Lightlike Hypersurfaces of an Indeﬁnite Nearly Trans-Sasakian Manifold with an (ℓ,m)-type Connection

Chul Woo Lee; Jae Won Lee∗

doi:10.5666/KMJ.2020.60.2.223

Article

Kyungpook Mathematical Journal 2020; 60(2): 223-238

Published online June 30, 2020

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

Go to

Abstract
1. Introduction
2. On (ℓ, m)-type Connections
3. Recurrents and Lie Recurrents
4. Indefinite Nearly Trans-Sasakian Manifolds
5. Indefinite Nearly Generalized Sasakian Space Forms
References

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

Go to

Abstract
1. Introduction
2. On (ℓ, m)-type Connections
3. Recurrents and Lie Recurrents
4. Indefinite Nearly Trans-Sasakian Manifolds
5. Indefinite Nearly Generalized Sasakian Space Forms
References

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

(1.1)

({\bar{\nabla}}_{\bar{X}} \bar{g}) (\bar{Y}, \bar{Z}) = - ℓ {θ (\bar{Y}) \bar{g} (\bar{X}, \bar{Z}) + θ (\bar{Z}) \bar{g} (\bar{X}, \bar{Y})} - m {θ (\bar{Y}) \bar{g} (J \bar{X}, \bar{Z}) + θ (\bar{Z}) \bar{g} (J \bar{X}, \bar{Y})},

(1.2)

\bar{T} (\bar{X}, \bar{Y}) = ℓ {θ (\bar{Y}) \bar{X} - θ (\bar{X}) \bar{Y}} + m {θ (\bar{Y}) J \bar{X} - θ (\bar{X}) J \bar{Y}}

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

(1.3)

{\bar{\nabla}}_{\bar{X}} \bar{Y} = {\tilde{\nabla}}_{\bar{X}} \bar{Y} + θ (\bar{Y}) {ℓ \bar{X} + m J \bar{X}} .

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

Go to

Abstract
1. Introduction
2. On (ℓ, m)-type Connections
3. Recurrents and Lie Recurrents
4. Indefinite Nearly Trans-Sasakian Manifolds
5. Indefinite Nearly Generalized Sasakian Space Forms
References

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

T M = T M^{⊥} \oplus_{o r t h} S (T M),

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

\bar{g} (ξ, N) = 1, \bar{g} (N, N) = \bar{g} (N, X) = 0, \forall X \in S (T M) .

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

T \bar{M} = T M \oplus t r (T M) = {T M^{⊥} \oplus t r (T M)} \oplus_{o r t h} S (T M) .

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

(2.1)

{\bar{\nabla}}_{X} Y = \nabla_{X} Y + B (X, Y) N,

(2.2)

{\bar{\nabla}}_{X} N = - A_{N} X + τ (X) N;

(2.3)

\nabla_{X} P Y = \nabla_{X}^{*} P Y + C (X, P Y) ξ,

(2.4)

\nabla_{X} ξ = - A_{ξ}^{*} 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, A_N and $A_{ξ}^{*}$ 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

(2.5)

\begin{array}{l} J^{2} \bar{X} = - \bar{X} + θ (\bar{X}) ζ, J ζ = 0, θ \circ J = 0, θ (ζ) = 1, \\ θ (\bar{X}) = \bar{g} (ζ, \bar{X}), \bar{g} (J \bar{X}, J \bar{Y}) = \bar{g} (\bar{X}, \bar{Y}) - θ (\bar{X}) θ (\bar{Y}) . \end{array}

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 D_o and D with respect to J, i.e., J(D_o) = D_o and J(D) = D, such that

\begin{matrix} S (T M) = {J (T M^{⊥}) \oplus J (t r (T M))} \oplus_{o r t h} D_{o}, \\ D = T M^{⊥} \oplus_{o r t h} J (T M^{⊥}) \oplus_{o r t h} D_{o} . \end{matrix}

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

T M = D \oplus J (t r (T M)) .

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

(2.6)

U = - J N, V = - J ξ, 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

(2.7)

J X = F X + 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

(2.8)

F^{2} X = - X + u (X) U + θ (X) ζ .

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

(2.9)

(\nabla_{X} g) (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) \bar{g} (J X, Z) + θ (Z) \bar{g} (J X, Y)},

(2.10)

T (X, Y) = ℓ {θ (Y) X - θ (X) Y} + m {θ (Y) F X - θ (X) F Y},

(2.11)

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

(2.12)

B (X, ξ) = 0, B (ξ, X) = 0.

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

(2.13)

B (X, Y) = g (A_{ξ}^{*} X, Y) + m u (X) θ (Y),

(2.14)

C (X, P Y) = g (A_{N} X, P Y) + {ℓ η (X) + m v (X)} θ (P Y),

(2.15)

\bar{g} (A_{ξ}^{*} X, N) = 0, \bar{g} (A_{N} X, N) = 0.

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

(2.16)

A_{ξ}^{*} ξ = 0, {\bar{\nabla}}_{X} ξ = - A_{ξ}^{*} X - τ (X) ξ .

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

(2.17)

(\nabla_{X} F) ξ = - \nabla_{X} V + F (A_{ξ}^{*} X) - τ (X) V,

(2.18)

(\nabla_{X} F) V = - F \nabla_{X} V - A_{ξ}^{*} X - τ (X) ξ .

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

(2.19)

(\nabla_{X} v) (Y) = m θ (Y) η (X) - ℓ θ (Y) v (X) + B (X, U) η (Y) + g (Y, \nabla_{X} U) .

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

(2.20)

v (\nabla_{X} U) = 0, u (\nabla_{X} V) = 0.

3. Recurrents and Lie Recurrents

Go to

Abstract
1. Introduction
2. On (ℓ, m)-type Connections
3. Recurrents and Lie Recurrents
4. Indefinite Nearly Trans-Sasakian Manifolds
5. Indefinite Nearly Generalized Sasakian Space Forms
References

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

(3.1)

(\nabla_{X} F) Y = ω (X) F Y .

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

(3.2)

F \nabla_{X} V + A_{ξ}^{*} X + {ω (X) + τ (X)} ξ = 0.

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

(3.3)

\nabla_{X} V - F (A_{ξ}^{*} X) - {ω (X) - τ (X)} V = 0.

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

(3.4)

u (\nabla_{X} V) = 0, θ (\nabla_{X} V) = 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

(3.5)

(ℒ_{X} F) Y = σ (X) F Y,

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

(3.6)

(ℒ_{X} F) Y = [X, F Y] - F [X, Y] .

The structure tensor field F is called Lie parallel if ℒ_XF = 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

(3.7)

(\nabla_{X} F) Y = \nabla_{F Y} X - F \nabla_{Y} X + σ (X) F Y .

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

(3.8)

\nabla_{ξ} X = - F (\nabla_{X} V - \nabla_{V} X) - A_{ξ}^{*} X - {σ (X) + τ (X)} ξ .

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

(3.9)

F \nabla_{ξ} X = \nabla_{X} V - \nabla_{V} X - F (A_{ξ}^{*} X) - {σ (X) - τ (X)} V .

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

(3.10)

u (\nabla_{X} V - \nabla_{V} X) = 0, θ (\nabla_{X} V - \nabla_{V} X) = 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

Go to

Abstract
1. Introduction
2. On (ℓ, m)-type Connections
3. Recurrents and Lie Recurrents
4. Indefinite Nearly Trans-Sasakian Manifolds
5. Indefinite Nearly Generalized Sasakian Space Forms
References

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

(4.1)

({\tilde{\nabla}}_{\bar{X}} J) \bar{Y} + ({\tilde{\nabla}}_{\bar{Y}} J) \bar{X} = α {2 \bar{g} (\bar{X}, \bar{Y}) ζ - θ (\bar{Y}) \bar{X} - θ (\bar{X}) \bar{Y}} - β {θ (\bar{Y}) J \bar{X} + θ (\bar{X}) J \bar{Y}} .

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

(4.2)

\begin{array}{l} ({\bar{\nabla}}_{\bar{X}} J) \bar{Y} + ({\bar{\nabla}}_{\bar{Y}} J) \bar{X} = (m - α) {θ (\bar{Y}) \bar{X} + θ (\bar{X}) \bar{Y}} \\ - (ℓ + β) {θ (\bar{Y}) J \bar{X} + θ (\bar{X}) J \bar{Y}} + 2 {α \bar{g} (\bar{X}, \bar{Y}) - m θ (\bar{X}) θ (\bar{Y})} ζ . \end{array}

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

(4.3)

\nabla_{ζ} ζ = ℓ ζ, 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 $A_{ξ}^{*} X = ρ P X$ or equivalently

(4.4) $B (X, Y) = ρ g (X, Y) .$

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 A_NX = γPX or equivalently

(4.5) $C (X, P Y) = γ g (X, P Y) .$

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

(4.6)

({\bar{\nabla}}_{X} J) Y = (\nabla_{X} F) Y - u (Y) A_{N} X + B (X, Y) U + {(\nabla_{X} u) (Y) + u (Y) τ (X) + B (X, F Y)} N .

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

(4.7)

\begin{array}{l} (\nabla_{X} F) Y + (\nabla_{Y} F) X = (m - α) {θ (Y) X + θ (X) Y} \\ - (ℓ + β) {θ (Y) F X + θ (X) F Y} \\ + 2 {α g (X, Y) - m θ (X) θ (Y)} ζ \\ + u (X) A_{N} Y + u (Y) A_{N} X - 2 B (X, Y) U \\ + m {θ (Y) u (X) - θ (X) u (Y)} U . \end{array}

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

(4.8)

{\begin{array}{l} B (U, V) = C (V, V), & B (U, ζ) + C (V, ζ) = 2 (m - α), \\ B (U, U) = C (U, V), & v (\nabla_{U} V) = - τ (U), \\ C (U, ζ) = 0, & 2 C (V, ζ) + C (ζ, V) = 2 m - 3 α, \\ B (U, ζ) = C (V, ζ) + C (ζ, V) + α, \\ B (U, ζ) = m + θ (A_{ξ}^{*} U), & θ (\nabla_{ξ} U) = θ (A_{ξ}^{*} U), \end{array}

where ∇ is the induced connection from ∇̄.

Proof

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

(\nabla_{X} F) U = - F \nabla_{X} U, (\nabla_{X} F) V = - F \nabla_{X} V - A_{ξ}^{*} X - τ (X) ξ .

From these two equations, we obtain

(\nabla_{U} F) V + (\nabla_{V} F) U = - F (\nabla_{U} V + \nabla_{V} U) - A_{ξ}^{*} U - τ (U) ξ .

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

F (\nabla_{U} V + \nabla_{V} U) + A_{ξ}^{*} U + τ (U) ξ = - 2 α ζ - A_{N} V + 2 B (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 (\nabla_{U} ζ + \nabla_{ζ} U) = - A_{N} ζ + {α - 2 m + 2 B (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

(4.9)

2 B (U, ζ) - C (ζ, V) = 2 m - 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 (\nabla_{V} ζ + \nabla_{ζ} V) = - A_{ξ}^{*} ζ + 2 B (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 ((\nabla_{X} F) Y, N) = (\nabla_{X} v) (Y) - v (Y) τ (X) + g (A_{N} X, F Y) .

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

\begin{array}{l} (\nabla_{X} v) Y + (\nabla_{Y} v) 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 (A_{N} X, F Y) - g (A_{N} Y, F X) . \end{array}

Substituting (2.19) into the last equation, we have

\begin{array}{c} B (X, U) η (Y) + B (Y, U) η (X) + g (Y, \nabla_{X} U) + g (X, \nabla_{Y} U) \\ = - 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 (A_{N} X, F Y) - g (A_{N} Y, F X) . \end{array}

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 (\nabla_{ξ} U),

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

B (U, ζ) = m + θ (\nabla_{ξ} U) .

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 $A_{ξ}^{*}$ and A_N satisfy

(4.10)

\begin{matrix} A_{ξ}^{*} V = 0, A_{N} V = - 2 α ζ, A_{N} ξ = 0, θ (A_{ξ}^{*} U) = 0, \\ θ (\nabla_{ξ} U) = 0, A_{N} X = C (X, V) U - 2 α v (X) ζ . \end{matrix}

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

(4.11)

2 α {g (X, Y) - θ (X) θ (Y)} ζ + u (X) A_{N} Y + u (Y) A_{N} X - 2 B (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

(4.12)

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

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

2 B (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

(4.13)

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): $A_{ξ}^{*} V = 0$ . Taking X = U and Y = V to (4.11) and using (4.13), we get (4.10): A_N V = −2αζ. Also, taking X = U and Y = ξ to (4.11) and using (2.12), we get (4.10): A_N 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): $θ (A_{ξ}^{*} U) = θ (\nabla_{ξ} U) = 0 .$

Taking Y = U to (4.12), we obtain

2 B (X, U) + m θ (X) = u (A_{N} X) + u (X) u (A_{N} U) .

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

A_{N} X - u (A_{N} X) U + u (X) {A_{N} U - u (A_{N} U) U} + 2 α v (X) ζ = 0.

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

A_{N} X = u (A_{N} X) U - 2 α v (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. $A_{ξ}^{*}$ and A_N 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. $A_{ξ}^{*}$ and A_N 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

(4.14)

F (\nabla_{X} U - \nabla_{U} X) = 0.

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

(4.15)

v (\nabla_{U} X) = 0, τ (U) = 0,

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

(4.16)

\nabla_{ξ} U = - A_{ξ}^{*} U .

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

θ (\nabla_{ξ} U) = - θ (A_{ξ}^{*} U) .

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

(4.17)

θ (\nabla_{ξ} U) = θ (A_{ξ}^{*} 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

(4.18)

v (\nabla_{ξ} ζ) = - m .

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

v (\nabla_{ξ} X) = η (\nabla_{X} V - \nabla_{V} X) - 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

Go to

Abstract
1. Introduction
2. On (ℓ, m)-type Connections
3. Recurrents and Lie Recurrents
4. Indefinite Nearly Trans-Sasakian Manifolds
5. Indefinite Nearly Generalized Sasakian Space Forms
References

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

(5.1)

\begin{array}{l} \bar{R} (X, Y) Z = R (X, Y) Z + B (X, Z) A_{N} Y - B (Y, Z) A_{N} X \\ + {(\nabla_{X} B) (Y, Z) - (\nabla_{Y} B) (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 (F Y, Z) - θ (Y) B (F X, Z)]} N, \end{array}

(5.2)

\begin{array}{l} R (X, Y) P Z = R * (X, Y) P Z + C (X, P Z) A_{ξ}^{*} Y - C (Y, P Z) A_{ξ}^{*} X \\ + {(\nabla_{X} C) (Y, P Z) - (\nabla_{Y} C) (X, P Z) \\ - τ (X) C (Y, P Z) + τ (Y) C (X, P Z) \\ - ℓ [θ (X) C (Y, P Z) - θ (Y) C (X, P Z)] \\ - m [θ (X) C (F Y, P Z) - θ (Y) C (F X, P Z)]} ξ . \end{array}

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

(5.3)

\begin{array}{l} \tilde{R} (\bar{X}, \bar{Y}) \bar{Z} = f_{1} {\bar{g} (\bar{Y}, \bar{Z}) \bar{X} - \bar{g} (\bar{X}, \bar{Z}) \bar{Y}} \\ + f_{2} {\bar{g} (\bar{X}, J \bar{Z}) J \bar{Y} - \bar{g} (\bar{Y}, J \bar{Z}) J \bar{X} + 2 \bar{g} (\bar{X}, J \bar{Y}) J \bar{Z}} \\ + f_{3} {θ (\bar{X}) θ (\bar{Z}) \bar{Y} - θ (\bar{Y}) θ (\bar{Z}) \bar{X} + \bar{g} (\bar{X}, \bar{Z}) θ (\bar{Y}) ζ - \bar{g} (\bar{Y}, \bar{Z}) θ (\bar{X}) ζ}, \end{array}

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

f_{1} = \frac{c + 3}{4}, f_{2} = f_{3} = \frac{c - 1}{4}; f_{1} = \frac{c - 3}{4}, f_{2} = f_{3} = \frac{c + 1}{4}; f_{1} = f_{2} = f_{3} = \frac{c}{4}

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

(5.4)

\begin{array}{l} \bar{R} (\bar{X}, \bar{Y}) \bar{Z} = \tilde{R} (\bar{X}, \bar{Y}) \bar{Z} \\ + {ℓ ({\bar{\nabla}}_{\bar{X}} θ) (\bar{Z}) + [\bar{X} ℓ + m^{2} θ (\bar{X})] θ (\bar{Z})} \bar{Y} \\ - {ℓ ({\bar{\nabla}}_{\bar{Y}} θ) (\bar{Z}) + [\bar{Y} ℓ + m^{2} θ (\bar{Y})] θ (\bar{Z})} \bar{X} \\ + {m ({\bar{\nabla}}_{\bar{X}} θ) (\bar{Z}) + [\bar{X} m - ℓ m θ (\bar{X})] θ (\bar{Z})} J \bar{Y} \\ - {m ({\bar{\nabla}}_{\bar{Y}} θ) (\bar{Z}) + [\bar{Y} m - ℓ m θ (\bar{Y})] θ (\bar{Z})} J \bar{X} \\ + m θ (\bar{Z}) {({\bar{\nabla}}_{\bar{X}} J) \bar{Y} - ({\bar{\nabla}}_{\bar{Y}} J) \bar{X}} . \end{array}

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

(5.5)

\begin{array}{l} R (X, Y) Z = B (Y, Z) A_{N} X - B (X, Z) A_{N} Y \\ + {ℓ ({\bar{\nabla}}_{X} θ) (Z) + [X ℓ + m^{2} θ (X)] θ (Z)} Y \\ - {ℓ ({\bar{\nabla}}_{Y} θ) (Z) + [Y ℓ + m^{2} θ (Y)] θ (Z)} X \\ + {m ({\bar{\nabla}}_{X} θ) (Z) + [X m - ℓ m θ (X)] θ (Z)} F Y \\ - {m ({\bar{\nabla}}_{Y} θ) (Z) + [Y m - ℓ m θ (Y)] θ (Z)} F X \\ + m θ (Z) {(\nabla_{X} F) Y - (\nabla_{Y} F) X + u (X) A_{N} Y - u (Y) A_{N} X + m [θ (Y) u (X) - θ (X) u (Y)] U} \\ + f_{1} {g (Y, Z) X - g (X, Z) Y} \\ + f_{2} {\bar{g} (X, J Z) F Y - \bar{g} (Y, J Z) F X + 2 \bar{g} (X, J Y) F Z} \\ + f_{3} {[θ (X) Y - θ (Y) X] θ (Z) + [g (X, Z) θ (Y) - g (Y, Z) θ (X)] ζ}, \end{array}

(5.6)

\begin{array}{l} (\nabla_{X} B) (Y, Z) - (\nabla_{Y} B) (X, Z) \\ + {τ (X) - ℓ θ (X)} B (Y, Z) - {τ (Y) - ℓ θ (Y)} B (X, Z) \\ - m {θ (X) B (F Y, Z) - θ (Y) B (F X, Z)} \\ = {m ({\bar{\nabla}}_{X} θ) (Z) + [X m - ℓ m θ (X)] θ (Z)} u (Y) \\ - {m ({\bar{\nabla}}_{Y} θ) (Z) + [Y m - ℓ m θ (Y)] θ (Z)} u (X) \\ + m θ (Z) {(\nabla_{X} u) Y - (\nabla_{Y} u) X + u (Y) τ (X) - u (X) τ (Y) + B (X, F Y) - B (Y, F X)} \\ + f_{2} {\bar{g} (X, J Z) u (Y) - \bar{g} (Y, J Z) u (X) + 2 \bar{g} (X, J Y) u (Z)}, \end{array}

(5.7)

\begin{array}{l} (\nabla_{X} C) (Y, P Z) - (\nabla_{Y} C) (X, P Z) \\ - {τ (X) + ℓ θ (X)} C (Y, P Z) + {τ (Y) + ℓ θ (Y)} C (X, P Z) \\ - m {θ (X) C (F Y, P Z) - θ (Y) C (F X, P Z)} \\ = {ℓ ({\bar{\nabla}}_{X} θ) (P Z) + [X ℓ + m^{2} θ (X)] θ (P Z)} η (Y) \\ - {ℓ ({\bar{\nabla}}_{Y} θ) (P Z) + [Y ℓ + m^{2} θ (Y)] θ (P Z)} η (X) \\ + {m ({\bar{\nabla}}_{X} θ) (P Z) + [X m - ℓ m θ (X)] θ (P Z)} v (Y) \\ - {m ({\bar{\nabla}}_{Y} θ) (P Z) + [Y m - ℓ m θ (Y)] θ (P Z)} v (X) \\ + m θ (P Z) {(\nabla_{X} v) Y - (\nabla_{Y} v) X - v (Y) τ (X) + v (X) τ (Y) + g (A_{N} X, F Y) - g (A_{N} Y, F X)} \\ + f_{1} {g (Y, P Z) η (X) - g (X, P Z) η (Y)} \\ + f_{2} {\bar{g} (X, J P Z) v (Y) - \bar{g} (Y, J P Z) v (X) + 2 \bar{g} (X, J Y) v (P Z) \\ + f_{3} {θ (X) η (Y) - θ (Y) η (X)} θ (P Z), \end{array}

due to the following equations:

\begin{matrix} \bar{g} (({\bar{\nabla}}_{X} J) Y, ξ) = (\nabla_{X} u) (Y) + u (Y) τ (X) + B (X, F Y), \\ \bar{g} (({\bar{\nabla}}_{X} J) Y, N) = (\nabla_{X} v) (Y) - v (Y) τ (X) + g (A_{N} X, F Y) . \end{matrix}

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

\begin{array}{l} R (X, Y) ξ = - \nabla_{X}^{*} (A_{ξ}^{*} Y) + \nabla_{Y}^{*} (A_{ξ}^{*} X) + A_{ξ}^{*} [X, Y] - τ (X) A_{ξ}^{*} Y + τ (Y) A_{ξ}^{*} X \\ + {C (Y, A_{ξ}^{*} X) - C (X, A_{ξ}^{*} Y) - 2 d τ (X, Y)} ξ . \end{array}

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

R (X, Y) ξ = θ (A_{ξ}^{*} X) {ℓ Y + m F Y} - θ (A_{ξ}^{*} Y) {ℓ X + m F X} + f_{2} {u (Y) F X - u (X) F Y - 2 \bar{g} (X, J Y) V} .

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

(5.8)

f_{2} {u (Y) v (X) - u (X) v (Y)} = g (A_{N} Y, A_{ξ}^{*} X) - g (A_{N} X, A_{ξ}^{*} Y) - 2 d τ (X, Y) .

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

(5.9)

({\bar{\nabla}}_{X} θ) (U) = - θ (\nabla_{X} U), ({\bar{\nabla}}_{X} θ) (ξ) = θ (A_{ξ}^{*} 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 $A_{ξ}^{*}$ and A_N 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

(\nabla_{X} C) (Y, U) = - C (Y, \nabla_{X} U) .

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

\begin{array}{l} C (X, \nabla_{Y} U) - C (Y, \nabla_{X} U) = ({\bar{\nabla}}_{X} θ) (U) {ℓ η (Y) + m v (Y)} - ({\bar{\nabla}}_{Y} θ) (U) {ℓ η (X) + m v (X)} \\ + (f_{1} + f_{2}) {v (Y) η (X) - v (X) η (Y)} \end{array}

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

C (ξ, \nabla_{V} U) - C (V, \nabla_{ξ} U) = ℓ θ (\nabla_{V} U) + f_{1} + f_{2} .

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

\begin{array}{l} C (ξ, \nabla_{V} U) = g (A_{N} ξ, \nabla_{V} U) + ℓ θ (\nabla_{V} U) = ℓ θ (\nabla_{V} U), \\ C (V, \nabla_{ξ} U) = g (A_{N} V, \nabla_{ξ} U) + m θ (\nabla_{ξ} U) = - m θ (\nabla_{ξ} U) = 0. \end{array}

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 $A_{ξ}^{*}$ .

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 (A_{ξ}^{*} U, A_{ξ}^{*} U) = 3 f_{2} .

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

(5.10)

B (U, U) = 0.

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

(\nabla_{ξ} B) (U, U) = 2 g (A_{ξ}^{*} U, A_{ξ}^{*} U) .

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

(\nabla_{U} B) (ξ, U) = g (A_{ξ}^{*} U, A_{ξ}^{*} 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 (A_{ξ}^{*} U, A_{ξ}^{*} U) = 3 f_{2} .

If M is a Hopf lightlike hypersurface of M̄ (c), that is, $A_{ξ}^{*} U = λ U$ for some smooth function λ, then $g (A_{ξ}^{*} U, A_{ξ}^{*} U) = 0$ . 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

f_{2} = 0, d τ (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

(5.11)

A_{ξ}^{*} 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

\begin{array}{l} f_{1} = ℓ θ (\nabla_{U} V - \nabla_{V} U) - 2 m (m - α), \\ f_{2} = ℓ θ (\nabla_{V} U - \nabla_{U} V) + m (m - α), \\ f_{3} = ℓ θ (\nabla_{U} V - \nabla_{V} U) - 2 m (m - α) - ζ ℓ + ℓ^{2} . \end{array}

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

(5.12)

2 m = 3 α, B (U, ζ) = α, B (ζ, U) = α - m, θ (\nabla_{ξ} U) = α - m .

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

(5.13)

({\bar{\nabla}}_{X} θ) (ζ) = - ℓ θ (X), ({\bar{\nabla}}_{X} θ) (V) = - θ (\nabla_{X} V),

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

\begin{matrix} f_{1} - f_{3} = ζ ℓ - ℓ^{2}, f_{1} + 2 f_{2} = - ℓ θ (\nabla_{U} V), \\ f_{1} + f_{2} = - m (m - α) - ℓ θ (\nabla_{V} U) . \end{matrix}

From these equations, we have our theorem.

References

Go to

Abstract
1. Introduction
2. On (ℓ, m)-type Connections
3. Recurrents and Lie Recurrents
4. Indefinite Nearly Trans-Sasakian Manifolds
5. Indefinite Nearly Generalized Sasakian Space Forms
References

NS. Ageshe, and MR. Chafle. A semi-symmetric non-metric connection on a Riemannian manifold. Indian J Pure Appl Math., 23(6)(1992), 399-409.
P. Alegre, DE. Blair, and A. Carriazo. Generalized Sasakian space forms. Israel J Math., 141(2004), 157-183.
Călin. C. Contributions to geometry of CR-submanifold, Thesis, University of Iasi, Romania, 1998.
KL. Duggal, and A. Bejancu. Lightlike submanifolds of semi-Riemannian manifolds and applications, , Kluwer Acad. Publishers, Dordrecht, 1996.
DH. Jin. Geometry of lightlike hypersurfaces of an indefinite Sasakian manifold. Indian J Pure Appl Math., 41(4)(2010), 569-581.
DH. Jin. Lightlike hypersurfaces of an indefinite trans-Sasakian manifold with a non-metric φ-symmetric connection. Bull Korean Math Soc., 53(6)(2016), 1771-1783.
DH. Jin. Special lightlike hypersurfaces of indefinite Kaehler manifolds. Filomat., 30(7)(2016), 1919-1930.
DH. Jin. Lightlike hypersurfaces of an indefinite trans-Sasakian manifold with an (ℓ, m)-type connection. J Korean Math Soc., 55(5)(2018), 1075-1089.
DH. Jin. Lightlike hypersurfaces of an indefinite nearly trans-Sasakian manifold. An Ştiinţ Univ Al I Cuza Iaşi Mat (NS)., 65(2019), 195-210.