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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

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We study a lightlike hypersurface M of an indefinite nearly trans-Sasakian manifold M̄ with an (ℓ, m)-type connection such that the structure vector field ζ of M̄ is tangent to M. In particular, we focus on such lightlike hypersurfaces M for which the structure tensor field F is either recurrent or Lie recurrent, or such that M itself is totally umbilical or screen totally umbilical.

Keywords: (ℓ,m)-type connection, recurrent, Lie recurrent, lightlike hypersurface, indefi,nite nearly trans-Sasakian manifold.

1. Introduction

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A linear connection ∇̄ on a semi-Riemannian manifold (M̄, ḡ) is called an (ℓ, m)-type connection if there exist two smooth functions ℓ and m such that

(∇¯X¯g¯) (Y¯,Z¯)=-ℓ{θ(Y¯)g¯(X¯,Z¯)+θ(Z¯)g¯(X¯,Y¯)}-m{θ(Y¯)g¯(JX¯,Z¯)+θ(Z¯)g¯(JX¯,Y¯)},T¯(X¯,Y¯)=ℓ{θ(Y¯)X¯-θ(X¯)Y¯}+m{θ(Y¯)JX¯-θ(X¯)JY¯}

for any vector fields X̄, Ȳ, Z̄ on M̄, where T̄ is the torsion tensor of ∇̄ and J is a (1, 1)-type tensor field and θ is a 1-form associated with a smooth vector field ζ by θ( X̄ ) = ḡ(X̄, ζ). Throughout this paper, we set (ℓ, m) ≠ (0, 0) and denote by X̄, Ȳ and Z̄ the smooth vector fields on M̄.

The notion of (ℓ, m)-type connection was introduced by Jin [8]. In the case (ℓ, m) = (1, 0), this connection ∇̄ becomes a semi-symmetric non-metric connection. The notion of a semi-symmetric non-metric connection on a Riemannian manifold was introduced by Ageshe-Chafle [1]. In the case (ℓ, m) = (0, 1), this connection ∇̄ becomes a non-metric φ-symmetric connection such that φ(X̄, Ȳ) = ḡ(JX̄, Ȳ ). The notion of the non-metric φ-symmetric connection was introduced by Jin [6].

Remark 1.1.([8])

Denote by ∇̃ a unique Levi-Civita connection of a semi-Riemannian manifold ( M̄, ḡ) with respect to ḡ. Then a linear connection ∇̄ on ( M̄, ḡ) is an (ℓ, m)-type connection if and only if ∇̄ satisfies

∇¯X¯Y¯=∇˜X¯Y¯+θ(Y¯){ℓX¯+mJX¯}.

The subject of study in this paper is lightlike hypersurfaces of an indefinite nearly trans-Sasakian manifold M̄ = (M̄, ζ, θ, J, ḡ) with an (ℓ, m)-type connection subject to the conditions: (1) the tensor field J and the 1-form θ, defined by (1.1) and (1.2 are identical with the indefinite nearly trans-Sasakian structure tensor J and the structure 1-form θ of M̄, respectively, and (2) the structure vector field ζ of M̄ is tangent to M.

Călin [3] proved that if the structure vector field ζ of M̄ is tangent to M, then it belongs to S(TM), we assume this in this paper.

2. On (ℓ, m)-type Connections

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A hypersurface M of a semi-Riemannian manifold (M̄, ḡ) is a lightlike hypersurface if its normal bundle TM⊥ is a vector subbundle of the tangent bundle TM. There exists a screen distribution S(TM) such that

TM=TM⊥⊕orthS(TM),

where ⊕orth denotes the orthogonal direct sum. It is known from [4] that, for any null section ξ of TM⊥ on a coordinate neighborhood , there exists a unique null section N of a unique lightlike vector bundle tr(TM), of rank 1, in the orthogonal complement S(TM)⊥ of S(TM) in M̄ satisfying

g¯(ξ,N)=1,         g¯(N,N)=g¯(N,X)=0,         ∀X∈S(TM).

In this case, the tangent bundle T M̄ of M̄ can be decomposed as follows:

TM¯=TM⊕tr(TM)={TM⊥⊕tr(TM)}⊕orthS(TM).

We call tr(TM) and N the transversal vector bundle and the null transversal vector field with respect to the screen distribution S(TM), respectively.

In the following, we denote by X, Y and Z smooth vector fields on M, unless otherwise specified. Let ∇̄ be an (ℓ, m)-type connection on M̄ defined by (1.3) and P the projection morphism of TM on S(TM). As ζ belongs to S(TM), from (1.1) we have ḡ(∇̄XN, ξ) + ḡ(N, ∇̄Xξ) = 0. Thus the local Gauss and Weingarten formulae of M and S(TM) are given by

∇¯XY=∇XY+B(X,Y)N,∇¯XN=-ANX+τ(X)N;∇XPY=∇X*PY+C(X,PY)ξ,∇Xξ=-Aξ*X-τ(X)ξ,

where ∇ and ∇* are the linear connections on TM and S(TM), respectively, B and C are the local second fundamental forms on TM and S(TM), respectively, AN and Aξ* are the shape operators, and τ is a 1-form.

An odd dimensional semi-Riemannian manifold (M̄, ḡ) is said to be an indefinite almost contact metric manifold [5, 6] if there exist a structure set {J, ζ, θ, ḡ}, where J is a (1, 1)-type tensor field, ζ is a vector field, θ is a 1-form and ḡ is the semi-Riemannian metric on M̄ such that

J2X¯=-X¯+θ(X¯)ζ,         Jζ=0,         θ∘J=0,         θ(ζ)=1,θ(X¯)=g¯(ζ,X¯),         g¯(JX¯,JY¯)=g¯(X¯,Y¯)-θ(X¯)θ(Y¯).

It is known [5, 6] that, for any lightlike hypersurface M of an indefinite almost contact metric manifold M̄ such that the structure vector field ζ of M̄ is tangent to M, J(TM⊥) and J(tr(TM)) are subbundles of S(TM), of rank 1, such that J(TM⊥)∩J(tr(TM)) = {0}. Thus there exist two non-degenerate almost complex distributions Do and D with respect to J, i.e., J(Do) = Do and J(D) = D, such that

S(TM)={J(TM⊥)⊕J(tr(TM))}⊕orthDo,D=TM⊥⊕orthJ(TM⊥)⊕orthDo.

In this case, the decomposition form of TM is reformed as follows:

TM=D⊕J(tr(TM)).

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

U=-JN,         V=-Jξ,         u(X)=g(X,V),         v(X)=g(X,U).

Denote by S̄ the projection morphism of TM on D. Any vector field X of M is expressed as X = S̄X + u(X)U. Applying J to this form, we have

JX=FX+u(X)N,

where F is a tensor field of type (1, 1) globally defined on M by FX = J S̄X. Applying J to (2.7) and using (2.5) and (2.6), we have

F2X=-X+u(X)U+θ(X)ζ.

Using (1.1), (1.2), (2.1) and (2.7), we see that

(∇Xg) (Y,Z)=B(X,Y)η(Z)+B(X,Z)η(Y)-ℓ{θ(Y)g(X,Z)+θ(Z)g(X,Y)}-m{θ(Y)g¯(JX,Z)+θ(Z)g¯(JX,Y)},T(X,Y)=ℓ{θ(Y)X-θ(X)Y}+m{θ(Y)FX-θ(X)FY},B(X,Y)-B(Y,X)=m{θ(Y)u(X)-θ(X)u(Y)},

where T is the torsion tensor with respect to the induced connection ∇ on M and η is a 1-form such that η(X) = ḡ(X, N).

From the fact that B(X, Y) = ḡ(∇̄XY, ξ), we know that B is independent of the choice of the screen distribution S(TM) and satisfies

B(X,ξ)=0,         B(ξ,X)=0.

The local second fundamental forms are related to their shape operators by

B(X,Y)=g(Aξ*X,Y)+mu(X)θ(Y),C(X,PY)=g(ANX,PY)+{ℓη(X)+mv(X)}θ(PY),g¯(Aξ*X,N)=0,         g¯(ANX,N)=0.

As S(TM) is non-degenerate, taking X = ξ to (2.13), we obtain

Aξ*ξ=0,         ∇¯Xξ=-Aξ*X-τ(X)ξ.

Applying ∇X to Fξ = −V and FV = ξ by turns and using (2.5), we have

(∇XF)ξ=-∇XV+F(Aξ*X)-τ(X)V,(∇XF)V=-F∇XV-Aξ*X-τ(X)ξ.

Applying ∇X to v(Y ) = g(Y, U) and using (2.9), we obtain

(∇Xv) (Y)=mθ(Y)η(X)-ℓθ(Y)v(X)+B(X,U)η(Y)+g(Y,∇XU).

Applying ∇X to g(U, U) = 0 and g(V, V ) = 0 and using (2.9), we get

v(∇XU)=0,         u(∇XV)=0.

3. Recurrents and Lie Recurrents

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Definition 3.1.([7])

The structure tensor field F of M is said to be recurrent if there exists a 1-form ω on M such that

(∇XF)Y=ω(X)FY.

Theorem 3.2

Let M be a lightlike hypersurface of an indefinite almost contact metric manifold M̄ with an (ℓ, m)-type connection ∇̄ such that ζ is tangent to M. If F is recurrent, then F is parallel with respect to the induced connection ∇ from ∇̄.

Proof

Comparing (2.18) with (3.1) in which we replace Y with V, we obtain

F∇XV+Aξ*X+{ω(X)+τ(X)}ξ=0.

Also, comparing (2.17) with (3.1), taking Y = ξ, we obtain

∇XV-F(Aξ*X)-{ω(X)-τ(X)}V=0.

Taking the scalar product with V and ζ to (3.3), we have

u(∇XV)=0,         θ(∇XV)=0.

Applying F to (3.2) and using (2.8) and (3.4) and then, comparing this result with (3.3), we get ω = 0. Thus F is parallel with respect to ∇.

Definition 3.3.([7])

The structure tensor field F of M is called Lie recurrent if there exists a 1-form θ on M such that

(ℒXF)Y=σ(X)FY,

where ℒX denotes the Lie derivative on M with respect to X, that is,

(ℒXF)Y=[X,FY]-F[X,Y].

The structure tensor field F is called Lie parallel if ℒXF = 0.

Theorem 3.4

Let M be a lightlike hypersurface of an indefinite almost contact metric manifold M̄ with an (ℓ, m)-type connection ∇̄ such that ζ is tangent to M. If F is Lie recurrent, then F is Lie parallel.

Proof

As the induced connection ∇ from ∇̄ is torsion-free, from (3.5) and (3.6) we have

(∇XF)Y=∇FYX-F∇YX+σ(X)FY.

Comparing (2.18) with (3.7), taking Y = V, we obtain

∇ξX=-F(∇XV-∇VX)-Aξ*X-{σ(X)+τ(X)}ξ.

Also, comparing (2.17) with (3.7), taking Y = ξ, we obtain

F∇ξX=∇XV-∇VX-F(Aξ*X)-{σ(X)-τ(X)}V.

Taking the scalar product with V and ζ to (3.9), we obtain

u(∇XV-∇VX)=0,         θ(∇XV-∇VX)=0.

Applying F to (3.8) and using (2.8) and (3.10) and then, comparing this result with (3.9), we have σ = 0. Thus F is Lie parallel.

4. Indefinite Nearly Trans-Sasakian Manifolds

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Definition 4.1.([9])

An indefinite almost contact metric manifold M̄ is called an indefinite nearly trans-Sasakian manifold if {J, ζ, θ, ḡ} satisfies

(∇˜X¯J)Y¯+(∇˜Y¯J)X¯=α{2g¯(X¯,Y¯)ζ-θ(Y¯)X¯-θ(X¯)Y¯}-β{θ(Y¯)JX¯+θ(X¯)JY¯}.

where ∇̃ is the Levi-Civita connection of M̄. We say that the set {J, ζ, θ, ḡ} is an indefinite nearly trans-Sasakian structure of type (α, β).

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

α=1,   β=0;         α=0,   β=1;         α=β=0,respectively.

Replacing the Levi-Civita connection ∇̃ by the (ℓ, m)-type connection ∇̄ given by (1.3), the equation (4.1) is reduced to

(∇¯X¯J)Y¯+(∇¯Y¯J)X¯=(m-α){θ(Y¯)X¯+θ(X¯)Y¯}-(ℓ+β){θ(Y¯)JX¯+θ(X¯)JY¯}+2{αg¯(X¯,Y¯)-mθ(X¯)θ(Y¯)}ζ.

Applying ∇̄ζ to ḡ(ζ, ζ) = 1 and using (1.1), we have θ(∇̄ζζ) = ℓ. Taking X̄ = Ȳ = ζ to (4.2), we obtain (∇̄ζJ)ζ = 0. It follows that J(∇̄ζζ) = 0. Applying J to this equation and using (2.5) and the fact that θ(∇̄ζζ) = ℓ, we have ∇̄ζζ = ℓζ. From this equation, (2.1) and (2.3), we obtain

∇ζζ=ℓζ,         B(ζ,ζ)=0,         C(ζ,ζ)=0.

Definition 4.2.([4])

A lightlike hypersurface M of (M̄, ḡ) is said to be

  • totally umbilical if there is a smooth function ρ on a coordinate neighborhood in M such that Aξ*X=ρPX or equivalently

    B(X,Y)=ρg(X,Y).

    In case ρ = 0 on , we say that M is totally geodesic.

  • screen totally umbilical if there exist a smooth function γ on a coordinate neighborhood such that ANX = γPX or equivalently

    C(X,PY)=γg(X,PY).

    In case γ = 0 on , we say that M is screen totally geodesic.

Theorem 4.3

Let M be a lightlike hypersurface of an indefinite nearly trans-Sasakian manifold M̄ with an (ℓ, m)-type connection such that the structure vector field ζ of M̄ is tangent to M.

  • 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.

Proof

(1) If M is totally umbilical, then, taking X = Y = ζ to (4.4) and using (4.3), we have ρ = 0. Thus M is totally geodesic. On the other hand, since B = 0, taking X = U and Y = ζ to (2.11), we see that m = 0.

(2) If M is screen totally umbilical, then, taking X = PY = ζ to (4.5) and using (4.3), we have γ = 0. Thus M is screen totally geodesic.

Applying ∇̄X to JY = FY + u(Y )N and using (2.3), we have

(∇¯XJ)Y=(∇XF)Y-u(Y)ANX+B(X,Y)U+{(∇Xu) (Y)+u(Y)τ(X)+B(X,FY)}N.

Substituting (4.6) into (4.2) and using (2.7) and (2.11), we obtain

(∇XF)Y+(∇YF)X=(m-α){θ(Y)X+θ(X)Y}-(ℓ+β){θ(Y)FX+θ(X)FY}+2{αg(X,Y)-mθ(X)θ(Y)}ζ+u(X)ANY+u(Y)ANX-2B(X,Y)U+m{θ(Y)u(X)-θ(X)u(Y)}U.

Lemma 4.4

Let M be a lightlike hypersurface of an indefinite nearly trans-Sasakian manifold M̄ with an (ℓ, m)-type connection ∇̄ such that the structure vector field ζ of M̄ is tangent to M. Then we have

{B(U,V)=C(V,V),B(U,ζ)+C(V,ζ)=2(m-α),B(U,U)=C(U,V),v(∇UV)=-τ(U),C(U,ζ)=0,2C(V,ζ)+C(ζ,V)=2m-3α,B(U,ζ)=C(V,ζ)+C(ζ,V)+α, B(U,ζ)=m+θ(Aξ*U),         θ(∇ξU)=θ(Aξ*U),

where ∇ is the induced connection from ∇̄.

Proof

Applying ∇X to FU = 0 and FV = ξ by turns, we obtain

(∇XF)U=-F∇XU,   (∇XF)V=-F∇XV-Aξ*X-τ(X)ξ.

From these two equations, we obtain

(∇UF)V+(∇VF)U=-F(∇UV+∇VU)-Aξ*U-τ(U)ξ.

Comparing this result with (4.7), taking X = U and Y = V, we have

F(∇UV+∇VU)+Aξ*U+τ(U)ξ=-2αζ-ANV+2B(U,V)U.

Taking the scalar product with V, ζ, U and N to this and using (2.13), (2.14), (2.20) and η(∇XPY ) = C(X, PY ), we get (4.8).

By direct calculation from FU = 0, Fζ = 0 and (4.7), we obtain

F(∇Uζ+∇ζU)=-ANζ+{α-2m+2B(U,ζ)}U.

Taking the scalar product with U and V to this by turns and using (2.5), (2.7), (2.14) and η(∇Uζ + ∇ζU) = C(U, ζ) + C(ζ, U), we get (4.8) and

2B(U,ζ)-C(ζ,V)=2m-α.

Substituting (4.8) into (4.9), we have (4.8).

By directed calculation from FV = ξ, Fζ = 0 and (4.7), we obtain

F(∇Vζ+∇ζV)=-Aξ*ζ+2B(V,ζ)U-(m-α)V+{ℓ+β-τ(ζ)}ξ.

Taking the scalar product with U and using (2.3), (2.11) and (2.13), we get (4.8): B(U, ζ) = C(V, ζ) + C(ζ, V ) + α.

Taking X = U and Y = ζ to (2.13), we have (4.8). On the other hand, applying ∇̄ X to v(Y ) = g(FY, N) and using (1.1), (2.1) and (2.2), we get

g((∇XF)Y,N)=(∇Xv) (Y)-v(Y)τ(X)+g(ANX,FY).

Taking the scalar product with N to (4.7), we obtain

(∇Xv)Y+(∇Yv)X=(m-α){θ(Y)η(X)+θ(X)η(Y)}-(ℓ+β){θ(Y)v(X)+θ(X)v(Y)}+v(Y)τ(X)+v(X)τ(Y)-g(ANX,FY)-g(ANY,FX).

Substituting (2.19) into the last equation, we have

B(X,U)η(Y)+B(Y,U)η(X)+g(Y,∇XU)+g(X,∇YU)=-α{θ(Y)η(X)+θ(X)η(Y)}-β{θ(Y)v(X)+θ(X)v(Y)}+v(Y)τ(X)+v(X)τ(Y)-g(ANX,FY)-g(ANY,FX).

Taking X = ζ and Y = ξ to this and using (2.11) and (2.12), we have

B(U,ζ)-C(ζ,V)=m-α-θ(∇ξU),

due to (2.14). Substituting this equation into (4.9), we obtain

B(U,ζ)=m+θ(∇ξU).

Comparing this equation with (4.8), we have (4.8).

Lemma 4.5

Let M be a lightlike hypersurfac of an indefinite nearly trans-Sasakian manifold M̄ with an (ℓ, m)-type connection ∇̄ such that ζ is tangent to M. If one of the following three conditions is satisfied,

  • (∇XF)Y + (∇Y F)X = 0,

  • F is parallel with respect to the induced connection ∇ on M, that is, ∇XF = 0,

  • F is recurrent,

then α = m and β = −ℓ. The shape operators Aξ*and AN satisfy

Aξ*V=0,         ANV=-2αζ,         ANξ=0,         θ(Aξ*U)=0,θ(∇ξU)=0,         ANX=C(X,V)U-2αv(X)ζ.
Proof

(1) Assume that (∇XF)Y + (∇Y F)X = 0. Taking the scalar product with N to (4.7) and using (2.15), we have

(m-α){θ(Y)η(X)+θ(X)η(Y)}=ℓ+β){θ(Y)v(X)+θ(X)v(Y)}.

Taking X = ξ, Y = ζ and X = V, Y = ζ in this equation, we obtain α = m and β = −ℓ. As α = m and β = −ℓ, (4.7) is reduced to

2α{g(X,Y)-θ(X)θ(Y)}ζ+u(X)ANY+u(Y)ANX-2B(X,Y)U+m{θ(Y)u(X)-θ(X)u(Y)}U=0.

Taking the scalar product with V to (4.11), we have

2B(X,Y)=u(Y)u(ANX)+u(X)u(ANY)+m{θ(Y)u(X)-θ(X)u(Y)}.

Taking Y = V in this equation and using (2.14), we obtain

2B(X,V)=u(X)C(V,V).

Replacing X by U to this equation, we have 2B(U, V ) = C(V, V ). Comparing this result with (4.8), we have C(V, V ) = 0. Thus we obtain

B(U,V)=C(V,V)=0,         B(X,V)=0.

Using (2.11) and (4.13), we see that B(V, X) = 0. From this, (2.13) and the fact that S(TM) is non-degenerate, we have (4.10): Aξ*V=0. Taking X = U and Y = V to (4.11) and using (4.13), we get (4.10): AN V = −2αζ. Also, taking X = U and Y = ξ to (4.11) and using (2.12), we get (4.10): AN ξ = 0. Taking X = V and Y = ζ to (2.14) and using (4.10) and the fact that m = α, we obtain C(V, ζ) = −m. From this result and (4.8), we have B(U, ζ) = m. Thus, from (4.8) we get (4.10): θ(Aξ*U)=θ(∇ξU)=0.

Taking Y = U to (4.12), we obtain

2B(X,U)+mθ(X)=u(ANX)+u(X)u(ANU).

Replacing Y by U to (4.11) and using the last equation, we get

ANX-u(ANX)U+u(X){ANU-u(ANU)U}+2αv(X)ζ=0.

Taking X = U to this, we have ANU = u(ANU)U. Thus we have

ANX=u(ANX)U-2αv(X)ζ.

(2) If F is parallel with respect to ∇, then (∇XF)Y + (∇Y F)X = 0. By item (1), we see that α = m and β = −ℓ. Aξ* and AN satisfy (4.10).

(3) If F is recurrent, then F is parallel with respect to ∇ by Theorem 3.2. By item (2), we see that α = m and β = −ℓ. Aξ* and AN satisfy (4.10).

Theorem 4.6

Let M be a lightlike hypersurface of an indefinite nearly trans-Sasakian manifold M̄ with an (ℓ, m)-type connection ∇̄ such that ζ is tangent to M. If F is Lie recurrent, then M̄ is an indefinite nearly β-Kenmotsu manifold with an (ℓ, m)-type connection ∇̄.

Proof

If F is Lie recurrent, then F is Lie parallel, i.e., σ = 0, by Theorem 3.4. Replacing Y by U to (3.7), we have (∇XF)U = −F∇UX. Applying ∇X to FU = 0, we get (∇XF)U = −F∇XU. Therefore we have

F(∇XU-∇UX)=0.

Taking the scalar product with N to this and using (2.20), we obtain

v(∇UX)=0,         τ(U)=0,

due to (4.8). Taking X = U to (3.8) and using (4.14) and (4.15), we get

∇ξU=-Aξ*U.

Taking the scalar product with ζ to this equation, we have

θ(∇ξU)=-θ(Aξ*U).

Comparing this with (4.8) and using (2.11) and (4.8), we have

θ(∇ξU)=θ(Aξ*U)=0,         B(U,ζ)=m,         B(ζ,U)=0.

Applying ∇ξ to g(U, ζ) = 0 and using (2.9) and (4.17), we obtain

v(∇ξζ)=-m.

Taking the scalar product with U to (3.8), we have

v(∇ξX)=η(∇XV-∇VX)-B(X,U).

Replacing X by ζ to this and using (2.4), (4.17) and (4.18), we have

C(ζ,V)=C(V,ζ)-m.

As B(U, ζ) = m, from (4.9), we obtain

C(ζ,V)=α,         C(V,ζ)=m+α.

Substituting the last two results into (4.8), we get α = 0. Thus M̄ is an indefinite nearly β-Kenmotsu manifold with an (ℓ, m)-type connection.

5. Indefinite Nearly Generalized Sasakian Space Forms

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Denote by R̄, R and R* the curvature tensors of the (ℓ, m)-type connection ∇̄ of M̄ and the induced connections ∇ and ∇* on M and S(TM), respectively. Using the Gauss-Weingarten formulae for M and S(TM), we obtain two Gauss equations for M and S(TM) such that

R¯(X,Y)Z=R(X,Y)Z+B(X,Z)ANY-B(Y,Z)ANX+{(∇XB) (Y,Z)-(∇YB) (X,Z)+τ(X)B(Y,Z)-τ(Y)B(X,Z)-ℓ[θ(X)B(Y,Z)-θ(Y)B(X,Z)]-m[θ(X)B(FY,Z)-θ(Y)B(FX,Z)]}N,R(X,Y)PZ=R*(X,Y)PZ+C(X,PZ)Aξ*Y-C(Y,PZ)Aξ*X+{(∇XC) (Y,PZ)-(∇YC) (X,PZ)-τ(X)C(Y,PZ)+τ(Y)C(X,PZ)-ℓ[θ(X)C(Y,PZ)-θ(Y)C(X,PZ)]-m[θ(X)C(FY,PZ)-θ(Y)C(FX,PZ)]}ξ.

Definition 5.1

An indefinite nearly trans-Sasakian manifold M̄ is said to be a indefinite nearly generalized Sasakian space form, denoted by M̄ (f1, f2, f3), if there exist three smooth functions f1, f2 and f3 on M̄ such that

R˜(X¯,Y¯)Z¯=f1{g¯(Y¯,Z¯)X¯-g¯(X¯,Z¯)Y¯}+f2{g¯(X¯,JZ¯)JY¯-g¯(Y¯,JZ¯)JX¯+2g¯(X¯,JY¯)JZ¯}+f3{θ(X¯)θ(Z¯)Y¯-θ(Y¯)θ(Z¯)X¯+g¯(X¯,Z¯)θ(Y¯)ζ-g¯(Y¯,Z¯)θ(X¯)ζ},

where R̃ is the curvature tensors of the Levi-Civita connection ∇̃ of M̄.

The notion of (Riemannian) generalized Sasakian space form was introduced by Alegre et. al. [2]. Sasakian, Kenmotsu and cosymplectic space form are important kinds of generalized Sasakian space forms such that

f1=c+34,f2=f3=c-14;         f1=c-34,f2=f3=c+14;         f1=f2=f3=c4

respectively, where c is a constant J-sectional curvature of each space forms.

By direct calculations from (1.2), (1.3) and (2.5), we have

R¯(X¯,Y¯)Z¯=R˜(X¯,Y¯)Z¯+{ℓ(∇¯X¯θ) (Z¯)+[X¯ℓ+m2θ(X¯)]θ(Z¯)}Y¯-{ℓ(∇¯Y¯θ) (Z¯)+[Y¯ℓ+m2θ(Y¯)]θ(Z¯)}X¯+{m(∇¯X¯θ) (Z¯)+[X¯m-ℓmθ(X¯)]θ(Z¯)}JY¯-{m(∇¯Y¯θ) (Z¯)+[Y¯m-ℓmθ(Y¯)]θ(Z¯)}JX¯+mθ(Z¯) {(∇¯X¯J)Y¯-(∇¯Y¯J)X¯}.

Comparing the tangential, transversal and radical components of the left-right terms of (5.4) such that X̄ = X, Ȳ = Y and Z̄ = Z and using (2.11), (2.15), (4.6), (5.1), (5.2), (5.3) and the last two equations, we obtain

R(X,Y)Z=B(Y,Z)ANX-B(X,Z)ANY+{ℓ(∇¯Xθ) (Z)+[Xℓ+m2θ(X)]θ(Z)}Y-{ℓ(∇¯Yθ) (Z)+[Yℓ+m2θ(Y)]θ(Z)}X+{m(∇¯Xθ) (Z)+[Xm-ℓmθ(X)]θ(Z)}FY-{m(∇¯Yθ) (Z)+[Ym-ℓmθ(Y)]θ(Z)}FX+mθ(Z) {(∇XF)Y-(∇YF)X+u(X)ANY-u(Y)ANX+m[θ(Y)u(X)-θ(X)u(Y)]U}+f1{g(Y,Z)X-g(X,Z)Y}+f2{g¯(X,JZ)FY-g¯(Y,JZ)FX+2g¯(X,JY)FZ}+f3{[θ(X)Y-θ(Y)X]θ(Z)+[g(X,Z)θ(Y)-g(Y,Z)θ(X)]ζ},(∇XB) (Y,Z)-(∇YB) (X,Z)+{τ(X)-ℓθ(X)}B(Y,Z)-{τ(Y)-ℓθ(Y)}B(X,Z)-m{θ(X)B(FY,Z)-θ(Y)B(FX,Z)}={m(∇¯Xθ) (Z)+[Xm-ℓmθ(X)]θ(Z)}u(Y)-{m(∇¯Yθ) (Z)+[Ym-ℓmθ(Y)]θ(Z)}u(X)+mθ(Z){(∇Xu)Y-(∇Yu)X+u(Y)τ(X)-u(X)τ(Y)+B(X,FY)-B(Y,FX)}+f2{g¯(X,JZ)u(Y)-g¯(Y,JZ)u(X)+2g¯(X,JY)u(Z)},(∇XC) (Y,PZ)-(∇YC) (X,PZ)-{τ(X)+ℓθ(X)}C(Y,PZ)+{τ(Y)+ℓθ(Y)}C(X,PZ)-m{θ(X)C(FY,PZ)-θ(Y)C(FX,PZ)}={ℓ(∇¯Xθ) (PZ)+[Xℓ+m2θ(X)]θ(PZ)}η(Y)-{ℓ(∇¯Yθ) (PZ)+[Yℓ+m2θ(Y)]θ(PZ)}η(X)+{m(∇¯Xθ) (PZ)+[Xm-ℓmθ(X)]θ(PZ)}v(Y)-{m(∇¯Yθ) (PZ)+[Ym-ℓmθ(Y)]θ(PZ)}v(X)+mθ(PZ){(∇Xv)Y-(∇Yv)X-v(Y)τ(X)+v(X)τ(Y)+g(ANX,FY)-g(ANY,FX)}+f1{g(Y,PZ)η(X)-g(X,PZ)η(Y)}+f2{g¯(X,JPZ)v(Y)-g¯(Y,JPZ)v(X)+2g¯(X,JY)v(PZ)+f3{θ(X)η(Y)-θ(Y)η(X)}θ(PZ),

due to the following equations:

g¯((∇¯XJ)Y,ξ)=(∇Xu) (Y)+u(Y)τ(X)+B(X,FY),g¯((∇¯XJ)Y,N)=(∇Xv) (Y)-v(Y)τ(X)+g(ANX,FY).

Using the Gauss-Weingarten formulae for S(TM), we obtain the following Codazzi equations for S(TM) such that

R(X,Y)ξ=-∇X*(Aξ*Y)+∇Y*(Aξ*X)+Aξ*[X,Y]-τ(X)Aξ*Y+τ(Y)Aξ*X+{C(Y,Aξ*X)-C(X,Aξ*Y)-2dτ(X,Y)}ξ.

Replacing Z by ξ to (5.5) and using (2.12) and (5.9), we have

R(X,Y)ξ=θ(Aξ*X){ℓY+mFY}-θ(Aξ*Y){ℓX+mFX}+f2{u(Y)FX-u(X)FY-2g¯(X,JY)V}.

Comparing the radical components of the last two equations, we obtain

f2{u(Y)v(X)-u(X)v(Y)}=g(ANY,Aξ*X)-g(ANX,Aξ*Y)-2dτ(X,Y).

Applying ∇̄X to θ(U) = 0 and θ(ξ) = 0 and using (2.16), we obtain

(∇¯Xθ) (U)=-θ(∇XU),         (∇¯Xθ) (ξ)=θ(Aξ*X).

Theorem 5.2

Let M be a lightlike hypersurface of an indefinite nearly generalized Sasakian space form M̄ (f1, f2, f3) with an (ℓ, m)-type connection ∇̄ such that ζ is tangent to M. If one of the following conditions is satisfied ;

  • (∇XF)Y + (∇Y F)X = 0,

  • F is parallel with respect to the induced connection ∇, that is, ∇XF = 0,

  • F is recurrent,

then f1 + f2 = 0 and f2 = 2dτ(U, V ).

Proof

If one of the items (1)~(3) is satisfied, then Aξ* and AN satisfy (4.10). Taking the scalar product with U to (4.10) and using (2.14), we have

C(X,U)=0.

Applying ∇X to C(Y, U) = 0 and using the last equation, we have

(∇XC) (Y,U)=-C(Y,∇XU).

Substituting the last two equations into (5.7) with PZ = U, we obtain

C(X,∇YU)-C(Y,∇XU)=(∇¯Xθ) (U){ℓη(Y)+mv(Y)}-(∇¯Yθ) (U){ℓη(X)+mv(X)}+(f1+f2){v(Y)η(X)-v(X)η(Y)}

Taking Y = V and X = ξ to this and using (4.10) and (5.9), we get

C(ξ,∇VU)-C(V,∇ξU)=ℓθ(∇VU)+f1+f2.

By using (2.14), (4.10) and the fact that m = α, we see that

C(ξ,∇VU)=g(ANξ,∇VU)+ℓθ(∇VU)=ℓθ(∇VU),C(V,∇ξU)=g(ANV,∇ξU)+mθ(∇ξU)=-mθ(∇ξU)=0.

From the last three equations, we get f1 + f2 = 0. Taking Y = V and X = U to (5.8) and using (4.10), we have f2 = 2dτ(U, V )

Definition 5.3

A lightlike hypersurface M is said to be a Hopf lightlike hypersurface if the structure vector field U is an eigenvector of Aξ*.

Theorem 5.4

Let M be a lightlike hypersurface of an indefinite nearly generalized Sasakian space form M̄ (f1, f2, f3) with an (ℓ, m)-type connection such that ζ is tangent to M and F is Lie recurrent. Then

g(Aξ*U,Aξ*U)=3f2.

If M is a Hopf lightlike hypersurface of M̄ (c), then f2 = 0.

Proof

Taking the scalar product with U to (4.16) and using (2.20), we get

B(U,U)=0.

Applying ∇ξ to (5.10) and using (2.11), (2.13), (4.16) and (4.17), we have

(∇ξB) (U,U)=2g(Aξ*U,Aξ*U).

Applying ∇U to B(ξ, U) = 0 and using (2.4) and (2.11)~(2.13), we have

(∇UB) (ξ,U)=g(Aξ*U,Aξ*U),

due to (4.17). Taking X = ξ, Y = U and Z = U to (5.6) and using (2.12), (4.17), (5.9), (5.10) and the last two equations, we obtain

g(Aξ*U,Aξ*U)=3f2.

If M is a Hopf lightlike hypersurface of M̄ (c), that is, Aξ*U=λU for some smooth function λ, then g(Aξ*U,Aξ*U)=0. Thus f2 = 0.

Theorem 5.5

Let M be a totally umbilical lightlike hypersurface of an indefinite nearly generalized Sasakian space form M̄ (f1, f2, f3) with an (ℓ, m)-type connection such that ζ is tangent to M. Then

f2=0,         dτ(U,V)=0.
Proof

If M is totally umbilical, then B = 0 and m = 0 by (1) of Theorem 4.3. As B = m = 0 and S(TM) is non-degenerate, (2.13) is reduced

Aξ*X=0.

Taking X = ξ and Y = Z = U to (5.6) and using (4.8), (5.9) and (5.11), we get f2 = 0. Taking X = U and Y = V to (5.8) and using (5.11), we have dτ(U, V ) = 0. Thus we have our theorem.

Theorem 5.6

Let M be a screen totally umbilical lightlike hypersurface of an indefinite nearly generalized Sasakian space form M̄ (f1, f2, f3) with an (ℓ, m)-type connection such that ζ is tangent to M. Then

f1=ℓθ(∇UV-∇VU)-2m(m-α),f2=ℓθ(∇VU-∇UV)+m(m-α),f3=ℓθ(∇UV-∇VU)-2m(m-α)-ζℓ+ℓ2.
Proof

If M is screen totally umbilical, then C = 0 by (2) of Theorem 4.3. As C = 0, from (2.11) and (4.8), we have

2m=3α,         B(U,ζ)=α,         B(ζ,U)=α-m,         θ(∇ξU)=α-m.

Applying ∇̄X to θ(ζ) = 1 and θ(V ) = 0, we have

(∇¯Xθ) (ζ)=-ℓθ(X),         (∇¯Xθ) (V)=-θ(∇XV),

due to θ(∇̄Xζ) = ℓθ(X). Taking (1) X = ξ, Y = PZ = ζ; (2)X = ξ, Y = U, PZ = V; (3)X = ξ, Y = V, PZ = U to (5.7) and using (2.19), (5.9), (5.13) and (5.12), we have

f1-f3=ζℓ-ℓ2,         f1+2f2=-ℓθ(∇UV),f1+f2=-m(m-α)-ℓθ(∇VU).

From these equations, we have our theorem.

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