpha \overline{g}(\overline{X},J\overline{Y}).$$

Let $(M,g)$ be a hypersurface of $\overline{M}$. Denote by $TM$ and $T{M}^{\perp }$ the tangent and normal bundles of M, respectively. Then, there exists a screen distribution $S\left(TM\right)$ on M [10] such that

$$TM=T{M}^{\perp }{\oplus }_{orth}S\left(TM\right),$$

where ${\oplus }_{orth}$ denotes the orthogonal direct sum. Throughout this article, we assume that $F\left(M\right)$ is the algebra of smooth functions on M and $\Gamma \left(E\right)$ is the $F\left(M\right)$-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 $\xi \in \Gamma \left(T{M}^{\perp }{|}_{\mathcal{U}}\right)$ on a coordinate neighborhood $\mathcal{U}\subset M$, there exists a unique null transversal vector field N of a unique transversal vector bundle $tr\left(TM\right)$ in $S{\left(TM\right)}^{\perp }$ [10] satisfying

$$\overline{g}(\xi ,N)=1,\overline{g}(N,N)=\overline{g}(N,X)=0,\forall X\in \Gamma \left(S\left(TM\right)\right).$$

Then, we have the decomposition of the tangent bundle $T\overline{M}$ of $\overline{M}$ as follows:

$$T\overline{M}=TM\oplus tr\left(TM\right)=\{T{M}^{\perp }\oplus tr\left(TM\right)\}{\oplus }_{orth}S\left(TM\right).$$

Let $P:TM\to S\left(TM\right)$ be the projection morphism. Then, we have the local Gauss–Weingarten formulas of M and $S\left(TM\right)$ as follows:

$$\begin{array}{c}{\overline{\nabla }}_{X}Y={\nabla }_{X}Y+B(X,Y)N,\hfill \end{array}$$

(6)

$$\begin{array}{c}{\overline{\nabla }}_{X}N=-A_{{}_N}X+\tau \left(X\right)N,\hfill \end{array}$$

(7)

$$\begin{array}{c}{\nabla }_{X}PY={\nabla }_X^{*}PY+C(X,PY)\xi ,\hfill \end{array}$$

(8)

$$\begin{array}{c}{\nabla }_X\xi =-A_{\xi }^{*}X-\tau \left(X\right)\xi ,\hfill \end{array}$$

(9)

respectively, where $\nabla \left({\nabla }^{*}\right)$ is the induced linear connection on $TM\left(S\left(TM\right),\mathrm{resp}.\right)$, $B\left(C\right)$ is the local second fundamental form on $TM\left(S\left(TM\right),\mathrm{resp}.\right)$, $A_{{}_N}\left(A_{\xi }^{*}\right)$ is the shape operator on $TM\left(S\left(TM\right),\mathrm{resp}.\right)$, and $\tau$ is a 1-form on $TM$. Then, it is well known that ∇ is a semi-symmetric non-metric connection and

$$\begin{array}{c}\left({\nabla }_{X}g\right)(Y,Z)=B(X,Y)\eta \left(Z\right)+B(X,Z)\eta \left(Y\right),\hfill \end{array}$$

(10)

$$\begin{array}{c}T(X,Y)=\theta \left(Y\right)X-\theta \left(X\right)Y.\hfill \end{array}$$

(11)

B is symmetric on $TM$, where T is the torsion tensor with respect to the induced connection ∇ on M and $\eta (•)=\overline{g}(•,N)$ is a 1-form on $TM$.

$B(X,Y)=\overline{g}({\overline{\nabla }}_{X}Y,\xi )$ implies that B is independent of the choice of the screen distribution $S\left(TM\right)$, and we have

$$B(X,\xi )=0.$$

(12)

Moreover, two local second fundamental forms B and C for $TM$ and $S\left(TM\right)$ give the relations with their shape operators, respectively, as follows:

$$\begin{array}{c}B(X,Y)=g(A_{\xi }^{*}X,Y),\overline{g}(A_{\xi }^{*}X,N)=0,\hfill \end{array}$$

(13)

$$\begin{array}{c}C(X,PY)=g(A_{{}_N}X,PY),\overline{g}(A_{{}_N}X,N)=0.\hfill \end{array}$$

(14)

From (13), $A_{\xi }^{*}$ is a $S\left(TM\right)$-valued real self-adjoint operator and satisfies

$$A_{\xi }^{*}\xi =0.$$

(15)

3. Semi-Symmetric Metric Connections

Let M be a lightlike hypersurface of an indefinite almost contact metric manifold $\overline{M}$, and denote by $J\left(T{M}^{\perp }\right)$ and $J\left(tr\right(TM\left)\right)$ sub-bundles of $S\left(TM\right)$, of rank 1 [11], respectively. Now we assume that the structure vector field $\zeta$ is tangent to M. Cǎlin [12] proved that if $\zeta \in \Gamma \left(TM\right)$, then$\zeta \in \Gamma \left(S\left(TM\right)\right)$. Then, there exist two non-degenerate almost complex distributions $D_o\left(\mathrm{i}.\mathrm{e}.,J\left(D_o\right)=D_o\right)$ and $D\left(\mathrm{i}.\mathrm{e}.,J\left(D\right)=D\right)$ with respect to J such that

$$\begin{array}{c}S\left(TM\right)=J\left(T{M}^{\perp }\right)\oplus J\left(tr\left(TM\right)\right){\oplus }_{orth}{D}_o,\hfill \\ D=T{M}^{\perp }{\oplus }_{orth}J\left(T{M}^{\perp }\right){\oplus }_{orth}{D}_o.\hfill \end{array}$$

From these two distributions, we have a decomposition of $TM$ as follows:

$$TM=D\oplus J\left(tr\right(TM\left)\right).$$

(16)

Consider two null vector fields U and V and their 1-forms u and v such that

$$U=-JN,V=-J\xi ,u\left(X\right)=g(X,V),v\left(X\right)=g(X,U).$$

(17)

Denote by $S:TM\to D$ the projection morphism of $TM$ on D. $X\in \Gamma \left(TM\right)$ is expressed as $X=SX+u\left(X\right)U$. Then, it is obtained

$$JX=FX+u\left(X\right)N,$$

(18)

where F is the structure tensor field of type (1, 1) globally defined on M by $FX=JSX$.

Applying J to (18) with (17) and (18), we have

$$F^{2}X=-X+u\left(X\right)U+\theta \left(X\right)\zeta .$$

(19)

Here, the vector field U is called the structure vector field of M.

Replacing Y by $\zeta$ in (6) with (5) and (18), one gets

$$\begin{array}{c}{\nabla }_X\zeta =-\alpha FX+(\beta +1)\{X-\theta \left(X\right)\zeta \},\hfill \end{array}$$

(20)

$$\begin{array}{c}B(X,\zeta )=-\alpha u\left(X\right).\hfill \end{array}$$

(21)

From the covariant derivative of $\overline{g}(\zeta ,N)=0$ in terms of X with (5), (7), and (14), it is obtained that

$$C(X,\zeta )=-\alpha v\left(X\right)+(\beta +1)\eta \left(X\right).$$

(22)

Applying ${\overline{\nabla }}_X$ to (17) and (18) and using (4), (6), and (7), we get

$$\begin{array}{c}B(X,U)=C(X,V),\hfill \end{array}$$

(23)

$$\begin{array}{c}{\nabla }_{X}U=F\left(A_{{}_N}X\right)+\tau \left(X\right)U-\{\alpha \eta \left(X\right)+(\beta +1)v\left(X\right)\}\zeta ,\hfill \end{array}$$

(24)

$$\begin{array}{c}{\nabla }_{X}V=F\left(A_{\xi }^{*}X\right)-\tau \left(X\right)V-(\beta +1)u\left(X\right)\zeta ,\hfill \end{array}$$

(25)

$$\begin{array}{c}\left({\nabla }_{X}F\right)\left(Y\right)=u\left(Y\right)A_{{}_N}X-B(X,Y)U+\alpha \{g(X,Y)\zeta -\theta \left(Y\right)X\}\hfill \\ +(\beta +1)\{\overline{g}(JX,Y)\zeta -\theta \left(Y\right)FX\},\hfill \end{array}$$

(26)

$$\begin{array}{c}\left({\nabla }_{X}u\right)\left(Y\right)=-u\left(Y\right)\tau \left(X\right)-B(X,FY)-(\beta +1)\theta \left(Y\right)u\left(X\right),\hfill \end{array}$$

(27)

$$\begin{array}{c}\left({\nabla }_{X}v\right)\left(Y\right)=v\left(Y\right)\tau \left(X\right)-g(A_{{}_N}X,FY)\hfill \\ -\left\{\alpha \eta \right(X)+(\beta +1\left)v\right(X\left)\right\}\theta \left(Y\right).\hfill \end{array}$$

(28)

Theorem 1.

Let M be a lightlike hypersurface of an indefinite trans-Sasakian manifold $\overline{M}$ with a semi-symmetric metric connection. If either $\nabla U=0$ or $\nabla V=0$, then $\tau =0$ and $\overline{M}$ is an indefinite Kenmotsu manifold. That is, $\alpha =0$ and $\beta =-1$.

Proof. 

(1) If $\nabla U=0$, then, taking the scalar product with $\zeta$ and V to (24) by turns, it is obtained

$$\alpha =0,\beta =-1,\tau =0.$$

As $\alpha =0$ and $\beta =-1$, $\overline{M}$ is an indefinite Kenmotsu manifold. Applying F to (24): $F\left(A_{{}_N}X\right)=0$ and using (19) and (22), it is obtained that

$$A_{{}_N}X=u\left(A_{{}_N}X\right)U.$$

(29)

(2) If $\nabla V=0$, then, taking the scalar product with $\zeta$ and U to (25) by turns, we have $\beta =-1$ and $\tau =0$. Applying F to (25): $F\left(A_{\xi }^{*}X\right)=0$ and using (19) and (21), one gets

$$A_{\xi }^{*}X=-\alpha u\left(X\right)\zeta +u\left(A_{\xi }^{*}X\right)U.$$

Taking the scalar product with U to the above equation, we have

$$B(X,U)=0.$$

(30)

Replacing X by $\zeta$ in (30) and using (21), we have $\alpha =0$. Hence, $\overline{M}$ 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

$$\left({\nabla }_{X}F\right)Y=\varpi \left(X\right)FY.$$

A lightlike hypersurface M of an indefinite trans-Sasakian manifold $\overline{M}$ 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 $\overline{M}$ with a semi-symmetric metric connection. Then

(1) 

$\alpha =0$ and $\beta =-1$ (i.e., $\overline{M}$ is an indefinite Kenmotsu manifold),

(2) 

F is parallel in terms of the induced connection ∇ on M,

(3) 

D and $J\left(tr\right(TM\left)\right)$ are parallel distributions on M, and

(4) 

M is locally a product manifold ${\mathcal{C}}_{{}_U}\times M^{♯}$, where ${\mathcal{C}}_{{}_U}$ is a null curve tangent to $J\left(tr\right(TM\left)\right)$ and $M^{♯}$ is a leaf of the distribution D.

Proof. 

(1) From (26), we have

$$\begin{array}{c}\varpi \left(X\right)FY=u\left(Y\right)A_{{}_N}X-B(X,Y)U+\alpha \{g(X,Y)\zeta -\theta \left(Y\right)X\}\hfill \\ +(\beta +1)\{\overline{g}(JX,Y)\zeta -\theta \left(Y\right)FX\}.\hfill \end{array}$$

(31)

Setting $Y=\zeta$ in (31) with (3) and (21), it is obtained that

$$\alpha \{-X+u(X)U+\theta (X\left)\zeta \right\}-(\beta +1)FX=0.$$

Taking $X=\xi$ to this equation and using the fact that $F\xi =-V$, we have

$$-\alpha \xi +(\beta +1)V=0.$$

Taking the scalar product with N and U to the above equation by turns, we get

$$\alpha =0,\beta =-1.$$

(32)

Therefore, $\overline{M}$ is an indefinite Kenmotsu manifold.

(2) Taking Y by $\xi$ to (31) and using (12), we get $\varpi \left(X\right)V=0$. It follows that $\varpi =0$. Thus, F is parallel with respect to the connection ∇.

(3) Taking the scalar product with V to (31), it is obtained that

$$B(X,Y)=u\left(Y\right)u\left(A_{{}_N}X\right).$$

Setting $Y=V$ and $Y=F{Z}_o$,$Z_o\in \Gamma \left(D_o\right)$ to the above equation by turns with the fact that $u\left(F{Z}_o\right)=0$ as $F{Z}_o=J{Z}_o\in \Gamma \left(D_o\right)$, we have

$$B(X,V)=0,B(X,F{Z}_o)=0.$$

(33)

Generally, from (6), (9), (13), and (25), we derive

$$\begin{array}{c}g({\nabla }_X\xi ,V)=-B(X,V),g({\nabla }_{X}V,V)=0,\hfill \\ g({\nabla }_X{Z}_o,V)=B(X,F{Z}_o),\forall Z_o\in \Gamma \left(D_o\right).\hfill \end{array}$$

From these equations and (33), we see that

$${\nabla }_{X}Y\in \Gamma \left(D\right),\forall X\in \Gamma \left(TM\right),\forall Y\in \Gamma \left(D\right),$$

and hence D is a parallel distribution on M.

On the other hand, setting $Y=U$ in (31) with (32), we have

$$A_{{}_N}X=B(X,U)U.$$

(34)

Using $FU=0$ in (34), it is obtained that

$$F\left(A_{{}_N}X\right)=0.$$

Using this result and (32), Equation (24) is reduced to

$${\nabla }_{X}U=\tau \left(X\right)U.$$

(35)

It follows that

$${\nabla }_{X}U\in \Gamma \left(J\left(tr\left(TM\right)\right)\right),\forall X\in \Gamma \left(TM\right),$$

and hence $J\left(tr\right(TM\left)\right)$ is parallel on M.

(4) From (16), D and $J\left(tr\right(TM\left)\right)$ are parallel. By the decomposition theorem [14], M is locally a product manifold ${\mathcal{C}}_{{}_U}\times M^{♯}$, where ${\mathcal{C}}_{{}_U}$ is a null curve tangent to $J\left(tr\right(TM\left)\right)$ and $M^{♯}$ is a leaf of D. ☐

Definition 3.

The structure tensor field F of M is said to be Lie-recurrent [13] if

$$\left({\mathcal{L}}_{{}_X}F\right)Y=\vartheta \left(X\right)FY,$$

for some 1-form ϑ on M, where ${\mathcal{L}}_{{}_X}$ denotes the Lie derivative on M with respect to X. That is,

$$\left({\mathcal{L}}_{{}_X}F\right)Y=[X,FY]-F[X,Y].$$

F is said to be Lie-parallel if ${\mathcal{L}}_{{}_X}F=0$. A lightlike hypersurface M of an indefinite trans-Sasakian manifold $\overline{M}$ 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 $\overline{M}$ with a semi-symmetric metric connection. Then, the following statements are satisfied:

(1) 

F is Lie-parallel,

(2) 

$\alpha =0$ and $\overline{M}$ is an indefinite β-Kenmotsu manifold,

(3) 

$\tau =-\beta \theta$ on $TM$, and

(4) 

$A_{\xi }^{*}U=0$ and $A_{\xi }^{*}V=0$.

Proof. 

(1) From (11) and $\theta \left(FY\right)=0$, it is obtained that

$$\vartheta \left(X\right)FY=\left({\nabla }_{X}F\right)Y-{\nabla }_{FY}X+F{\nabla }_{Y}X+\theta \left(Y\right)FX.$$

(26) implies that

$$\begin{array}{c}\vartheta \left(X\right)FY=-{\nabla }_{FY}X+F{\nabla }_{Y}X+u\left(Y\right)A_{{}_N}X-B(X,Y)U\hfill \\ +\alpha \{g(X,Y)\zeta -\theta \left(Y\right)X\}+(\beta +1)\overline{g}(JX,Y)\zeta -\beta \theta \left(Y\right)FX.\hfill \end{array}$$

(36)

Taking $Y=\xi$ in (36) with (12), we have

$$-\vartheta \left(X\right)V={\nabla }_{V}X+F{\nabla }_{\xi }X+(\beta +1)u\left(X\right)\zeta .$$

(37)

Taking the scalar product with both V and $\zeta$ in (37) by turns, we get

$$u\left({\nabla }_{V}X\right)=0,\theta \left({\nabla }_{V}X\right)=-(\beta +1)u\left(X\right).$$

(38)

Replacing Y by V in (36) and using $\theta \left(V\right)=0$, we have

$$\vartheta \left(X\right)\xi =-{\nabla }_{\xi }X+F{\nabla }_{V}X-B(X,V)U+\alpha u\left(X\right)\zeta .$$

Applying F to the above equation with (19) and (38), it is obtained that

$$\vartheta \left(X\right)V={\nabla }_{V}X+F{\nabla }_{\xi }X+(\beta +1)u\left(X\right)\zeta .$$

Comparing the above equation with (37), we get $\vartheta =0$. Therefore, F is Lie-parallel.

(2) Replacing X by U in (36) and using (14), (17), (19), (22)–(24), and $FU=0$ and $F\zeta =0$, it is obtained that

$$\begin{array}{c}u\left(Y\right)A_{{}_N}U-F\left(A_{{}_N}FY\right)-A_{{}_N}Y-\tau \left(FY\right)U\hfill \\ +\left\{\alpha v\right(Y)+(\beta +1\left)\eta \right(Y\left)\right\}\zeta -\alpha \theta \left(Y\right)U=0.\hfill \end{array}$$

(39)

Taking the scalar product with $\zeta$ into (39) and using (22), it is obtained that $\alpha v\left(Y\right)=0$, and hence, $\alpha =0$. That is, $\overline{M}$ is an indefinite $\beta$-Kenmotsu manifold.

(3) Taking the scalar product with N to (36) and using (14)${}_2$, we have

$$-\overline{g}({\nabla }_{FY}X,N)+\overline{g}({\nabla }_{Y}X,U)=\beta \theta \left(Y\right)v\left(X\right),$$

(40)

because $\alpha =0$. Replacing X by $\xi$ in (40) and using (9) and (13), we get

$$B(X,U)=\tau \left(FX\right).$$

(41)

Taking $X=U$ to (41) and using (23) and $FU=0$, we have

$$C(U,V)=B(U,U)=0.$$

(42)

Taking the scalar product with V in (39) and using (14), (23), (42), and $\alpha =0$, it is obtained that

$$B(X,U)=-\tau \left(FX\right).$$

Comparing the above equation with (41), it is obtained that $\tau \left(FX\right)=0$.

Replacing X by V in (40) and using (25), we have

$$B(FY,U)+\beta \theta \left(Y\right)=-\tau \left(Y\right).$$

Taking $Y=U$ and $Y=\zeta$ and using $FU=F\zeta =0$, it is obtained that

$$\tau \left(U\right)=0,\tau \left(\zeta \right)=-\beta .$$

(43)

Replacing X by $FY$ to $\tau \left(FX\right)=0$ and using (19) and (43), it is obtained that $\tau \left(X\right)=-\beta \theta \left(X\right)$. Thus, we have (3).

(4) As $\tau \left(FX\right)=0$, from (13) and (41), we have $g(A_{\xi }^{*}U,X)=0$. The non-degeneracy of $S\left(TM\right)$ implies $A_{\xi }^{*}U=0$. Replacing X by $\xi$ to (37) and using (15) and $\tau \left(FX\right)=0$, it is obtained that $A_{\xi }^{*}V=0$. ☐

Definition 4.

The structure vector field U is said to be principal [13] (with respect to the shape operator $A_{\xi }^{*}$) if there exists a smooth function κ such that

$$A_{\xi }^{*}U=\kappa U.$$

(44)

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

$$B(X,U)=\kappa v\left(X\right),C(X,V)=\kappa v\left(X\right).$$

(45)

Theorem 4.

Let M be a Hopf-lightlike hypersurface of an indefinite trans-Sasakian manifold with a semi-symmetric metric connection. Then, $\alpha =0$.

Proof. 

Replacing X by $\zeta$ in (45)${}_1$ and using (21), we get $\alpha =0$. ☐

5. Indefinite Generalized Sasakian Space Forms

For the curvature tensors $\overline{R},R$, and $R^{*}$ of the semi-symmetric metric connection $\overline{\nabla }$ on $\overline{M}$, and the induced linear connections ∇ and ${\nabla }^{*}$ on M and $S\left(TM\right)$, respectively, two Gauss equations for M and $S\left(TM\right)$ follow as

$$\begin{array}{c}\overline{R}(X,Y)Z=R(X,Y)Z+B(X,Z)A_{{}_N}Y-B(Y,Z)A_{{}_N}X\hfill \\ +\{\left({\nabla }_{X}B\right)(Y,Z)-\left({\nabla }_{Y}B\right)(X,Z)+\tau \left(X\right)B(Y,Z)\hfill \\ -\tau \left(Y\right)B(X,Z)+B\left(T\right(X,Y),Z)\}N,\hfill \end{array}$$

(46)

$$\begin{array}{c}R(X,Y)PZ=R^{*}(X,Y)PZ+C(X,PZ)A_{\xi }^{*}Y-C(Y,PZ)A_{\xi }^{*}X\hfill \\ +\{\left({\nabla }_{X}C\right)(Y,PZ)-\left({\nabla }_{Y}C\right)(X,PZ)-\tau \left(X\right)C(Y,PZ)\hfill \\ +\tau \left(Y\right)C(X,PZ)+C\left(T\right(X,Y),PZ)\}\xi ,\hfill \end{array}$$

(47)

respectively.

Definition 5.

An indefinite generalized Sasakian space form $\overline{M}(f_1,f_2,f_3)$ [15] is an indefinite trans-Sasakian manifold $(\overline{M},J,\zeta ,\theta ,\overline{g})$ with

$$\begin{array}{c}\tilde{R}(X,Y)Z=f_1\{\overline{g}(\overline{Y},\overline{Z})\overline{X}-\overline{g}(\overline{X},\overline{Z})\overline{Y}\}\\ +f_2\{\overline{g}(\overline{X},J\overline{Z})J\overline{Y}-\overline{g}(\overline{Y},J\overline{Z})J\overline{X}+2\overline{g}(\overline{X},J\overline{Y})J\overline{Z}\}\\ +f_3\{\theta \left(\overline{X}\right)\theta \left(\overline{Z}\right)\overline{Y}-\theta \left(\overline{Y}\right)\theta \left(\overline{Z}\right)\overline{X}\\ +\overline{g}(\overline{X},\overline{Z})\theta \left(\overline{Y}\right)\zeta -\overline{g}(\overline{Y},\overline{Z})\theta \left(\overline{X}\right)\zeta \}\end{array}$$

(48)

for some three smooth functions $f_1,f_2$ and $f_3$ on $\overline{M}$, where $\tilde{R}$ denote the curvature tensor of the Levi-Civita connection $\tilde{\nabla }$ on $\overline{M}$.

Note that Sasakian$\left(f_1=\frac{c+3}{4},f_2=f_3=\frac{c-1}{4}\right)$, Kenmotsu$\left(f_1=\frac{c-3}{4},f_2=f_3=\frac{c+1}{4}\right)$, and cosymplectic$\left(f_1=f_2=f_3=\frac{c}{4}\right)$ 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

$$\begin{array}{c}\overline{R}(\overline{X},\overline{Y})\overline{Z}=\tilde{R}(\overline{X},\overline{Y})\overline{Z}+\overline{g}(\overline{X},\overline{Z}){\overline{\nabla }}_{\overline{Y}}\zeta -\overline{g}(\overline{Y},\overline{Z}){\overline{\nabla }}_{\overline{X}}\zeta \\ +\{\left({\overline{\nabla }}_{\overline{X}}\theta \right)\left(\overline{Z}\right)-\overline{g}(\overline{X},\overline{Z})\}\overline{Y}-\{\left({\overline{\nabla }}_{\overline{Y}}\theta \right)\left(\overline{Z}\right)-\overline{g}(\overline{Y},\overline{Z})\}\overline{X}.\end{array}$$

(49)

Taking the scalar product with $\xi$ and N in (49) by turns and substituting (46) and (48) to the resulting equations and using (5) and (47), we get

$$\begin{array}{c}\left({\nabla }_{X}B\right)(Y,Z)-\left({\nabla }_{Y}B\right)(X,Z)\hfill \\ +\left\{\tau \right(X)-\theta (X\left)\right\}B(Y,Z)-\left\{\tau \right(Y)-\theta (Y\left)\right\}B(X,Z)\hfill \\ +\alpha \left\{u\right(Y\left)g\right(X,Z)-u(X\left)g\right(Y,Z\left)\right\}\hfill \\ =f_2\{u\left(Y\right)\overline{g}(X,JZ)-u\left(X\right)\overline{g}(Y,JZ)+2u\left(Z\right)\overline{g}(X,JY)\},\hfill \end{array}$$

(50)

$$\begin{array}{c}\left({\nabla }_{X}C\right)(Y,PZ)-\left({\nabla }_{Y}C\right)(X,PZ)\hfill \\ -\left\{\tau \right(X)+\theta (X\left)\right\}C(Y,PZ)+\left\{\tau \right(Y)+\theta (Y\left)\right\}C(X,PZ)\hfill \\ -\{\left({\overline{\nabla }}_X\theta \right)\left(PZ\right)+\beta g(X,PZ)\}\eta \left(Y\right)\hfill \\ +\{\left({\overline{\nabla }}_Y\theta \right)\left(PZ\right)+\beta g(Y,PZ)\}\eta \left(X\right)\hfill \\ +\alpha \left\{v\right(Y\left)g\right(X,PZ)-v(X\left)g\right(Y,PZ\left)\right\}\hfill \\ =f_1\{g(Y,PZ)\eta \left(X\right)-g(X,PZ)\eta \left(Y\right)\}\hfill \\ +f_2\{v\left(Y\right)\overline{g}(X,JPZ)-v\left(X\right)\overline{g}(Y,JPZ)+2v\left(PZ\right)\overline{g}(X,JY)\}\hfill \\ +f_3\{\theta \left(X\right)\eta \left(Y\right)-\theta \left(Y\right)\eta \left(X\right)\}\theta \left(PZ\right).\hfill \end{array}$$

(51)

Theorem 5.

Let M be a lightlike hypersurface of an indefinite generalized Sasakian space form $\overline{M}(f_1,f_2,f_3)$ with a semi-symmetric metric connection. Then, $\alpha ,\beta ,f_1,f_2$, and $f_3$ satisfy that α is a constant on M, $\alpha \beta =0$, and

$$f_1-f_2={\alpha }^2-{\beta }^2,f_1-f_3={\alpha }^2-{\beta }^2-\zeta \beta .$$

Proof. 

From the covariant derivative of $\theta \left(V\right)=0$ with respect to X and (6) and (25), it is obtained that

$$\left({\overline{\nabla }}_X\theta \right)\left(V\right)=(\beta +1)u\left(X\right).$$

(52)

Applying ${\nabla }_X$ to (23): $B(Y,U)=C(Y,V)$ and using (21)–(25), we get

$$\begin{array}{c}\left({\nabla }_{X}B\right)(Y,U)=\left({\nabla }_{X}C\right)(Y,V)-2\tau \left(X\right)C(Y,V)\hfill \\ -\alpha (\beta +1)\left\{u\right(Y\left)v\right(X)-u(X\left)v\right(Y\left)\right\}\hfill \\ -{\alpha }^{2}u\left(Y\right)\eta \left(X\right)-{(\beta +1)}^{2}u\left(X\right)\eta \left(Y\right)\hfill \\ -g(A_{\xi }^{*}X,F\left(A_{{}_N}Y\right))-g(A_{\xi }^{*}Y,F\left(A_{{}_N}X\right)).\hfill \end{array}$$

Substituting this equation and (23) into (50) with $Z=U$, we have

$$\begin{array}{c}\left({\nabla }_{X}C\right)(Y,V)-\left({\nabla }_{Y}C\right)(X,V)\hfill \\ -\left\{\tau \right(X)+\theta (X\left)\right\}C(Y,V)+\left\{\tau \right(Y)+\theta (Y\left)\right\}C(X,V)\hfill \\ -\alpha (2\beta +1)\left\{u\right(Y\left)v\right(X)-u(X\left)v\right(Y\left)\right\}\hfill \\ -\{{\alpha }^2-{(\beta +1)}^2\}\{u\left(Y\right)\eta \left(X\right)-u\left(X\right)\eta \left(Y\right)\}\hfill \\ =f_2\{u\left(Y\right)\eta \left(X\right)-u\left(X\right)\eta \left(Y\right)+2\overline{g}(X,JY)\}.\hfill \end{array}$$

Comparing the above equation with (51) such that $PZ=V$ and using (52), it is obtained that

$$\begin{array}{c}\{f_1-f_2-{\alpha }^2+{\beta }^2\}\{u\left(Y\right)\eta \left(X\right)-u\left(X\right)\eta \left(Y\right)\}\hfill \\ =2\alpha \beta \left\{u\right(Y\left)v\right(X)-u(X\left)v\right(Y\left)\right\}.\hfill \end{array}$$

Taking $Y=U$, $X=\xi$ and $Y=U,X=V$ to the above equation by turns, it is obtained that

$$f_1-f_2={\alpha }^2-{\beta }^2,\alpha \beta =0.$$

(53)

From the covariant derivative of $\theta \left(\zeta \right)=1$ with respect to X, (5) implies

$$\left({\overline{\nabla }}_X\theta \right)\left(\zeta \right)=0.$$

(54)

From the covariant derivative of $\eta \left(Y\right)=\overline{g}(Y,N)$ with respect to X, (7) implies

$$\left({\nabla }_X\eta \right)\left(Y\right)=-g(A_{{}_N}X,Y)+\tau \left(X\right)\eta \left(Y\right).$$

(55)

Applying ${\nabla }_Y$ to (22) and using (20), (22), (28), and (55), we get

$$\begin{array}{c}\left({\nabla }_{X}C\right)(Y,\zeta )=-\left(X\alpha \right)v\left(Y\right)+\left(X\beta \right)\eta \left(Y\right)\hfill \\ -\alpha \{v\left(Y\right)\tau \left(X\right)-g(A_{{}_N}X,FY)-g(A_{{}_N}Y,FX)\hfill \\ -\alpha \theta \left(Y\right)\eta \left(X\right)+\theta \left(X\right)v\left(Y\right)-\theta \left(Y\right)v\left(X\right)\}\hfill \\ +(\beta +1)\{\tau \left(X\right)\eta \left(Y\right)-g(A_{{}_N}X,Y)-g(A_{{}_N}Y,X)\hfill \\ +(\beta +1)\theta \left(X\right)\eta \left(Y\right)\}.\hfill \end{array}$$

Substituting this and (22) into (51) with $PZ=\zeta$ and using (54), we get

$$\begin{array}{c}-\left(X\alpha \right)v\left(Y\right)+\left(Y\alpha \right)v\left(X\right)+\left(X\beta \right)\eta \left(Y\right)-\left(Y\beta \right)\eta \left(X\right)\hfill \\ =(f_1-f_3-{\alpha }^2+{\beta }^2)\{\theta \left(Y\right)\eta \left(X\right)-\theta \left(X\right)\eta \left(Y\right)\}.\hfill \end{array}$$

Taking $Y=\zeta ,X=\xi$ and $Y=U,X=V$ to this by turns, it is obtained that

$$f_1-f_3={\alpha }^2-{\beta }^2-\zeta \beta ,U\alpha =0.$$

Applying ${\nabla }_Y$ to (21) and using (20), (21), and (27), we have

$$\begin{array}{c}\left({\nabla }_{X}B\right)(Y,\zeta )=-\left(X\alpha \right)u\left(Y\right)-(\beta +1)B(X,Y)\hfill \\ +\alpha \left\{u\right(Y\left)\tau \right(X)+\theta (Y\left)u\right(X)-\theta (X\left)u\right(Y)\hfill \\ +B(X,FY)+B(Y,FX)\}.\hfill \end{array}$$

Substituting this equation and (21) into (50) with $Z=\zeta$, it is obtained that

$$\left(X\alpha \right)u\left(Y\right)=\left(Y\alpha \right)u\left(X\right).$$

Taking $Y=U$, we get $X\alpha =0$. It follows that $\alpha$ is a constant on M. ☐

Definition 6.

(a) A screen distribution $S\left(TM\right)$ is said to be totally umbilical [10] in M if

$$C(X,PY)=\gamma g(X,Y)$$

for some smooth function γ on a neighborhood $\mathcal{U}$. In particular, case $S\left(TM\right)$ is totally geodesic in M if $\gamma =0$.

(b) A lightlike hypersurface M is said to be screen conformal [11] if

$$C(X,PY)=\phi B(X,Y)$$

(56)

for some non-vanishing smooth function φ on a neighborhood $\mathcal{U}$.

Theorem 6.

Let M be a lightlike hypersurface of an indefinite generalized Sasakian space form $\overline{M}(f_1,f_2,f_3)$ with a semi-symmetric metric connection. If one of the following five conditions is satisfied,

(1) 

M is recurrent,

(2) 

$S\left(TM\right)$ is totally umbilical,

(3) 

M is screen conformal,

(4) 

$\nabla U=0$, and

(5) 

$\nabla V=0$,

then $\overline{M}(f_1,f_2,f_3)$ is an indefinite Kenmotsu space form such that

$$\alpha =0,\beta =-1;f_1=-1,f_2=f_3=0.$$

Proof. 

Applying ${\overline{\nabla }}_X$ to $\theta \left(U\right)=0$ and using (6) and (24), it is obtained

$$\left({\overline{\nabla }}_X\theta \right)\left(U\right)=\alpha \eta \left(X\right)+(\beta +1)v\left(X\right).$$

(57)

(a) Theorem 2 implies that $\alpha =0$ and $\beta =-1$. By directed calculation from (35), it is obtained that

$$R(X,Y)U=2d\tau (X,Y)U.$$

(58)

On the other hand, since $\alpha =0$ and $\beta =-1$, we have ${\overline{\nabla }}_X\zeta =0$ 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

$$\begin{array}{c}R(X,Y)Z=B(Y,Z)A_{{}_N}X-B(X,Z)A_{{}_N}Y\hfill \\ +\left({\overline{\nabla }}_X\theta \right)\left(Z\right)Y-\left({\overline{\nabla }}_Y\theta \right)\left(Z\right)X\hfill \\ +(f_1+1)\{g(Y,Z)X-g(X,Z)Y\}\hfill \\ +f_2\{\overline{g}(X,JZ)FY-\overline{g}(Y,JZ)FX+2\overline{g}(X,JY)FZ\}\hfill \\ +f_3\{\theta \left(X\right)\theta \left(Z\right)Y-\theta \left(Y\right)\theta \left(Z\right)X\hfill \\ +\overline{g}(X,Z)\theta \left(Y\right)\zeta -\overline{g}(Y,Z)\theta \left(X\right)\zeta \}.\hfill \end{array}$$

Setting $Z=U$ in the above equation and using (57) and (58), we get

$$\begin{array}{c}2d\tau (X,Y)U=B(Y,U)A_{{}_N}X-B(X,U)A_{{}_N}Y\hfill \\ +(f_1+1)\{v\left(Y\right)X-v\left(X\right)Y\}\hfill \\ +f_2\{\eta \left(X\right)FY-\eta \left(Y\right)FX\}\hfill \\ +f_3\{v\left(X\right)\theta \left(Y\right)-v\left(Y\right)\theta \left(X\right)\}\zeta .\hfill \end{array}$$

Taking the scalar product with N to the above equation and using (14)${}_2$, we get

$$2f_2\{v\left(Y\right)u\left(X\right)-v\left(X\right)u\left(Y\right)\}.$$

It follows that $f_2=0$. Thus, $f_1+1=f_2=f_3=0$.

(b) Since $S\left(TM\right)$ is totally umbilical, (22) is reduced to

$$\gamma \theta \left(X\right)=-\alpha v\left(X\right)+(\beta +1)\eta \left(X\right).$$

Taking $X=\zeta$, $X=V$, and $X=\xi$ to this equation by turns, we get $\gamma =0,\alpha =0$, and $\beta =-1$, respectively. As $\gamma =0$, $S\left(TM\right)$ is totally geodesic in M. As $\alpha =0$ and $\beta =-1$, $\overline{M}$ is an indefinite Kenmotsu manifold and $f_1+1=f_2=f_3$ by Theorem 5.

Taking $PZ=V$ in (51) and using (52) and the result: $C=0$, we have

$$f_2\{u\left(Y\right)\eta \left(X\right)-u\left(X\right)\eta \left(Y\right)+2\overline{g}(X,JY)\}=0.$$

Taking $X=\xi$ and $Y=U$, we get $f_2=0$. Thus, $f_1=-1$ and $f_2=f_3=0$, and $\overline{M}(f_1,f_2,f_3)$ is an indefinite Kenmotsu space form with $c=-1$.

(c) Taking $PY=\zeta$ in (56) and using (21) and (22), we get

$$\alpha v\left(X\right)-(\beta +1)\eta \left(X\right)=\alpha \phi u\left(X\right).$$

Taking $X=V$ and $X=\xi$ by turns, we have $\alpha =0$ and $\beta =-1$, respectively. Thus, $\overline{M}$ is an indefinite Kenmotsu manifold and we get $f_1+1=f_2=f_3$.

Applying ${\nabla }_X$ to $C(Y,PZ)=\phi B(Y,PZ)$, we have

$$\left({\nabla }_{X}C\right)(Y,PZ)=\left(X\phi \right)B(Y,PZ)+\phi \left({\nabla }_{X}B\right)(Y,PZ).$$

Substituting this equation into (51) and using (50), we have

$$\begin{array}{c}\{X\phi -2\phi \tau (X\left)\right\}B(Y,PZ)-\{Y\phi -2\phi \tau (Y\left)\right\}B(X,PZ)\hfill \\ -\{\left({\overline{\nabla }}_X\theta \right)\left(PZ\right)-g(X,PZ)\}\eta \left(Y\right)+\{\left({\overline{\nabla }}_Y\theta \right)\left(PZ\right)-g(Y,PZ)\}\eta \left(X\right)\hfill \\ =f_1\{g(Y,PZ)\eta \left(X\right)-g(X,PZ)\eta \left(Y\right)\}\hfill \\ +f_2\{[v\left(Y\right)-\phi u\left(Y\right)]\overline{g}(X,JPZ)-[v\left(X\right)-\phi u\left(X\right)]\overline{g}(Y,JPZ)\hfill \\ +2[v\left(PZ\right)-\phi u\left(PZ\right)]\overline{g}(X,JY)\}+f_3\{\theta \left(X\right)\eta \left(Y\right)-\theta \left(Y\right)\eta \left(X\right)\}\theta \left(PZ\right).\hfill \end{array}$$

Replacing Y by $\xi$ in the above equation, it is obtained that

$$\begin{array}{c}\{\xi \phi -2\phi \tau \left(\xi \right)\}B(X,PZ)+\left({\overline{\nabla }}_X\theta \right)\left(PZ\right)\hfill \\ -g(X,PZ)-\left({\overline{\nabla }}_{\xi }\theta \right)\left(PZ\right)\eta \left(X\right)\hfill \\ =f_{1}g(X,PZ)+f_2\{v\left(X\right)-\phi u\left(X\right)\}u\left(PZ\right)\hfill \\ +2f_2\{v\left(PZ\right)-\phi u\left(PZ\right)\}u\left(X\right)-f_3\theta \left(X\right)\theta \left(PZ\right).\hfill \end{array}$$

Taking $X=V,PZ=U$ and then $X=U,PZ=V$ to the above equation by turns and using (52), (57), and the fact that $f_1+1=f_2$, we have

$$\begin{array}{c}\{\xi \phi -2\phi \tau \left(\xi \right)\}B(V,U)=2f_2,\hfill \\ \{\xi \phi -2\phi \tau \left(\xi \right)\}B(U,V)=3f_2,\hfill \end{array}$$

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 $\overline{M}(f_1,f_2,f_3)$ is an indefinite Kenmotsu space form such that $c=-1$.

(d) Theorem 1 implies $\tau =0,\alpha =0,\beta =-1$, 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 ${\nabla }_X$ to $C(Y,U)=0$ and using ${\nabla }_{X}U=0$, we have

$$\left({\nabla }_{X}C\right)(Y,U)=0.$$

Substituting the last two equations into (51) with $PZ=U$ and using (57) and the fact that $f_1+1=f_2$, we have

$$2f_2\{v\left(Y\right)\eta \left(X\right)-v\left(X\right)\eta \left(Y\right)\}=0.$$

Taking $X=V$ and $Y=\xi$, we get $f_2=0$. Thus $f_1+1=f_2=f_3=0$ and $\overline{M}(f_1,f_2,f_3)$ is an indefinite Kenmotsu space form such that $c=-1$.

(e) Theorem 1 implies $\tau =0,\alpha =0,\beta =-1$ and (30). Thus $f_1+1=f_2=f_3$ by Theorem 5.

From (23) and (30), we get

$$C(X,V)=0.$$

Applying ${\nabla }_X$ to $C(Y,V)=0$ and using the fact that ${\nabla }_{X}V=0$, we have

$$\left({\nabla }_{X}C\right)(Y,V)=0.$$

Substituting these into (51) with $PZ=V$ and using (52), we get

$$f_2\{u\left(Y\right)\eta \left(X\right)-u\left(X\right)\eta \left(Y\right)+2\overline{g}(X,JY)\}=0.$$

Taking $U=U$ and $X=\xi$, we have $f_2=0$. Thus, $f_1+1=f_2=f_3=0$ and $\overline{M}(f_1,f_2,f_3)$ is an indefinite Kenmotsu space form with $c=-1$. ☐

Theorem 7.

Let M be a lightlike hypersurface of an indefinite generalized Sasakian space form $\overline{M}(f_1,f_2,f_3)$ with a semi-symmetric non-metric connection. If M is a Lie-recurrent or Hopf lightlike hypersurface, then $\overline{M}$ is an indefinite β-Kenmotsu space form with

$$f_1=-{\beta }^2,f_2=0,f_3=\zeta \beta .$$

Proof. 

(a) Theorem 3 implies $\alpha =0$ and

$$B(X,U)=0.$$

(59)

Applying ${\nabla }_X$ to $B(Y,U)=0$ and using (21) and (24), we have

$$\left({\nabla }_{X}B\right)(Y,U)=-B(Y,F\left(A_{{}_N}X\right)).$$

Setting $Z=U$ in the last two equations into (50), we have

$$\begin{array}{c}B(X,F\left(A_{{}_N}Y\right))-B(Y,F\left(A_{{}_N}X\right))\hfill \\ =f_2\{u\left(Y\right)\eta \left(X\right)-u\left(X\right)\eta \left(Y\right)+2\overline{g}(X,JY)\}.\hfill \end{array}$$

Taking $X=\xi$ and $Y=U$ to the above equation and using (12) and (59), it is obtained that $f_2=0$.

Therefore, Theorem 5 implies

$$f_1=-{\beta }^2,f_2=0,f_3=\zeta \beta .$$

(b) Applying ${\nabla }_Y$ to (45)${}_1$ and using (21), (24), and (28), it is obtained that

$$\begin{array}{c}\left({\nabla }_{X}B\right)(Y,U)=\left(X\kappa \right)v\left(Y\right)-B(Y,F\left(A_{{}_N}X\right))\hfill \\ -\kappa \{(\beta +1)\theta \left(Y\right)v\left(X\right)+g(A_{{}_N}X,FY)\},\hfill \end{array}$$

because $\alpha =0$. Substituting this equation and (45)${}_1$ into (50), we have

$$\begin{array}{c}\left(X\kappa \right)v\left(Y\right)-\left(Y\kappa \right)v\left(X\right)+B(X,F\left(A_{{}_N}Y\right))-B(Y,F\left(A_{{}_N}X\right))\hfill \\ +\kappa \left\{\beta \right[\theta \left(X\right)v\left(Y\right)-\theta \left(Y\right)v\left(X\right)]+\tau (X\left)v\right(Y)-\tau (Y\left)v\right(X)\hfill \\ +g(A_{{}_N}Y,FX)-g(A_{{}_N}X,FY)\}\hfill \\ =f_2\{u\left(Y\right)\eta \left(X\right)-u\left(X\right)\eta \left(Y\right)+2\overline{g}(X,JY)\}.\hfill \end{array}$$

Taking $Y=U$ and $X=\xi$ 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

$$f_1=-{\beta }^2,f_2=0,f_3=\zeta \beta .$$

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.