2. Almost contact manifolds with B-metric

DOI: 10.24193/subbmath.2019.4.12

Geometric characteristics and properties of a two-parametric family of Lie groups with

almost contact B-metric structure of the smallest dimension

Miroslava Ivanova and Lilko Dospatliev

Abstract. Almost contact B-metric manifolds of the lowest dimension 3 are con- structed by a two-parametric family of Lie groups. Our purpose is to determine the class of considered manifolds in a classification of almost contact B-metric manifolds and their most important geometric characteristics and properties.

Mathematics Subject Classification (2010):53C15, 53C50, 53D15.

Keywords:Almost contact B-metric manifold, Lie group, Lie algebra, indefinite metric.

1. Introduction

The study of the differential geometry of the almost contact B-metric manifolds has initiated in [5]. The geometry of these manifolds is a natural extension of the geometry of the almost complex manifolds with Norden metric [3, 6] in the case of odd dimension. Almost contact B-metric manifolds are investigated and studied for example in [5, 11, 12, 14, 15, 17, 18, 20].

Here, an object of special interest are the Lie groups considered as three- dimensional almost contact B-metric manifolds. For example of such investigation see [19].

The aim of the present paper is to make a study of the most important geometric characteristics and properties of a family of Lie groups with almost contact B-metric structure of the lowest dimension 3, belonging to the main vertical classes. These classes areF₄andF₅, where the fundamental tensorF is expressed explicitly by the metricg, the structure (ϕ, ξ, η) and the vertical components of the Lee forms θ and θ^∗, i.e. in this case the Lee forms are proportional to η at any point. These classes contain some significant examples as the time-like sphere ofg and the light cone of

the associated metric ofgin the complex Riemannian space, considered in [5], as well as the Sasakian-like manifolds studied in [7].

The paper is organized as follows. In Sec. 2, we give some necessary facts about almost contact B-metric manifolds. In Sec. 3, we construct and study a family of Lie groups as three-dimensional manifolds of the considered type.

2. Almost contact manifolds with B-metric

Let (M, ϕ, ξ, η, g) be a (2n+ 1)-dimensionalalmost contact B-metric manifold, i.e. (ϕ, ξ, η) is a triplet of a tensor (1,1)-fieldϕ, a vector fieldξand its dual 1-formη called an almost contact structure and the following identities holds:

ϕξ= 0, ϕ²=−Id +η⊗ξ, η◦ϕ= 0, η(ξ) = 1, where Id is the identity. The B-metricgis pseudo-Riemannian and satisfies

g(ϕx, ϕy) =−g(x, y) +η(x)η(y)

for arbitrary tangent vectorsx, y∈T_pM at an arbitrary pointp∈M [5].

Further, x, y,z,w will stand for arbitrary vector fields onM or vectors in the tangent space at an arbitrary point inM.

Let us note that the restriction of a B-metric on the contact distributionH = ker(η) coincides with the corresponding Norden metric with respect to the almost complex structure and the restriction of ϕ on H acts as an anti-isometry on the metric onH which is the restriction ofg onH.

The associated metric ˜g ofg onM is given by ˜g(x, y) =g(x, ϕy) +η(x)η(y). It is a B-metric, too. Hence, (M, ϕ, ξ, η,˜g) is also an almost contact B-metric manifold.

Both metricsgand ˜g are indefinite of signature (n+ 1, n).

The structure group of (M, ϕ, ξ, η, g) isG × I, whereI is the identity on span(ξ) andG=GL(n;C)∩ O(n, n).

The (0,3)-tensor F on M is defined by F(x, y, z) = g (∇xϕ)y, z

, where ∇ is the Levi-Civita connection ofg. The tensorF has the following properties:

F(x, y, z) =F(x, z, y) =F(x, ϕy, ϕz) +η(y)F(x, ξ, z) +η(z)F(x, y, ξ).

A classification of the almost contact B-metric manifolds is introduced in [5], where eleven basic classesFi (i= 1,2, . . . ,11) are characterized with respect to the properties of F. The special classF0 is defined by the condition F(x, y, z) = 0 and is contained in each of the other classes. Hence, F0 is the class of almost contact B-metric manifolds with∇-parallel structures, i.e. ∇ϕ=∇ξ=∇η=∇g=∇˜g= 0.

Let gij, i, j ∈ {1,2, . . . ,2n+ 1}, be the components of the matrix of g with respect to a basis{ei}²ⁿ⁺¹_i=1 ={e1, e2, . . . , e2n+1}ofTpM at an arbitrary pointp∈M, and g^ij – the components of the inverse matrix of (gij). The Lee forms associated withF are defined as follows:

θ(z) =g^ijF(ei, ej, z), θ^∗(z) =g^ijF(ei, ϕej, z), ω(z) =F(ξ, ξ, z).

In [12], thesquare norm of∇ϕis introduced by:

k∇ϕk²=g^ijg^ksg (∇e_iϕ)ek, ∇e_jϕ es

. (2.1)

If (M, ϕ, ξ, η, g) is anF0-manifold then the square norm of∇ϕis zero, but the inverse implication is not always true. An almost contact B-metric manifold satisfying the conditionk∇ϕk²= 0 is called anisotropic-F0-manifold. The square norms of∇η and

∇ξ are defined in [13] by:

k∇ηk²=g^ijg^ks(∇e_iη)ek ∇e_jη

es, k∇ξk²=g^ijg ∇e_iξ,∇e_jξ

. (2.2) Let R be the curvature tensor of type (1,3) of Levi-Civita connection ∇, i.e.

R(x, y)z=∇x∇yz− ∇y∇xz− ∇[x,y]z. The corresponding tensor ofRof type (0,4) is defined byR(x, y, z, w) =g(R(x, y)z, w).

The Ricci tensor ρ and the scalar curvature τ for R as well as their as- sociated quantities are defined by the following traces ρ(x, y) = g^ijR(e_i, x, y, e_j), τ=g^ijρ(e_i, e_j),ρ^∗(x, y) =g^ijR(e_i, x, y, ϕe_j) andτ^∗=g^ijρ^∗(e_i, e_j), respectively.

An almost contact B-metric manifold is called Einstein if the Ricci tensor is proportional to the metric tensor, i.e.ρ=λg, λ∈R.

Letαbe a non-degenerate 2-plane (section) inTpM. It is known from [20] that the special 2-planes with respect to the almost contact B-metric structure are: atotally real section ifαis orthogonal to itsϕ-imageϕα andξ, aϕ-holomorphic section ifα coincides withϕα and aξ-section ifξlies onα.

The sectional curvaturek(α;p)(R) of αwith an arbitrary basis {x, y} at pre- gardingRis defined by

k(α;p)(R) = R(x, y, y, x)

g(x, x)g(y, y)−g(x, y)². (2.3) It is known from [12] that a linear connectionD is called a natural connection on an arbitrary manifold (M, ϕ, ξ, η, g) if the almost contact structure (ϕ, ξ, η) and the B-metricg (consequently also ˜g) are parallel with respect to D, i.e. Dϕ=Dξ= Dη =Dg = D˜g = 0. In [18], it is proved that a linear connection D is natural on (M, ϕ, ξ, η, g) if and only ifDϕ=Dg= 0. A natural connection exists on any almost contact B-metric manifold and coincides with the Levi-Civita connection if and only if the manifold belongs toF0.

LetT be the torsion tensor of D, i.e.T(x, y) =Dxy−Dyx−[x, y].The corre- sponding tensor ofT of type (0,3) is denoted by the same letter and is defined by the conditionT(x, y, z) =g(T(x, y), z).

In [15], it is introduced a natural connection ˙D on (M, ϕ, ξ, η, g) in all basic classes by

D˙xy=∇xy+¹₂

(∇xϕ)ϕy+ (∇xη)y·ξ −η(y)∇xξ. (2.4) This connection is called a ϕB-connection in [16]. It is studied for the main classes F₁,F₄,F₅,F₁₁ in [15, 10, 11]. Let us note that the ϕB-connection is the odd- dimensional analogue of the B-connection on the almost complex manifold with Nor- den metric, studied for the classW1in [4].

In [17], a natural connection ¨Dis called aϕ-canonical connection on (M, ϕ, ξ, η, g) if its torsion tensor ¨T satisfies the following identity:

T¨(x, y, z)−T¨(x, z, y)−T¨(x, ϕy, ϕz) + ¨T(x, ϕz, ϕy)

=η(x)n

T¨(ξ, y, z)−T(ξ, z, y)¨ −T¨(ξ, ϕy, ϕz) + ¨T(ξ, ϕz, ϕy)o +η(y)n

T(x, ξ, z)¨ −T¨(x, z, ξ)−η(x) ¨T(z, ξ, ξ)o

−η(z)n

T¨(x, ξ, y)−T¨(x, y, ξ)−η(x) ¨T(y, ξ, ξ)o .

It is established that the ϕB-connection and the ϕ-canonical connection coincide if and only if (M, ϕ, ξ, η, g) is in the classF₁⊕ F₂⊕ F₄⊕ F₅⊕ F₆⊕ F₈⊕ F₉⊕ F₁₀⊕ F₁₁. In [8] it is determined the class of all three-dimensional almost contact B-metric manifolds. It isF1⊕ F4⊕ F5⊕ F8⊕ F9⊕ F10⊕ F11.

3. A family of Lie groups as three-dimensional (F

⊕ F

)-manifolds

In this section we study three-dimensional real connected Lie groups with almost contact B-metric structure. On a three-dimensional connected Lie groupGwe take a global basis of left-invariant vector fields{e₀, e₁, e₂}onG.

We define an almost contact structure on Gby

ϕe0=o, ϕe1=e2, ϕe2=−e1, ξ=e0;

η(e0) = 1, η(e1) =η(e2) = 0, (3.1) whereois the zero vector field and define a B-metric on Gby

g(e0, e0) =g(e1, e1) =−g(e2, e2) = 1,

g(e0, e1) =g(e0, e2) =g(e1, e2) = 0. (3.2) We consider the Lie algebragonG, determined by the following non-zero commuta- tors:

[e0, e1] =−be1−ae2, [e0, e2] =ae1−be2, [e1, e2] = 0, (3.3) where a, b ∈ R. We verify immediately that the Jacobi identity for g is satisfied.

Hence,Gis a 2-parametric family of Lie groups with corresponding Lie algebra g.

Theorem 3.1. Let(G, ϕ, ξ, η, g)be a three-dimensional connected Lie group with almost contact B-metric structure determined by (3.1), (3.2) and (3.3). Then it belongs to the class F4⊕ F5.

Proof. The well-known Koszul equality for the Levi-Civita connection∇ofg 2g(∇_eie_j, e_k) =g([e_i, e_j], e_k) +g([e_k, e_i], e_j) +g([e_k, e_j], e_i) (3.4) implies the following form of the componentsF_ijk=F(e_i, e_j, e_k) ofF:

2Fijk =g([ei, ϕej]−ϕ[ei, ej], ek) +g(ϕ[ek, ei]−[ϕek, ei], ej) +g([ek, ϕej]−[ϕek, ej], ei).

(3.5)

Using (3.5) and (3.3) for the non-zero componentsFijk, we get:

F101=F110=−F202=−F220=a,

F102=F120=F201=F210=b. (3.6) Immediately we establish that the components in (3.6) satisfy the condition F = F⁴+F⁵ which means that the manifold belongs to F4⊕ F5. Here, the components F^sofF in the basic classesFs(s= 4,5) have the following form (see [8])

F₄(x, y, z) =¹₂θ₀n

x¹ y⁰z¹+y¹z⁰

−x² y⁰z²+y²z⁰ ,

1

2θ₀=F₁₀₁=F₁₁₀=−F202=−F220; F₅(x, y, z) =¹₂θ^∗₀

x¹ y⁰z²+y²z⁰

+x² y⁰z¹+y¹z⁰ ,

1

2θ^∗₀=F₁₀₂=F₁₂₀=F₂₀₁=F₂₁₀.

(3.7)

whereθ0=θ(e0) andθ₀^∗=θ^∗(e0) are determined byθ0= 2a,θ₀^∗= 2b. Therefore, the induced three-dimensional manifold (G, ϕ, ξ, η, g) belongs to the classF4⊕ F5 from the mentioned classification. It is anF0-manifold if and only if (a, b) = (0,0) holds.

Obviously, (G, ϕ, ξ, η, g) belongs toF4,F5 andF0if and only if the parameters θ^∗₀ vanishes if the manifold belongs to F4, and θ0 vanishes if it belong to F5, and θ0=θ^∗₀ vanishes if it belong toF0, respectively.

According to the above, the commutators in (3.3) take the form [e₀, e₁] =−¹₂(θ^∗₀e₁+θ₀e₂), [e₀, e₂] = ¹₂(θ₀e₁−θ₀^∗e₂),

[e1, e2] = 0, (3.8)

in terms of the basic components of the Lee formsθ andθ^∗. According to Theorem 3.1 and the consideration in [9], we can remark that the Lie algebra determined as above belongs to the typeBia(V II_h),h >0 of the Bianchi classification (see [1, 2]).

Using (3.4) and (3.3), we obtain the components of∇:

∇e1e0=be1+ae2, ∇e1e1=−be0, ∇e1e2=ae0,

∇e2e₀=−ae1+be₂, ∇e2e₁=ae₀, ∇e2e₂=be₀. (3.9) We denote byRijkl =R(ei, ej, ek, el) the components of the curvature tensorR, ρjk =ρ(ej, ek) of the Ricci tensor ρ, ρ^∗_jk =ρ^∗(ej, ek) of the associated Ricci tensor ρ^∗ and kij =k(ei, ej) of the sectional curvature for ∇ of the basic 2-plane αij with a basis {ei, ej}, where i, j ∈ {0,1,2}. On the considered manifold (G, ϕ, ξ, η, g) the basic 2-planesαij of special type are: a ϕ-holomorphic section —α12 andξ-sections

—α01,α02. Further, by (2.3), (3.2), (3.3) and (3.9), we compute

−R₀₁₀₁=R₀₂₀₂= ¹₂ρ₀₀=k₀₁=k₀₂=¹₄(θ²₀−θ₀^∗2), R₀₁₀₂=R₀₂₀₁=−ρ₁₂=−¹₂ρ^∗₀₀=−¹₂τ^∗=−¹₂θ₀θ₀^∗, R₁₂₁₂=ρ^∗₁₂=k₁₂=−¹₄(θ₀²+θ^∗2₀ ), ρ₁₁=−ρ₂₂=−¹₂θ^∗2₀ ,

τ= ¹₂(θ₀²−3θ^∗2₀ ).

(3.10)

The rest of the non-zero components ofR,ρandρ^∗ are determined by (3.10) and the propertiesRijkl=Rklij, Rijkl =−Rjikl=−Rijlk, ρjk=ρkjandρ^∗_jk=ρ^∗_kj.

Taking into account (2.1), (2.2), (3.1), (3.2) and (3.9), we have

k∇ϕk²=−2k∇ηk²=−2k∇ξk²=θ₀²−θ^∗2₀ . (3.11) Proposition 3.2. The following characteristics are valid for(G, ϕ, ξ, η, g):

1. The ϕB-connection D˙ (respectively, ϕ-canonical connection D) is zero in the¨ basis{e0, e₁, e₂}.

2. The manifold is an isotropic-F0-manifold if and only if the conditionθ₀ =±θ^∗₀ is valid.

3. The manifold is flat if and only if it belongs to F₀.

4. The manifold is Ricci-flat (respectively,∗-Ricci-flat) if and only if it is flat.

5. The manifold is scalar flat if and only if the condition θ0=±√

3θ^∗₀ holds.

6. The manifold is ∗-scalar flat if and only if it belongs to eitherF4 orF5. Proof. Using (2.4), (3.1) and (3.9), we get immediately the assertion (1). Equation (3.11) implies the assertion (2). The assertions (5), (3) and (6) hold, according to (3.10). On the three-dimensional almost contact B-metric manifold with the basis {e0, e1, e2}, bearing in mind the definitions of the Ricci tensorρand theρ^∗, we have

ρjk=R0jk0+R1jk1−R2jk2 ρ^∗_jk=R1kj2+R2jk1.

By virtue of the latter equalities, we get the assertion (4).

According to (3.6) and (3.10) we establish the truthfulness of the following

Proposition 3.3. The following properties are equivalent for the studied manifold (G, ϕ, ξ, η, g):

1. it belongs toF4; 2. it isη-Einstein;

3. the Lee formθ^∗ vanishes.

Using again (3.6) and (3.10)we establish the truthfulness of the following

Proposition 3.4. The following properties are equivalent for the studied manifold (G, ϕ, ξ, η, g):

1. it belongs toF₅; 2. it is Einstein;

3. it is a hyperbolic space form with k=−¹₄θ₀^∗2; 4. the Lee formθ vanishes.

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Miroslava Ivanova

Trakia University, Department of Informatics and Mathematics, 6000 Stara Zagora, Bulgaria e-mail:mivanova [email protected]

Lilko Dospatliev

Trakia University, Department of Pharmacology, Animal Physiology and Physiological Chemistry, 6000 Stara Zagora, Bulgaria

e-mail:[email protected]