Weakly ϕ-Symmetric Contact Metric Spaces

Domenico Perrone

Abstract The examples, that we denote by G_{w}, given in [2] of contact metric spaces which are weakly locally ϕ -symmetric, but not strongly, satisfy the stronger condition that their contact metric structure is homogeneous. In this paper we give the first example of weakly locally ϕ-symmetric space which is not homogeneous, consequently these spaces form a larger class. Moreover, we show that the examples G_{w} are the only 3-dimensional weakly, but not strongly, locally ϕ-symmetric spaces which have constant scalar curvature and vertical Ricci curvature.

Mathematics Subject Classification:53D10, 53C25, 53C30. Keywords and phrases: weakly ϕ-symmetric, strongly ϕ-symmetric, three-manifolds.

1 Introduction

A locally symmetric Sasakian manifold (or K-contact) manifold is of constant sectional curvature 1 (see [7],[10]). For this reason, Takahashi [9] introduced the notion of a locally ϕ-symmetric space. It is a Sasakian manifold satisfying the condition

(∇_{X}R)(Y, Z, V, W) = 0 (1.1)

for all vector fields X, Y, Z, V, W orthogonal to the characteristic vector field ξ. This notion has been for the most part explored only in the Sasakian context and it is not clear what the corresponding notion should be for a general contact metric manifold.

D.E.Blair, T. Koufogiorgos and R.Sharma [1] extended the Takahashi’s notion to a general contact metric manifold (M, η, g, ξ, ϕ) by using the same curvature condition (1.1); moreover they proved that a 3-dimensional contact metric manifold , with Qϕ = ϕQ, is weakly locally ϕ-symmetric iff it is of constant scalar curvature. In the Sasakian case, the condition (1.1) is satisfied if and only if all characteristic reflections are (local) isometries. Then E. Boeckx and L. Vanhecke [3] gave the following new generalization : a contact metric manifold is called locally ϕ-simmetric if and only if all characteristic reflections are (local) isometries. To distinguish between the two definitions, since the first is weaker than the second, following [2] we speak of weakly locally ϕ-symmetric spaces (for the first one) and strongly locally ϕ-symmetric (for the second one).

∗ Balkan Journal of Geometry and Its Applications, Vol.7, No.2, 2002, pp. 67-77.

\begin{matrix}c \\ ◯ \end{matrix} Balkan Society of Geometers, Geometry Balkan Press 2002.

In [4] was determined all 3-dimensional strongly locally ϕ-symmetric spaces and proved that a contact metric three-manifold is strongly locally ϕ-symmetric if and only if it is a locally homogeneous contact metric manifold satisfying the condition σ(X) = 0 ∀X ∈ Kerη, where σ(X) denotes the vertical Ricci curvature ρ(X, ξ) = g(Qξ, X). Recently E. Boeckx, P. Bunken and L. Vanhecke [2] give the first examples of contact metric spaces which are weakly locally ϕ-symmetric, but not strongly. These examples are non-unimodular Lie group of dimension three, that we denote by G_{w}, equipped with a left invariant contact metric structure which depends by a parameter w ∈ R, w < 0. We note that the parameter w is completely determined by the Webster scalar curvature (see Remark 4.1).

In this paper we show that the spaces G_{w} are the only weakly locally ϕ-symmetric, but not strongly, with constant scalar curvature and vertical Ricci curvature σ(X) (see Theorem 4.1). The examples G_{w} satisfy the stronger condition that their contact metric structure is homogeneous. So, one natural question is to see if there exist weakly locally ϕ-symmetric three-spaces which are not homogeneous. We give a positive answer to this question (see Theorem 4.3), consequently the weakly locally ϕ-symmetric spaces form a larger class. In the last section, we show that the unit tangent sphere bundle of a Riemannian two-manifold (M, G) is weakly locally ϕ-symmetric if and only if (M, G) has constant sectional curvature.

2 Preliminaries on contact metric manifolds

In this section we collect some basic facts about contact metric manifolds. All manifolds are assumed to be connected and smooth. A (2n + 1)-dimensional manifold M has an almost contact structure if it admits a vector field ξ (the characteristic field), a one-form η and a (1, 1)-tensor field ϕ satisfying

η(ξ) = 1, ϕ^{2} = −I + η ⊗ ξ.

Then one can always find a Riemannian metric g which is compatible with the structure, that is, such that

g(ϕX, ϕY) = g(X, Y) − η(X)η(Y)

for all vector fields X and Y. (ξ, η, ϕ, g) is called an almost contact metric structure and (M, ξ, η, ϕ, g) an almost contact metric manifold. If additionally it holds dη(X, Y) = g(X, ϕY), then (M, ξ, η, ϕ, g) is called a contact metric manifold. In what follows we denote by ∇ the Levi Civita connection and by R the corresponding Riemann curvature tensor given by R_{XY} = ∇_{[X,Y]} − [∇_{X}, ∇_{Y}] for all smooth vector fields X, Y. Moreover, we denote by ρ the Ricci tensor of type (0, 2), by Q the corresponding endomorphism field and by r the scalar curvature. We note that σ(X) := g(Qξ, X) = 0 ∀X ∈ Kerη iff Qξ is parallel to ξ, moreover Qϕ = ϕQ implies Qξ is parallel to ξ. The tensor h = \frac{1}{2}L_{ξ}ϕ, where L denotes the Lie derivative, is symmetric and satisfies −ϕh = ∇ξ + ϕ = hϕ. A contact metric space is said to be a K-contact manifold if ξ is a Killing vector field, or equivalently, h = 0. For a three-dimensional contact metric manifold, the Webster scalar curvature W (see [5]) and the ϕ−sectional curvatureH are given by

W = \frac{1}{8}(r − ρ(ξ, ξ) + 4) 2H = r − 4(1 − λ^{2}) = r − 2ρ(ξ, ξ), (2.1)

moreover, the contact metric structure is K-contact iff it is Sasakian.

Next, let (M, ξ, η, ϕ, g) be a three-dimensional contact metric manifold and m a point of M. Then there exists a local orthonormal basis {ξ, e_{1}, e_{2} = ϕe_{1}} of smooth eigenvectors of h in a neigborhood of m. Now, let U_{1} be the open subset of M where h ≠ 0 and let U_{2} be the open subset of points m ∈ M such that h = 0 in a neighborhood of m. U_{1} ∪ U_{2} is an open dense subset of M. On U_{1} we put he_{1} = λe_{1} and hence, from (2.4) we have he_{2} = −λe_{2} where λ is a non-vanishing smooth function. Then we have

Lemma 2.1 [4] On U_{1} we have

\begin{matrix}∇_{ξ}e_{1} = −ae_{2} ,  ∇_{ξ}e_{2} = ae_{1} , \\ ∇_{e_{1}}ξ = −(λ + 1)e_{2} ,  ∇_{e_{2}}ξ = −(λ − 1)e_{1} , \\ ∇_{e_{1}}e_{1} = \frac{1}{2λ}\{(e_{2})(λ) + A\}e_{2} , ∇_{e_{2}}e_{2} = \frac{1}{2λ}\{e_{1}(λ) + B\}e_{1} , \\ ∇_{e_{1}}e_{2} = −\frac{1}{2λ}\{(e_{2})(λ) + A\}e_{1} + (λ + 1)ξ , \\ ∇_{e_{2}}e_{1} = −\frac{1}{2λ}\{e_{1}(λ) + B\}e_{2} + (λ − 1)ξ , \end{matrix} (2.2)

[e_{1}, e_{2}] = −\frac{1}{2λ}\{(e_{2})(λ) + A)\}e_{1} + \frac{1}{2λ}\{(e_{1})(λ) + B\}e_{2} + 2ξ , (2.3)

where A = ρ(ξ, e_{1}), B = ρ(ξ, e_{2}) and a is a smooth function.

Finally, we recall that the components of the Ricci operator Q, with respect to {ξ, e_{1}, e_{2} = ϕe}, are given by (see [8])

\{ \begin{matrix}Qξ = 2(1 − λ^{2})ξ + Ae_{1} + Be_{2}, \\ Qe_{1} = Aξ + (\frac{r}{2} − 1 + λ^{2} + 2aλ)e_{1} + ξ(λ)e_{2}, \\ Qe_{2} = Bξ + ξ(λ)e_{1} + (\frac{r}{2} − 1 + λ^{2} − 2aλ)e_{2}, \end{matrix}

from which it follows easily

(∇_{ξ}Q)ξ = −4λξ(λ)ξ + {ξ(A) + aB}e_{1} + {ξ(B) − aA}e_{2}, (2.4)

(∇_{e_{1}}Q)e_{1} = \{e_{1}(A) + (λ + 1)ξ(λ) − \frac{B}{2λ}[e_{2}(λ) + A]\}ξ

+\{e_{1}(\frac{r}{2} + λ^{2} + 2aλ) − \frac{ξ(λ)}{λ}[e_{2}(λ) + A]\}e_{1} (2.5)

+{e_{1}ξ(λ) + 2a(e_{2})(λ) + (2a − λ − 1)A}e_{2},

(∇_{e_{2}}Q)e_{2} = \{e_{2}(B) + (λ − 1)ξ(λ) − \frac{A}{2λ}[e_{1}(λ) + B]\}ξ

+{e_{2}(ξ)λ − 2ae_{1}(λ) + (1 − λ − 2a)B}e_{1} (2.6)

+\{e_{2}(\frac{r}{2} + λ^{2} − 2aλ) − \frac{ξ(λ)}{λ}[e_{1}(λ) + B]\}e_{2},

(∇_{e_{1}}Q)e_{2} = \{e_{1}(B) + (λ + 1)(\frac{r}{2} + 3λ^{2} − 3 − 2aλ) +

\frac{A}{2λ}[e_{2}(λ) + A]\}ξ + \{e_{1}ξ(λ) + 2ae_{2}(λ) + A(2a − λ − 1)\}e_{1} (2.7)

+\{e_{1}(\frac{r}{2} + λ^{2} − 2aλ) − 2B(λ + 1) + \frac{ξ(λ)}{λ}[e_{2}(λ) + A]\}e_{2},

(∇_{e_{2}}Q)e_{1} = \{e_{2}(A) + (λ − 1)(\frac{r}{2} + 3λ^{2} − 3 + 2aλ) +

+\frac{B}{2λ}[e_{1}(λ) + B]\}ξ + \{e_{2}(\frac{r}{2} + λ^{2} + 2aλ) − 2A(λ − 1) (2.8)

+\frac{ξ(λ)}{λ}[(e_{1})(λ) + B]\}e_{1} + \{e_{2}ξ(λ) − 2ae_{1}(λ) + B(1 − 2a − λ)\}e_{2}.

3 A characterization of weakly locally ϕ-symmetric contact metric three-manifolds

In the sequel we denote by M a contact metric three-manifold and by (η, g, ϕ, ξ) its contact metric structure.

Lemma 3.1 A contact metric three-manifold M is weakly locally ϕ-symmetric if and only if

\{ \begin{matrix}e_{1}(H) = 2B(λ + 1) \\ e_{2}(H) = 2A(λ − 1). \end{matrix} (3.1)

where H is the ϕ-sectional curvature and λ is the eigenvalue corresponding to the eigenvector e_{1}.

Proof. From (1.1) follows that M is weakly locally ϕ-symmetric if and ony if

(∇_{V} R)(X, Y, Z) = g((∇_{V} R)(X, Y, Z), ξ)ξ

for any X, Y, Z, V ∈ Ker η. Since dimM = 3, we have the well-known formula

R(X, Y)Z = g(X, Z)QY − g(Y, Z)QX + ρ(X, Z)Y − ρ(Y, Z)X +

−\frac{r}{2}\{g(X, Z)Y − g(Y, Z)X\},

for all X, Y, Z vector fields on M. Therefore, we have

(∇_{e_{1}}R)(e_{1}, e_{2}, e_{1}) =  (∇_{e_{1}}Q)e_{2} + g((∇_{e_{1}}Q)e_{1}, e_{1})e_{2}

−g((∇_{e_{1}}Q)e_{2}, e_{1})e_{1} − e_{1}(\frac{r}{2})e_{2},

(∇_{e_{1}}R)(e_{1}, e_{2}, e_{2}) =  −(∇_{e_{1}}Q)e_{1} + g((∇_{e_{1}}Q)e_{1}, e_{2})e_{2}

−g((∇_{e_{1}}Q)e_{2}, e_{2})e_{2} + e_{1}(\frac{r}{2})e_{1},

(∇_{e_{2}}R)(e_{1}, e_{2}, e_{1}) =  (∇_{e_{2}}Q)e_{2} + g((∇_{e_{2}}Q)e_{1}, e_{1})e_{2}

−g((∇_{e_{2}}Q)e_{2}, e_{1})e_{1} − e_{2}(\frac{r}{2})e_{2},

(∇_{e_{2}}R)(e_{1}, e_{2}, e_{2}) =  −(∇_{e_{2}}Q)e_{1} + g((∇_{e_{2}}Q)e_{1}, e_{2})e_{2}

−g((∇_{e_{2}}Q)e_{2}, e_{2})e_{1} + e_{2}(\frac{r}{2})e_{1}.

Consequently (∇_{e_{1}}R)(e_{1}, e_{2}, e_{1}) and (∇_{e_{2}}R)(e_{1}, e_{2}, e_{1}) are parallel to ξ if and only if holds the following

\{ \begin{matrix}\frac{1}{2}e_{1}(r) = g(∇_{e_{1}}Q)e_{1}, e_{1}) + g(∇_{e_{1}}Q)e_{2}, e_{2}) \\ \frac{1}{2}e_{2}(r) = g(∇_{e_{2}}Q)e_{1}, e_{1}) + g(∇_{e_{2}}Q)e_{2}, e_{2}). \end{matrix} (3.2)

Imposing that the other components (∇_{e_{1}}R)(e_{1}, e_{2}, e_{2}) and (∇_{e_{2}}R)(e_{1}, e_{2}, e_{2}) are parallel to ξ we get the same condition (3.2). If M is Sasakian, then Qξ = 2ξ, Qe_{1} = (\frac{r}{2} − 1)e_{1}, Qe_{2} = \frac{r}{2} − 1)e_{2}, from which it follows

ξ(r) = g((∇_{ξ}Q)ξ, ξ) + g((∇_{e_{1}}Q)e_{1}, ξ) + g((∇_{e_{2}}Q)e_{2}, ξ) = 0,

and hence

r = const. ⟺ e_{1}(r) = e_{2}(r) = 0.

But r = 4 + 2H, so ξ(H) = ξ(r) = 0 and r = const. ⇔ H = const. ⇔ e_{1}(H) = e_{2}(H) = 0. Moreover (see [11]): M is locally ϕ-symmetric ⇔ r = const. Therefore, we get the statement of Lemma 3.1, since for M Sasakian A = B = 0.

Now assume that M is not Sasakian. From (2.5)-(2.8) we have

g((∇_{e_{1}}Q)e_{1}, e_{1}) = \frac{e_{1}(r)}{2} + e_{1}(λ^{2}) + e_{1}(2aλ) − \frac{ξ(λ)}{λ}\{e_{2}(λ) + A\},

g((∇_{e_{1}}Q)e_{2}, e_{2}) = \frac{e_{1}(r)}{2} + e_{1}(λ^{2}) − e_{1}(2aλ) + \frac{ξ(λ)}{λ}\{e_{2}(λ) + A\} − 2B(λ + 1),

g((∇_{e_{2}}Q)e_{1}, e_{1}) = \frac{e_{2}(r)}{2} + e_{2}(λ^{2}) + e_{2}(2aλ) + \frac{ξ(λ)}{λ}\{e_{2}(λ) + B\} − 2A(λ − 1),

g((∇_{e_{2}}Q)e_{2}, e_{2}) = \frac{e_{2}(r)}{2} + e_{2}(λ^{2}) − e_{2}(2aλ) − \frac{ξ(λ)}{λ}\{e_{1}(λ) + B\}.

Then, using 3.2, M is weakly locally ϕ-symmetric if and ony if

\{ \begin{matrix}\frac{1}{2}e_{1}(r) + 2e_{1}(λ^{2}) = 2B(λ + 1) \\ \frac{1}{2}e_{2}(r) + 2e_{2}(λ^{2}) = 2A(λ − 1). \end{matrix} (3.3)

Then, by (2.1), (3.3) is equivalent to (3.1).

Corollary 3.2 If Qξ is parallel to ξ, then M is weakly locally ϕ-symmetric if and only if it has constant ϕ-sectional curvature.

4 Main results

Theorem 4.1 Let M be a 3-dimensional contact metric manifold. Then M is weakly locally ϕ-symmetric with constant scalar curvature and vertical Ricci curvature if, and only if, either M is strongly locally ϕ-symmetric or it is locally isometric to a Lie group G_{w}.

Proof. The necessary condition is trivial. We show the sufficient condition. In the Sasakian case, the two definition are equivalent, so we have to consider only the non Sasakian case. Then the set U_{1} ≠ ∅ where we suppose λ < 0. Since r = const., from Lemma 3.1 we have

\{ \begin{matrix}2λe_{1}(λ) = B(λ + 1) \\ 2λe_{2}(λ) = A(λ − 1), \end{matrix} (4.1)

and hence

2λ[e_{1}, e_{2}](λ^{2}) = 2λ(λ − 1)e_{1}(A) − 2λ(λ + 1)e_{2}(B) + 2AB. (4.2)

Moreover, by Lemma 2.1 and (4.1), we have

2λ[e_{1}, e_{2}](λ^{2}) = −2AB + 8λ^{2}ξ(λ), (4.3)

and, by (4.1) and 0 = ξ(r) = g((∇_{ξ}Q)ξ, ξ) + g((∇_{e_{1}}Q)e_{1}, ξ) + g((∇_{e_{2}}Q)e_{2}, ξ), we get

8λ^{2}ξ(λ) = 4λe_{1}(A) + 4λe_{2}(B) − 2(Be_{2}(λ) + Ae_{1}(λ)) − 4AB

= 4λe_{1}(A) + 4λe_{2}(B) − 6AB. (4.4)

From (4.2) and (4.3) we have

4λ^{2}ξ(λ) = λ(λ − 1)e_{1}(A) − λ(λ + 1)e_{2}(B) + 2AB (4.5)

which, using (4.4), gives

λ(λ − 3)e_{1}(A) − λ(λ + 3)e_{2}(B) + 5AB = 0. (4.6)

Since ρ(ξ, e_{i}) = const., from (4.6) and (4.5) we get AB = 0 and ξ(λ) = 0. Now, we

consider separately the cases A = B = 0; A ≠ 0, B = 0; A = 0, B ≠ 0. Case A=B=0. In this case, (4.1) gives e_{1}(λ) = e_{2}(λ) = 0 and hence, since ξ(λ) = 0, we have λ constant. Now, using the formula

e_{i}(r) = g((∇_{ξ}Q)ξ, e_{i}) + g((∇_{e_{1}}Q)e_{1}, e_{i}) + g((∇_{e_{2}}Q)e_{2}, e_{i}), (4.7)

for i=1,2, and (2.4)-(2.6), since r and λ are constant, we get e_{1}(a) = e_{2}(a) = 0. Moreover, ξ(a) = [e_{1}, e_{2}](a) = 0. So, also a is constant. Then, applying Theorem 3.1 of [8] and theorem 5.1 of [4], we get that M is strongly locally locally ϕ-symmetric.

Case A=0, B≠ 0. From (4.1) we have e_{2}(λ) = 0 and e_{1}(λ) = \frac{B}{2λ}(λ + 1). But, see lamma 2.1, (a+λ−1)e_{1}(λ) = [ξ, e_{2}](λ) = 0. Therefore either a = 1−λ or e_{1}(λ) = 0. Assume a = 1 − λ. Then lemma 2.1 gives

\begin{matrix}∇_{ξ}e_{1} = (λ − 1)e_{2} , & ∇_{ξ}e_{2} = (1 − λ)e_{1} , \\ ∇_{e_{1}}ξ = −(λ + 1)e_{2} , & ∇_{e_{2}}ξ = −(λ − 1)e_{1} , \\ ∇_{e_{1}}e_{1} = 0, & ∇_{e_{2}}e_{2} = \frac{B(1 + 3λ)}{4λ^{2}}e_{1} , \\ ∇_{e_{1}}e_{2} = (λ + 1)ξ, & ∇_{e_{2}}e_{1} = −\frac{B(1 + 3λ)}{4λ^{2}}e_{2} + (λ − 1)ξ, \end{matrix} (4.8)

and

[e_{1}, e_{2}] = \frac{B(1 + 3λ)}{4λ^{2}}e_{2} + 2ξ. (4.9)

Consequently using (4.8),

\begin{aligned}R(e_{1}, e_{2})e_{1} & = & −∇_{e_{1}}∇_{e_{2}}e_{1} + ∇_{e_{2}}∇_{e_{1}}e_{1} + ∇_{[e_{1}e_{2}]}e_{1} = \\ & = & \{−\frac{B^{2}}{16λ^{4}} (15λ^{2} + 16λ + 5) + (λ − 1)(λ + 3)\} e_{2} + Bξ. \end{aligned}

On the other hand 2g(R(e_{1}, e_{2})e_{1}, e_{2}) = 2H = r − 4(1 − λ^{2}), therefore we obtain

8λ^{4}{r + 2(λ − 1)^{2}} + B^{2}(15λ^{2} + 16λ + 5) = 0.

This equation, since B, r are constant, implies λ = const.(≠ 0) and hence, by (4.1), λ = −1 and a = 2. Assuming e_{1}(λ) = 0, we have λ = const. = −1. In this case

Qξ = Be_{2},  Qe_{1} = (\frac{r}{2} − 2a) e_{1},  Qe_{2} = Bξ + (\frac{r}{2} + 2a) e_{2},

and hence applying formula (4.7), for i=1,2, we get

e_{2}(a) = 0, 2e_{1}(a) = (2 − a)B. (4.10)

Moreover [e_{1}, e_{2}] = −\frac{B}{2} e_{2} + 2ξ, so (4.10) gives

−\frac{B}{2} e_{2}(a) + 2ξ(a) = [e_{1}, e_{2}](a) = e_{1}e_{2}(a) − e_{2}e_{1}(a) = −e_{2} \{\frac{2 − a}{2}B\} = 0,

from which we have ξ(a) = 0. Then by (2.2)

(a − 2)e_{1}(a) = [ξ, e_{2}](a) = ξe_{2}(a) − e_{1}ξ(a) = 0.

gives a = const., and by (4.10), a = 2. Thus, we have

[e_{1}, e_{2}] = −\frac{B}{2} e_{2} + 2ξ, [ξ, e_{2}] = 0, [e_{1}, ξ] = 2e_{2}.

So, M is locally isometric to a Lie group G_{w} (see [2],[8]). Case A≠0, B=0. We show that this case can not occur. A ≠ 0, B = 0 and λ < 0, by (4.1), imply

e_{1}(λ) = 0, e_{2}(λ) = \frac{A(λ − 1)}{2λ} ≠ 0.

Then computing R(e_{1}, e_{2})e_{1} as in the before case, we get λ = const. which contradicts e_{2}(λ) ≠ 0.

Corollary 4.2 A 3-dimensional homogeneous contact metric manifold is weakly, but not strongly, locally ϕ-symmetric iff it is locally isometric to a Lie group G_{w}.

Remark 4.1 (i) If in the proof of theorem 4.1 we assume λ > 0, then can not occur the case A = 0, B ≠ 0. (ii) The non unimodular Lie group G_{w} is associated to the Lie algebra

[e_{1}, e_{2}] = αe_{2} + 2ξ, [e_{1}, ξ] = 2e_{2}, [ξ, e_{2}] = 0,

hence it is determined by the Milnor’s isomorphism invariant D [6] given by: D =

−\frac{8γ}{α^{2}} = −\frac{16}{α^{2}} < 0. In our case α = −\frac{B}{2} . On the other hand, computing the Webster

scalar curvature of G_{w}, using (2.1), we find W = −\frac{α^{2}}{4} − \frac{1}{2} < 0. So D, and hence G_{w}, is determined by the Webster scalar curvature W.

Theorem 4.3 There exists a weakly locally ϕ-symmetric space with constant scalar curvature and non constant vertical Ricci curvature. In particular such space is neither locally homogeneous nor strongly locally ϕ-symmetric.

Proof. Consider the 3-dimensional manifold M_{1} = {x ∈ R^{3} : x_{1} ≠ 0}. In the sequel

we denote by ∂_{i}, i = 1, 2, 3, the partial derivative \frac{∂}{∂x_{i}}. Let η the 1-form defined by

η = x_{1}x_{2}dx_{1} + dx_{3}.

η is a contact form because

η ∧ dη = −x_{1}dx_{1} ∧ dx_{2} ∧ dx_{3}.

The characteristic vector field of (M_{1}, η) is ξ = ∂_{3}. In fact

η(∂_{3}) = 1, (dη)(∂_{3}, ·) = x_{1}dx_{2} ∧ dx_{1})(∂_{3}, ·) = 0.

It is not difficult to see that the contact distribution is generated by the global vector fields

e_{1} = −\frac{2}{x_{1}} ∂_{2}, e_{2} = ∂_{1} − \frac{4x_{3}}{x_{1}} ∂_{2} − x_{1}x_{2}∂_{3}.

The vector fields e_{1}, e_{2}, ξ satisfy

[ξ, e_{1}] = 0, [ξ, e_{2}] = 2e_{1}, [e_{1}, e_{2}] = 2ξ + \frac{1}{x_{1}} e_{1}. (4.11)

Now, consider the Riemannian metric g defined by

g(ξ, e_{1}) = g(ξ, e_{2}) = g(e_{1}, e_{2}) = 0, g(ξ, ξ) = g(e_{1}, e_{1}) = g(e_{2}, e_{2}) = 1,

and the tensor ϕ defined by

ϕ(ξ) = 0, ϕ(e_{1}) = e_{2}, ϕ(e_{2}) = −e_{1}.

The tensors η, g and ϕ satisfy

(dη)(ξ, e_{i}) = 0 = g(ξ, ϕe_{i}), (dη)(e_{i}, e_{i}) = 0 = g(e_{i}, ϕe_{i}),

(dη)(e_{1}, e_{2}) = \frac{1}{2}\{e_{1}η(e_{2}) − e_{2}η(e_{1}) − η([e_{1}, e_{2}])\} = −1 = g(e_{1}, ϕe_{2}).

Then (η, g, ϕ) is a contact metric structure on M_{1}. Moreover the tensor h satisfies

h(e_{1}) = \frac{1}{2}\{[ξ, e_{2}] − ϕ[ξ, e_{1}]\} = e_{1}, h(e_{2}) = hϕe_{1} = −ϕh(e_{1}) = −e_{2}.

Thus λ = +1 and (e_{1}, e_{2}, e_{3} = ξ) is an orthonormal ϕ-basis of eigenvector for h. Since (e_{1}, e_{2}, e_{3} = ξ) is an orthonormal basis, the Levi-Civita connection is defined by the formula

∇_{e_{i}}e_{j} = \frac{1}{2} ∑_{k} −\{g(e_{i}, [e_{j}, e_{k}]) + g(e_{j}, [e_{k}, e_{i}]) + g(e_{k}, [e_{i}, e_{j}])\}e_{k}.

Then, by (4.11), we get

\begin{matrix}∇_{ξ}ξ = 0, & ∇_{e_{1}}ξ = −2e_{2}, & ∇_{e_{2}}ξ = 0, \\ ∇_{ξ}e_{1} = −2e_{2}, & ∇_{e_{1}}e_{1} = −\frac{1}{x_{1}} e_{2}, & ∇_{e_{2}}e_{1} = 0, \\ ∇_{ξ}e_{2} = 2e_{1}, & ∇_{e_{1}}e_{2} = \frac{1}{x_{1}} e_{1} + 2ξ, & ∇_{e_{2}}e_{2} = 0. \end{matrix} (4.12)

Using (4.12) we obtain

R(e_{1}, e_{2})e_{1} = −4e_{2},  R(ξ, e_{1})e_{2} = 0,  R(ξ, e_{2})e_{1} = −\frac{2}{x_{1}} e_{2},

from which H = −4, B = 0, and A = −\frac{2}{x_{1}} . Moreover λ = 1, then

\{ \begin{matrix}e_{1}(H) = 0 = 2B(λ + 1) \\ e_{2}(H) = 0 = 2A(λ − 1), \end{matrix}

and hence, using Lemma 3.1, (M_{1}, η, g) is a weakly locally ϕ-symmetric. Of course such space is neither homogeneous nor strongly locally ϕ-symmetric because A is not a constant function. This conclude the proof. Remark 4.2 The main result of [5]

says that every compact and orientable three-manifold has a contact metric structure whose Webster scalar curvature W is either a constant ≤ 0 or it is strictly positive everywhere. Theorem 4.2 gives an example of non-compact contact metric threemanifold with W = const. = −\frac{1}{2} < 0 with the geometric property that the basis {e_{1}, e_{2}, ξ} is parallel along the integral curves of the vector field e_{2}.

5 The unit tangent sphere bundle of a surface

Let (M, G) be a 2-dimensional Riemannian manifold. Consider on M isothermal local coordinate (x_{1}, x_{2}) on M. Then the Riemannian metric G is given by

G = e^{2f}((dx_{1})^{2} + (dx_{2})^{2})

where f is a C^{∞} function on M. Let T M be the tangent sphere bundle. The immersion of the unit tangent sphere bundle T^{1}M = {z = (p, v) ∈ T M : e^{2f}((v_{1})^{2} + (v_{2})^{2}) = 1} into T M is defined by

(y_{1}, y_{2}, θ) ⟶ (x_{1}, x_{2}, v_{1}, v_{2}) = (y_{1}, y_{2}, e^{−f}cosθ, e^{−f}sinθ).

Let(η, g, ξ, ϕ) the standard contact metric structure on T^{1}M. Then ξ = 2ξ′ where ξ′ is geodesic flow given by

ξ′ = v_{1} \frac{∂}{∂y_{1}} + v_{2} \frac{∂}{∂y_{2}} + (v_{1}f_{2} − v_{2}f_{1})\frac{∂}{∂θ} .

where f_{1} = \frac{∂f}{∂x_{1}} and f_{2} = \frac{∂f}{∂x_{2}} . Moreover setting

e_{2} = 2\frac{∂}{∂θ} = 2 \{−v_{2} \frac{∂}{∂v_{1}} + v_{1} \frac{∂}{∂v_{2}} \} ,

and

e_{1} = 2U = 2 \{−v_{2} \frac{∂}{∂y_{1}} + v_{1} \frac{∂}{∂y_{2}} − (v_{2}f_{2} + v_{1}f_{1})\frac{∂}{∂θ} \}

then (ξ, e_{1}, e_{2} = ϕe_{1}) is a local orthonormal ϕ-basis of T^{1}M. Denote by ∇ the Levi-Civita connection of (T^{1}M, g). The Gaussian curvature k of (M, G) considered as a function on T^{1}M is defined by k(p, v) = k(p). Using the Christoffel symbols of (M, G), we find

\begin{matrix}∇_{ξ}ξ = ∇_{e_{1}}e_{1} = ∇_{e_{2}}e_{2} = 0, & ∇_{e_{1}}ξ = −∇_{ξ}e_{1} = −ke_{2}, \\ ∇_{e_{2}}e_{1} = (k − 2)ξ, ∇_{e_{2}}ξ = (2 − k)e_{1}, & ∇_{ξ}e_{2} = −ke_{1}, ∇_{e_{1}}e_{2} = kξ. \end{matrix}

Cosequently, we get

R(e_{1}, e_{2})e_{1} = −e_{1}(k)ξ + k^{2}e_{2}, R(ξ, e_{1})e_{2} = −ξ(k)ξ − e_{1}(k)e_{1}, R(ξ, e_{2})e_{1} = −ξ(k)ξ,

he_{1} = \frac{1}{2}\{[ξ, e_{2}] − ϕ[ξ, e_{1}]\} = (k − 1)e_{1}, he_{2} = −ϕhe_{1} = (1 − k)e_{2},

from which

\{ \begin{matrix}H = g(R(e_{1}, e_{2})e_{1}, e_{2}) = k^{2} \\ B = ρ(ξ, e_{2}) = g(R(ξ, e_{1})e_{2}, e_{1}) = −e_{1}(k) \\ A = ρ(ξ, e_{1}) = g(R(ξ, e_{2})e_{1}, e_{2}) = 0 \\ λ = k − 1. \end{matrix}

Then e_{2}(H) = e_{2}(k^{2}) = 0 = 2A(λ − 1) and

e_{1}(H) = 2B(λ + 1) ⇔ e_{1}(k^{2}) = 0.

Moreover 2ξ(k^{2}) = [e_{1}, e_{2}](k^{2}). So, by lemma 3.1, T^{1}M is weakly locally ϕ-symmetric if and only if (M, G) has constant curvature. Hence we get the following theorem.

Theorem 5.1 The unit tangent sphere bundle T^{1}M equipped with the standard contact metric structure is weakly locally ϕ-symmetric if and only if the base manifold has constant Gaussian curvature.

Remark 5.1. Let M(c) be a 2-dimensional Riemannian manifold of constant Gaussian curvature c. Then the universal covering of T^{1}(M) is a simply connected Lie group equipped with a left invariant contact metric structure, more precisely we get : SU(2) if c > 0, S̃L(2, R) if c < 0, Ẽ(2) if c = 0, the universal covering of the isometry groups of S^{2}, H^{2} and E^{2}, respectively. Acknowledgements. This paper was supported by funds of the University of Lecce and the M.U.R.S.T.

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Universita’ degli Studi di Lecce Dipartimento di Matematica Via Provinciale Lecce-Arnesano 73100 Lecce, Italy email: perrone@ilenic.unile.it