Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 43 (2002), No. 1, 181-193.

A Classification of Contact Metric 3-Manifolds with Constant ξ-sectional and ϕ-sectional Curvatures

F. Gouli-Andreou Ph. J. Xenos

Department of Mathematics, Aristotle University of Thessaloniki Thessaloniki 540 06, Greece e-mail: fgouli@mailhost.ccf.auth.gr

Mathematics Division - School of Technology, Aristotle University of Thessaloniki Thessaloniki 540 06, Greece e-mail: fxenos@vergina.eng.auth.gr

Abstract. We study the 3-dimensional contact metric manifolds equipped with constant ξ-sectional curvature and ϕ-sectional curvature or constant norm of the Ricci operator. MSC 2000: 53D10, 53C25, 53C15

1. Introduction

D. E. Blair in [2], [3] constructed a family of examples of (3 − τ)-manifolds which do not satisfy the condition Qϕ = ϕQ. The existence of these examples depends on the constancy of the ξ-sectional curvature. After this remark the following question raises:

Question 1: Does every (3 − τ)-manifold with constant ξ-sectional curvature satisfy the condition Qϕ = ϕQ? S. Tanno in [16] stated the problem about the existence of (2n+1)-dimensional contact metric manifolds of constant ϕ-sectional curvature, which are not Sasakian. Positive answers have been given by D. E. Blair, Th. Koufogiorgos and R. Sharma in [5], for 3-dimensional contact metric manifolds satisfying Qϕ = ϕQ, Th. Koufogiorgos in [14], for (κ, μ)-contact metric

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 2002 Heldermann Verlag

182 F. Gouli-Andreou, Ph. J. Xenos: A Classification of Contact Metric 3-Manifolds with …

manifolds of dimension greater than 3 and D. E. Blair, Th. Koufogiorgos and B. Papantoniou in [4] for (κ, μ)-contact metric manifolds of dimension 3. In [4] the authors, extending the Tanno’s problem showed that there exist (κ, μ)-contact metric manifolds of dimension 3 which do not belong to the class of the manifolds satisfying Qϕ = ϕQ.

Extending Tanno’s problem and the result of [4] we can state the following:

Question 2: Do there exist 3-dimensional contact metric manifolds of constant ϕ-sectional curvature, which do not belong to the class of (κ, μ)-contact metric manifolds?

Combination of the above mentioned questions leads us to the study of 3-dimensional contact metric manifolds of constant ξ-sectional and ϕ-sectional curvature.

The main goal of the present paper (Theorem 15) is the proof of the existence of two new classes of 3-dimensional contact metric manifolds with constant ξ-sectional and constant ϕ-sectional curvatures, which do not belong to the up to date well known classes ([4], [5]).

D. E. Blair, Th. Koufogiorgos and R. Sharma in [5] proved that a 3-dimensional contact metric manifold satisfying Qϕ = ϕQ is flat or Sasakian or a manifold with constant ϕsectional curvature k and constant ξ-sectional curvature −k. In the present paper we prove the converse and so we can state the argument: A non-flat, non-Sasakian 3-dimensional contact metric manifold satisfies Qϕ = ϕQ if and only if it has constant ϕ-sectional curvature k and constant ξ-sectional curvature −k.

Complete, conformally flat Riemannian manifolds with constant scalar curvature and the norm of the Ricci tensor bounded (respectively constant) were classified by Goldberg ([8]) in general dimension (respectively, by Cheng, Ishikawa and Shiohama [7] in dimension 3). On the other hand the first author and R. Sharma in [10] proved that a conformally flat, contact metric 3-manifold with Ricci curvature vanishing along the characteristic vector field ξ and the norm of its Ricci tensor being constant, is flat. Therefore, it is interesting to study 3-dimensional contact metric manifolds equipped with more general conditions: constant ξ-sectional curvature and constant norm of the Ricci operator along ξ.

2. Preliminaries

A contact metric manifold M^{2n+1} ≡ M^{2n+1}(ϕ, ξ, η, g) is a (2n + 1)-dimensional Riemannian manifold on which has been defined globally a (1, 1) tensor field ϕ, a vector field ξ (characteristic vector field), a 1-form η (contact form) and a Riemannian metric g (associated metric) which satisfy:

ϕ^{2} = −I + η ⊗ ξ, η(ξ) = 1, η(X) = g(X, ξ),
g(ϕX, ϕY) = g(X, Y) − η(X)η(Y), dη(X, Y) = g(X, ϕY)

for all vector fields X and Y on M^{2n+1}. The structure (ϕ, ξ, η, g) is called contact metric structure.

Denoting by L and R the Lie derivation and the curvature tensor respectively, we define the operators l and h by

l := R(., ξ)ξ, η := \frac{1}{2}L_{ξ}ϕ.

The tensors l and h are self-adjoint and satisfy

hξ = lξ = 0, η ∘ h = 0, T rh = T rhϕ = 0, hϕ + ϕh = 0.

On every contact metric manifold M^{2n+1} the following formulas hold

η ∘ ϕ = 0, ϕξ = 0, dη(ξ, X) = 0, ∇_{ξ}ϕ = 0,
∇_{X}ξ = −ϕX − ϕhX (⇒ ∇_{ξ}ξ = 0), ϕlϕ − l = 2(ϕ^{2} + h^{2}),  (1)
∇_{ξ}h = ϕ − ϕl − ϕh^{2}, T rl = g(Qξ, ξ) = 2n − trh^{2},

where ∇ is the Riemannian connection. On M^{2n+1} × R we can define an almost complex structure J by J(X, f \frac{d}{dt}) = (ϕX − fξ, η(X)\frac{d}{dt}), where f is a real-valued function. If J is integrable, then the contact metric structure is said to be normal and M^{2n+1} is called Sasakian. A 3-dimensional contact metric manifold is Sasakian if and only if h = 0, ([1]).

The sectional curvature K(X, ξ) of a plain section spanned by ξ and a vector field X orthogonal to ξ is called ξ-sectional curvature. The sectional curvature K(X, ϕX) of a plain section spanned by the vector field X (orthogonal to ξ) and ϕX is called ϕ-sectional curvature.

It is well known that on every 3-dimensional Riemannian manifold the curvature tensor R(X, Y)Z is given by

R(X, Y)Z = g(Y, Z)QX − g(X, Z)QY + g(QY, Z)X − g(QX, Z)Y
− \frac{S}{2} [g(Y, Z)X − g(X, Z)Y],  (2)

where Q is the Ricci operator, S(= T rQ) is the scalar curvature and X, Y and Z are arbitrary vector fields.

A 3-dimensional contact metric manifold satisfing ∇_{ξ}τ = 0, (τ = L_{ξ}g) is called (3 − τ)- manifold, ([11]).

A contact metric manifold M^{2n+1}(ϕ, ξ, η, g) is called (κ, μ)-contact metric manifold ([4]) if it satisfies the condition

R(X, Y)ξ = κ[η(Y)X − η(X)Y] + μ[η(Y)hX − η(X)hY],

where κ and μ are real constants and X, Y are vector fields on M^{2n+1}.

3. Auxiliary results

Let M^{3} be a 3-dimensional contact metric manifold. If e ∈ ker(η) is a unit eigenvector of h with eigenvalue λ, then ϕe is also an eigenvector of h with eigenvalue −λ. Hence, (e, ϕe, ξ) is an orthonormal frame on M^{3}.

Since e and ϕe are unit vector fields orthogonal to ξ, we see that

∇_{ξ}e = aϕe, ∇_{ξ}ϕe = −αe,

for some function a on M^{3}. The orthogonality of e, ϕe and ξ implies

∇_{e}e = bϕe, ∇_{ϕe}ϕe = ce, ∇_{e}ϕe = −be + (λ + 1)ξ, ∇_{ϕe}e = −cϕe + (λ − 1)ξ,

where b and c are functions on M^{3}. Finally, from (1) we have

∇_{e}ξ = −(1 + λ)ϕe, ∇_{ϕe}ξ = (1 − λ)e.

Therefore, we can state the following

Lemma 1. Let M^{3} be 3-dimensional contact metric manifold. Then, the following formulas hold:

∇_{ξ}e = aϕe, ∇_{ξ}ϕe = −αe, ∇_{e}e = bϕe, ∇_{ϕe}ϕe = ce,
∇_{e}ϕe = −be + (λ + 1)ξ, ∇_{ϕe}e = −cϕe + (λ − 1)ξ,  (3)
∇_{e}ξ = −(1 + λ)ϕe, ∇_{ϕe}ξ = (1 − λ)e,

where a, b and c are functions on M^{3}.

Proposition 2. Let M^{3} be 3-dimensional contact metric manifold of constant ξ-sectional curvature k. Then, M^{3} is (3 − τ)-manifold with constant T rl.

Proof. By straightforward computation using (3) and ∇_{ξ}ξ = 0 we obtain

le = (1 − λ^{2} − 2αλ)e + (ξ · λ)ϕe, lϕe = (1 − λ^{2} + 2αλ)ϕe + (ξ · λ)e,

and hence

1 − λ^{2} − 2αλ = k, 1 − λ^{2} + 2αλ = k.

Adding the above two relations we obtain 2(1 − λ^{2}) = 2k. Because of T rl = 2(1 − λ^{2}) ([5]) we have T rl =constant. Subtracting the same relations we obtain αλ = 0, that is α = 0 or λ = 0.

If λ = 0, then M^{3} is Sasakian, which is trivially (3 − τ)-manifold ([5]).

Suppose that a = 0. Taking into account that T rl = constant we obtain that ∇_{ξ}h = 0. This relation and ([11]) complete the proof. □

Proposition 2 and Theorem 3.2 of [12] imply the following

Corollary 3. Let M^{3} be a 3-dimensional, conformally flat, contact metric manifold of constant ξ-sectional curvature. Then, M^{3} is either flat or a Sasakian space form.

Proposition 2 and Theorem 3.1 of [14] imply the following

Corollary 4. Let M^{3} be a 3-dimensional contact metric manifold of constant ξ-sectional curvature satisfing R(e, ξ) · R = 0. Then, M^{3} is either flat or a Sasakian manifold.

Proposition 2 and Theorem 3.1 of [13] imply the following

Corollary 5. Let M^{3} be a 3-dimensional contact metric manifold of constant ξ-sectional curvature satisfing R(e, ξ) · C = 0. Then, M^{3} is either flat or a Sasakian manifold.

Proposition 2 and Theorem 5.1 of [11] imply the following Corollary 6. Let M^{3} be a 3-dimensional contact metric manifold with constant ξ-sectional curvature and η-parallel Ricci tensor. Then, M^{3} is either flat or a Sasakian space form.

Proposition 2 and Theorem 6.2 of [11] imply the following

Corollary 7. Let M^{3} be a 3-dimensional contact metric manifold with constant ξ-sectional curvature and cyclic η-parallel Ricci tensor. Then, M^{3} is either flat or a Sasakian manifold with constant scalar curvature or of constant ξ-sectional curvature k < 1 and constant ϕsectional curvature −k.

Lemma 1, Proposition 2 and [11] imply:

Lemma 8. Let M^{3} be a 3-dimensional contact metric manifold with constant ξ-sectional curvature. Then, the following formulas hold:

∇_{ξ}e = ∇_{ξ}ϕe = 0, ∇_{e}e = bϕe, ∇_{ϕe}ϕe = ce,
∇_{e}ϕe = −be + (λ + 1)ξ, ∇_{ϕe}e = −cϕe + (λ − 1)ξ,  (4)
∇_{e}ξ = −(1 + λ)ϕe, ∇_{ϕe}ξ = (1 − λ)e.

where a, b and c are functions on M^{3} and λ is a constant.

Proposition 2 and [6] (relations 2.16) yield

Lemma 9. Let M^{3} be a 3-dimensional contact metric manifold with constant ξ-sectional curvature. Then, the following formulas hold:

Qe = (λ^{2} + \frac{S}{2} − 1)e + 2λbξ, η(Qe) = 2λb,

Qϕe = (λ^{2} + \frac{S}{2} − 1)ϕe + 2λcξ, η(Qϕe) = 2λc,  (5)

Qξ = 2λbe + 2λcϕe + 2(1 − λ^{2})ξ.

Lemma 10. Let M^{3} be a 3-dimensional contact metric manifold with constant ξ-sectional curvature. Then, either l = 0, or the following relations are equivalent: b = 0, c = 0.

Proof. Suppose that l is not identically equal to zero on M^{3}. Let λ^{2} ≠ 1 on an open neighborhood U at a point p ∈ M^{3}, where l ≠ 0. Applying the Jacobi’s identity for the vector fields e, ϕe, ξ and taking into account the relation (4) we obtain

ξ · b = (λ − 1)c, ξ · c = (λ + 1)b.  (6)

Let b = 0 (or c = 0) on M^{3}. Then, from the first (or the second) of (6) we conclude that c = 0 (or b = 0) on U. So, c = 0, (b = 0) on M^{3}. □

Remark 11. On a 3-dimensional contact metric manifold M^{3}, we have b = c = 0 if and only if Qϕ = ϕQ, ([11]).

4. Main results

Theorem 12. Let M^{3} be a 3-dimensional contact metric manifold with constant ξ-sectional curvature. Then, either M^{3} is Sasakian or

ξ · ξ · ξ · S = 4(λ^{2} − 1)(ξ · S).  (7)

Proof. If l = 0 on M^{3}, then λ^{2} = 1 and ξ · ξ · ξ · S = 0 ([9]).

Suppose that M^{3} is not Sasakian and l is not identically equal to zero. So, let λ^{2} ≠ 0, 1 on an open neighborhood U of a point p ∈ M^{3}. Applying the second Bianchi’s identity for the vector fields e, ϕe and ξ we obtain

e · b + ϕe · c − \frac{1}{4λ}ξ · S = 2bc.  (8)

Differentiating the above equation along ξ and taking into account (6) we obtain

ξ · e · b + ξ · ϕe · c − \frac{1}{4λ}ξ · ξ · S = 2(λ − 1)c^{2} + 2(λ + 1)b^{2}.

Next, differentiating the first and the second equations of (6) with respect to e and ϕe respectively and adding the results we get

e · ξ · b + ϕe · ξ · c = (λ − 1)e · c + (λ + 1)ϕe · b.

Hence,

[ξ, e]b + [ξ, ϕe]c = \frac{1}{4λ}ξ · ξ · S + 2(λ − 1)c^{2} + 2(λ + 1)b^{2} + (1 − λ)e · c − (λ + 1)ϕe · b.

The above equation using (4) yields

(λ + 1)ϕe · b + (λ − 1)e · c = \frac{1}{8λ}ξ · ξ · S + λ(b^{2} + c^{2}) + b^{2} − c^{2}.  (9)

Differentiating again (9) along ξ and taking into account (6) and (8) we obtain

(λ + 1)ξ · ϕe · b + (λ − 1)ξ · e · c = \frac{1}{8λ}ξ · ξ · ξ · S + 4(λ^{2} − 1)bc.  (10)

As λ^{2} ≠ 1 on U we obtain from (6) and (8)

(λ + 1)ϕe · ξ · b + (λ − 1)e · ξ · c = (λ^{2} − 1)[\frac{1}{4λ}ξ · S + 2bc].

Subtracting the above equation from (10) and using (4) the seeking formula follows at once. □

Theorem 13. Let M^{3} be a 3-dimensional contact metric manifold with constant ξ-sectional curvature. If the norm of the Ricci operator is constant along ξ, then either Qϕ = ϕQ or l = 0 with constant scalar curvature and η(QX) = 0 for all eigenvectors X ∈ ker(η) of h with eigenvalue 1.

Proof. The square of the norm of the Ricci operator Q is T rQ^{2} = g(Q^{2}e, e) + g(Q^{2}ϕe, ϕe) + g(Q^{2}ξ, ξ) and is computed using (5) and turns out to be

(λ^{2} + \frac{S}{2} − 1)^{2} + 4λ^{2}(b^{2} + c^{2}) + 2(1 − λ^{2})^{2} = ψ,  (11)

where ψ is a smooth function on M^{3} being constant along ξ.

Suppose that l = 0. Then, λ^{2} = 1 and (11) yields

\frac{S^{2}}{4} + 4(b^{2} + c^{2}) = ψ.  (12)

Differentiating three times the equation (12) along ξ and taking into account (6) and (7) for λ = 1 we obtain respectively

S(ξ · S) + 32bc = 0,
S(ξ · ξ · S) + (ξ · S)^{2} + 64b^{2} = 0,  (13)
(ξ · S)(ξ · ξ · S) = 0.

Therefore, ξ · S = 0. or ξ · ξ · S = 0.

Supposing ξ · S = 0 from the first of (13) we have b = 0 or c = 0.

If b = 0, from (5) we obtain η(Qe) = 0.

If c = 0 then (6) implies b = 0 that is Qϕ = ϕQ. In this case the manifold is flat.

If ξ · ξ · S = 0 then from (13) we have ξ · S = 0 and b = 0. If M^{3} is Sasakian then it is known that we have Qϕ = ϕQ. Suppose that M^{3} is not Sasakian with l not identically equal to zero. So, let be λ^{2} ≠ 0, 1

on an open neighborhood U of a point p ∈ M^{3}. Hence, we can write the equation (11) in the form

b^{2} + c^{2} = \frac{ψ}{4λ^{2}} + \frac{(λ^{2} − 1)^{2}}{2λ^{2}} − \frac{(λ^{2} + \frac{S}{2} − 1)^{2}}{4λ^{2}} .

Differentiating the above equation along ξ and taking into account (6) we obtain

bc = −\frac{1}{16λ^{2}} (λ^{2} + \frac{S}{2} − 1)(ξ · S).  (14)

Differentiating two times the relation (14) with respect to ξ and using (6) and (14) we have

(ξ · S)[8(1 − λ^{2})(λ^{2} + \frac{S}{2} − 1) − 1 − ξ · ξ · S] = 0.

Hence,

ξ · S = 0 or ξ · ξ · S = 8(1 − λ^{2})(λ^{2} + \frac{S}{2} − 1) − 1.  (15)

Supposing ξ · S = 0, the equation (14) yields b = 0 or c = 0 on U and hence b = 0 or c = 0 on M^{3}. Both cases using (6) imply Qϕ = ϕQ.

If the second of (15) holds on U, differentiating this relation along ξ and using Theorem 12 we obtain ξ · S = 0 and therefore Qϕ = ϕQ. □

Proposition 14. Let M^{3} be a 3-dimensional non-Sasakian contact metric manifold with constant ξ-sectional curvature. If l is not identically equal to zero then the following formulas hold:

e · b = \frac{1}{8λ}ξ · S + bc + Φ,  (16)

ϕe · b = \frac{1}{16λ}ξ · ξ · S + \frac{1}{2}(1 − λ)(λ^{2} + \frac{S}{2} − 1) + b^{2},  (17)

e · c = −\frac{1}{16λ}ξ · ξ · S + \frac{1}{2}(1 + λ)(λ^{2} + \frac{S}{2} − 1) + c^{2},  (18)

ϕe · c = \frac{1}{8λ}ξ · S + bc − Φ.  (19)

where Φ is a smooth function on M^{3} such that

ξ · Φ = 0,  (20)

e · Φ = \frac{1}{16λ}[ϕe · ξ · ξ · S − 2b(ξ · ξ · S) + 2(e · ξ · S) − 4c(ξ · S) −
− 4λ(λ + 1)(ϕe · S)] + (λ + 1)(λ^{2} + \frac{S}{2} − 3)b + 4cΦ,  (21)

ϕe · Φ = \frac{1}{16λ}[e · ξ · ξ · S − 2c(ξ · ξ · S) − 2(ϕe · ξ · S) + 4b(ξ · S) +
+ 4λ(1 − λ)(e · S)] + (λ − 1)(λ^{2} + \frac{S}{2} − 3)c + 4bΦ.  (22)

Proof. Calculating R(e, ϕe)ξ firstly by straightforward computation using Lemma 8 and secondly from the relation (2) we obtain

ϕe · b + e · c = b^{2} + c^{2} + λ^{2} − 1 + \frac{S}{2} .  (23)

From (23) and (9) the relations (17) and (18) follow at once.

Differentiating (17) first with respect to ξ (respectively with respect to e) and secondly with respect to e (respectively with respect to ξ) and using (6) we have

ξ · e · ϕe · b = \frac{λ − 1}{4λ}e · ξ · S + 2(λ − 1)[e · (bc)]  (24)

respectively

e · ξ · ϕe · b = \frac{1}{16λ}(ξ · e · ξ · ξ · S) + \frac{1 − λ}{4}(ξ · e · S) +
+ 2b(ξ · e · b) + 2(λ − 1)c(e · b).  (25)

Differentiation of the relation (7) along e implies

\frac{1}{16λ}(e · ξ · ξ · ξ · S) = \frac{λ^{2} − 1}{4λ}(e · ξ · S).  (26)

Adding (25) and (26) and using Lemma 8 we obtain

ξ · e · ϕe · b = \frac{λ + 1}{16λ}(ϕe · ξ · ξ · S) + \frac{λ^{2} − 1}{4λ}(e · ξ · S) +
+ \frac{1 − λ}{4}(ξ · e · S) + 2b(ξ · e · b) + 2(λ − 1)c(e · b).  (27)

Subtraction of (24) from (27) yields

(λ + 1)ϕe · ϕe · b = \frac{λ + 1}{16λ}(ϕe · ξ · ξ · S) + \frac{1 − λ^{2}}{4} (ϕe · S) +
+ 2b(ξ · e · b) + 2(1 − λ)b(e · c).  (28)

On the other hand differentiation of (17) with respect to ϕe using λ^{2} ≠ 1 (since l ≠ 0) implies

(λ + 1)(ϕe · ϕe · b) = \frac{λ + 1}{16λ}(ϕe · ξ · ξ · S) + \frac{1 − λ^{2}}{4} (ϕe · S) + 2(λ + 1)b(ϕe · b).

Comparing the above relation with (28) we obtain

b = 0, ξ · e · b = (λ + 1)ϕe · b + (λ − 1)e · c.  (29)

If b = 0 Lemma 10 implies c = 0, therefore from Remark 11 we obtain Qϕ = ϕQ. In this case it has been proved ([5]) that S = constant, which means that (16) and (19) are trivial (Φ = 0).

Differentiating (18) first with respect to ξ (respectively to ϕe) and secondly with respect to ϕe (respectively to ξ) and following the technique used to prove the relation (29) we can show that either Qϕ = ϕQ or

ξ · ϕe · c = (λ + 1)ϕe · b + (λ − 1)e · c.  (30)

We suppose that the second of (29) and (30) hold on M^{3}.

Using (6), (17) and (18) we obtain

ξ · e · b = ξ · ϕe · c = \frac{1}{8λ}(ξ · ξ · S) + ξ · (bc).

From the above relation and (23) the relations (16) and (19) follow at once.

Now we compute [e, ϕe]b (respectively [e, ϕe]c) in two ways, first using (16) and (17) (respectively (18), (19)) as e · ϕe · b − ϕe · e · b (respectively e · ϕe · c − ϕe · e · c), and secondly through (4), (6), (16) and (17) as (∇_{e}ϕe − ∇_{ϕe}e)b (respectively (4), (6), (18) and (19) as (∇_{e}ϕe − ∇_{ϕe}e)c). Comparing the two resulting expressions we obtain (22) (respectively (21)). □

Theorem 15. Let M^{3} be a 3-dimensional contact metric manifold with constant ξ-sectional curvature k and constant ϕ-sectional curvature m. Then, one of the following conditions holds:

(i) M^{3} is Sasakian,  (ii) Qϕ = ϕQ, and m = −k,
(iii) l = 0,  (iv) k + m = \frac{2}{3},  (v) k + m = −2.

Proof. We suppose that M^{3} is a non-Sasakian manifold with l being not identically equal to zero.

It is known ([5]) that on every 3-dimensional contact metric manifold K(e, ϕe) = \frac{S}{2} −T rl. Hence, this relation and Proposition 2 imply that S = constant. In this case the relations (16), (17), (18), (19), (21) and (22) take the form:

e · b = bc + Φ,  (31)

ϕe · b = b^{2} + \frac{1 − λ}{2}(λ^{2} + \frac{S}{2} − 1),  (32)

e · c = c^{2} + \frac{1 + λ}{2}(λ^{2} + \frac{S}{2} − 1),  (33)

ϕe · c = bc − Φ,  (34)

e · Φ = (λ + 1)(λ^{2} + \frac{S}{2} − 3)b + 4cΦ,  (35)

ϕe · Φ = (λ − 1)(λ^{2} + \frac{S}{2} − 3)c + 4bΦ.  (36)

Computing [e, ϕe]Φ in two different ways (as in the last part of the proof of Proposition 14), using (4), (20), (35) and (36) we obtain

8Φ^{2} = (λ^{2} + \frac{S}{2} − 3)[−4(λ + 1)b^{2} + 4(λ − 1)c^{2} + (1 − λ^{2})(λ^{2} + \frac{S}{2} − 1)].  (37)

Differentiating (37) with respect to e (respectively to ϕe) and taking into account (31), (33),

(35) and (37) (respectively (32), (34), (36) and (37)) we have

(λ^{2} + \frac{S}{2} − 3)[−(λ + 1)b^{2}c + (λ − 1)c^{3} + \frac{1 − λ^{2}}{2} (λ^{2} + \frac{S}{2} − 1)c + (λ + 1)bΦ] = 0,

(λ^{2} + \frac{S}{2} − 3)[−(λ + 1)b^{3} + (λ − 1)bc^{2} + \frac{1 − λ^{2}}{2} (λ^{2} + \frac{S}{2} − 1)b + (λ − 1)cΦ] = 0.

Hence, either

λ^{2} + \frac{S}{2} − 3 = 0,

or

(λ + 1)bΦ = c[(λ + 1)b^{2} + (1 − λ)c^{2} + \frac{λ^{2} − 1}{2}(λ^{2} + \frac{S}{2} − 1)] = 0  (38)

and

(λ − 1)cΦ = b[(λ + 1)b^{2} + (1 − λ)c^{2} + \frac{λ^{2} − 1}{2}(λ^{2} + \frac{S}{2} − 1)] = 0.  (39)

Suppose that λ^{2}+ \frac{S}{2} −3 = 0, then using K(e, ϕe) = \frac{S}{2} −T rl, T rl = 2(1−λ^{2}) and K(e, ξ) = \frac{T rl}{2}, we obtain k + m = −2.

In this case using [16] we conclude that if k = −3 and m = 1, then M^{3} is Sasakian. Also, for k + m = −2 and m > 1 we obtain a new class of contact metric 3-manifolds, which does not belong to the (κ, μ)-contact metric manifolds, ([4]).

Suppose now that (38) and (39) hold. If b = 0 (respectively c = 0), then (6) implies c = 0 (respectively b = 0) and therefore Qϕ = ϕQ. In this case using [5] we have m = −k. If bc ≠ 0, multiplying (38) with b and (39) with c we obtain

Φ[(λ + 1)b^{2} + (1 − λ)c^{2}] = 0.

Case A: Φ = 0. The relation (37) yields

(λ + 1)b^{2} + (1 − λ)c^{2} + \frac{λ^{2} − 1}{4}(λ^{2} + \frac{S}{2} − 1) = 0.

On the other hand the relation (38) yields

(λ + 1)b^{2} + (1 − λ)c^{2} + \frac{λ^{2} − 1}{2}(λ^{2} + \frac{S}{2} − 1) = 0.

Comparing the last two relations we obtain either λ^{2} = 1, a contradiction because of the assumption that l is not identically equal to zero on M^{3}, or

λ^{2} + \frac{S}{2} − 1 = 0.

From Φ = 0, (31), (32), (33) and (34) we obtain

e · b = ϕe · c = bc, ϕe · b = b^{2}, ϕe · c = c^{2}.  (40)

Computing [e, ϕe]b in two ways (by use of (4) and (40)) and comparing the results we obtain ξ · b = 0. Hence, from the assumption λ^{2} ≠ 1 and (6) we obtain b = c = 0, a contradiction.

Case B:

Φ ≠ 0 and (λ + 1)b^{2} + (1 − λ)c^{2} = 0.  (41)

The relations (38), (39) and (41) with the assumption λ^{2} ≠ 1 yield

bΦ = \frac{λ − 1}{2}(λ^{2} + \frac{S}{2} − 1)c,  (42)

cΦ = \frac{λ + 1}{2}(λ^{2} + \frac{S}{2} − 1)b.  (43)

On the other hand (37) and (41) imply

8Φ^{2} = (λ^{2} + \frac{S}{2} − 3)(1 − λ^{2})(λ^{2} + \frac{S}{2} − 1).

Hence, Φ =constant. This conclusion and the relations (35) and (36) yield

4bΦ = (1 − λ)(λ^{2} + \frac{S}{2} − 3)c,  (44)

4cΦ = −(λ + 1)(λ^{2} + \frac{S}{2} − 3)b.  (45)

Comparing (42) with (44) or (43) with (45) we obtain

λ^{2} + \frac{S}{2} = \frac{5}{3}.

Taking into account the last relation, K(e, ϕe) = \frac{S}{2} − T rl, T rl = 2(1 − λ^{2}) and K(e, ξ) = \frac{T rl}{2}, we obtain k + m = \frac{2}{3}.

In this case using [16] we conclude that if k = 1 and m = −\frac{1}{3}, then M^{3} is Sasakian. Also, for k + m = \frac{2}{3} and m > −\frac{1}{3} we obtain a new class of contact metric 3-manifolds, which does not belong to the (κ, μ)-contact metric manifolds, ([4]). □

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[10] Gouli-Andreou, F. ; Sharma, R.: A class of conformally flat contact metric 3-manifolds. Preprint.

[11] Gouli-Andreou, F.; Xenos, Ph. J.: On 3-dimensional contact metric manifolds with ∇_{ξ}τ = 0. J. Geom. 62 (1998), 154–165. Zbl 0905.53024

[12] Gouli-Andreou, F.; Xenos, Ph. J.: Two classes of conformally flat contact metric 3manifolds. J. Geom. 64 (1999), 80–88. Zbl 0918.53015

[13] Gouli-Andreou, F.; Xenos, Ph. J.: On a type of contact metric 3-manifolds. Yokohama Math. J. 46 (1999), 109–118. Zbl 0956.53036

[14] Koufogiorgos, Th.: Contact Riemannian manifolds with constant ϕ-sectional curvature. Tokyo J. Math. 20(1) (1997), 13–22. Zbl 0882.53032

[15] Perrone, D.: Contact Riemannian manifolds satisfying R(X, ξ)·R = 0. Yokohama Math. J. 39 (1992), 141–149. Zbl 0777.53046

[16] Tanno, S.: Ricci curvatures of contact Riemannian manifolds. Tohoku Math. J. 40 (1988), 441–448. Zbl 0655.53035

Received November 29, 2000