Classification of Locally Symmetric Contact Metric Manifolds

Anna Maria Pastore

Abstract

We complete the classification of 5-dimensional locally symmetric contact metric manifolds stated by D. Blair and J.M. Sierra. Furthermore, in general dimension we prove the existence of a foliation with totally geodesic leaves locally isometric to a Riemannian product E^{m+1} × S^{m}(4).

Mathematics Subject Classification: 53C15, 53C35 Key Words: contact metric manifolds; locally symmetric spaces

Introduction

In [6], Z. Olszak proved that for dimensions 2n + 1 ≥ 5 there are not contact metric manifolds of constant curvature unless the constant is 1 and in this case the structure is Sasakian. On the other hand, in [7], S. Tanno proved that a locally symmetric Kcontact manifold is of constant curvature. Motivated by these results, D. Blair and J.M. Sierra proposed the question of classifying locally symmetric contact manifolds, and in [5] they studied the 5-dimensional case, proving the following theorem. Theorem. Let M be a complete 5-dimensional locally symmetric contact metric manifold. Then the simply-connected covering space is either S^{5}(1) or E^{3} × S^{2}(4) or H^{2}(k_{1}) × H^{2}(k_{2}) × R , where H^{2}(k_{i}) i = 1, 2 is the hyperbolic plane with constant negative curvature k_{i}.

However, whereas S^{5}(1) and E^{3} ×S^{2}(4) admit such a structure ([2], [3]), the problem of the existence in the third case remained still open. We recall also that the 3-dimensional case has been studied in [4] by Blair and Sharma who proved that a 3-dimensional locally symmetric contact metric manifold is of constant curvature +1 or 0 .

In this paper we prove that the third possibility in the theorem of Blair and Sierra has to be removed . Moreover, in the general case, we prove that a locally symmetric contact metric manifold M^{2n+1}, 2n + 1 > 5, admits a foliation whose leaves are totally geodesic and locally isometric to the Riemannian product E^{m+1} × S^{m}(4), for a suitable m.

Balkan Journal of Geometry and Its Applications, Vol.3, No.1, 1998, pp. 89-96 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

1 Preliminaries

We recall some results on contact metric manifolds and for more details we refer to [1],[3],[5].

A contact metric manifold M^{2n+1} is a C^{∞}-manifold with a global 1-form η such that η ∧(dη)^{n} ≠ 0. It is well known that there exists a unique vector field ξ on M^{2n+1} satisfying η(ξ) = 1 and dη(ξ, X) = 0. A manifold M^{2n+1} is said to be a contact metric manifold if it admits a contact metric structure (φ, ξ, η, g), where φ is a tensor field of type (1, 1) and g is an associated metric such that

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

Denoting by L the Lie-derivation operator, the tensor field h = \frac{1}{2}L_{ξ}φ is a symmetric operator which anticommutes with φ. Obviously, h(ξ) = 0 and if λ is an eigenvalue of h with eigenvector X, then −λ is an eigenvalue with eigenvector φ(X) . Moreover, we have h = 0 if and only if ξ is a Killing vector field and in this case M^{2n+1} is called a K-manifold.

We have the following formulas, for any vector field X on M^{2n+1}:

∇_{X}ξ = −φ(X) − φh(X) (1)

\frac{1}{2}(R_{ξX}ξ − φR_{ξφ(X)}ξ) = h^{2}(X) + φ^{2}(X) (2)

(∇_{ξ}h)(X) = φ(X) − h^{2}φ(X) − φR_{Xξ}ξ, (3)

where ∇ is the Levi-Civita connection and R its curvature tensor field, [5]. Furthermore, in [2], the following theorem is proved.

Theorem B. Let M^{2n+1} be a contact metric manifold and suppose that R(X, Y)ξ = 0 for all vector fields X and Y. Then M^{2n+1} is locally the product of a flat (n + 1)-dimensional manifold and an n-dimensional manifold of positive constant curvature 4.

Finally, supposing that M^{2n+1} is a locally symmetric contact metric manifold we have ∇_{ξ}h = 0, [3]. Consequently, (3) gives

R_{ξX}ξ = −X + η(X)ξ + h^{2}(X) (4)

and the following formulas hold for all orthogonal to ξ unit eigenvectors X, Y of h with eigenvalues λ, μ respectively, ([5] lemma 3.3):

\begin{aligned}(λ^{2} − μ^{2})g(∇_{φX}X, Y) & = (1 − λ)[(1 − λ)g(∇_{X}φX, Y) \\ & −2λg(∇_{Y} X, φX) + (1 + μ)g(∇_{X}X, φY)] \end{aligned} (5)

and

\begin{aligned}(φY)(λ^{2}) & = 2(Xμ)(1 − μ)g(φY, X) + 2(1 − μ^{2})g(∇_{X}φY, X) \\ & +2(1 − λ − μ + λμ)g(∇_{X}Y, φX) \\ & +4λ(1 − μ)g(∇_{Y} X, φX) \end{aligned} (6)

2 The five-dimensional case

Let M^{5} be a locally symmetric contact metric manifold. If the tensor field h vanishes, then M^{5} is a K-manifold of constant curvature +1 and it is realized by S^{5}(1) with the standard Sasakian structure, [6], [7].

Now, suppose that h ≠ 0. As discussed in section 4 of [5] , for any p ∈ M^{5} there exists a unit vector X ∈ T_{p}(M^{5}) such that g(X, ξ) = 0 and R_{Xξ}ξ = 0. Using (4), we have

h^{2}(X) − X = 0 (7)

and since h(ξ) = 0, the spectrum of the operator h is given by {0, λ, −λ, μ, −μ}. We suppose λ ≥ 0 , μ ≥ 0 and we denote by {ξ, e_{1}, e_{2}, e_{3}, e_{4}} the set of the corresponding eigenvectors. Writing X = ∑_{i=1}^{4} X^{i}e_{i}, and applying (7) we obtain that at least one of λ or μ must be 1, say μ. Moreover, φ(e_{1}) = e_{2} , φ(e_{3}) = e_{4} and the eigenvalues are constant along their eigenvectors.

Blair and Sierra distingueshed three cases:

i) λ = 1 ;

ii) λ = 0 ;

iii) λ ≠ 0, 1.

In their proof the first case implies that M^{5} is locally isometric to the Riemannian product E^{3} × S^{2}(4) via theorem B, the second one leeds to an empty class and the third one implies the local isometry of M^{5} with H^{2}(k_{1}) × H^{2}(k_{2}) × R.

Now, we shall prove that the third possibility has to be excluded, obtaining the following classification theorem.

Theorem 1 Let M be a complete 5-dimensional locally symmetric contact metric manifold. Then the simply-connected covering space is either S^{5}(1) or E^{3} × S^{2}(4).

Proof. Let us suppose λ ≠ 0, 1. In this hypothesis, Blair and Sierra proved the following results:

a) The distribution [+1] ⨁[−1] ⨁[ξ] is integrable with flat totally geodesic leaves. Here, [+1] and [−1] denote respectively the eigenspaces related to the eigenvalues +1 and −1 and [ξ] is the distribution spanned by ξ.

b) The Levi-Civita connection satisfies the following relations:

92 A.M.Pastore

∇_{ξ} = 0  ∇_{e_{4}} = 0
∇_{e_{1}}e_{1} = −β′_{1}e_{3}  ∇_{e_{1}}e_{2} = −γ′_{1}e_{3} − γ_{1}e_{4} + (1 + λ)ξ
∇_{e_{1}}e_{3} = β′_{1}e_{1} + γ′_{1}e_{2} ∇_{e_{1}}e_{4} = γ_{1}e_{2}
∇_{e_{1}}ξ = (−1 − λ)e_{2} ∇_{e_{2}}e_{1} = −β′_{2}e_{3} − β_{2}e_{4} − (1 − λ)ξ
∇_{e_{2}}e_{2} = −γ′_{2}e_{3}  ∇_{e_{2}}e_{3} = β′_{2}e_{1} + γ′_{2}e_{2}
∇_{e_{2}}e_{4} = β_{2}e_{1}  ∇_{e_{2}}ξ = (1 − λ)e_{1}
∇_{e_{3}}e_{1} = α_{3}e_{2}  ∇_{e_{3}}e_{2} = −α_{3}e_{1}
∇_{e_{3}}e_{3} = 0  ∇_{e_{3}}e_{4} = 2ξ
∇_{e_{3}}ξ = −2e_{4},

where β_{2} = −\frac{1 − λ}{1 + λ}γ_{1} , λα_{3} = −γ′_{1} , ξ(α_{3}) = 0.

c) R_{e_{1}e_{2}}ξ = −((1 + λ)γ′_{2} + (1 − λ)β′_{1})e_{3}.

d) The eigenvalue λ must be a non costant function, and ξ(λ) = 0, e_{4}(λ) = 0. First at all, we deduce some other formulas. Taking Y = e_{4} and X = e_{i}, i = 1, 2

in (6) we get

−e_{3}(λ^{2}) = 4(1 − λ)g(∇_{e_{1}}e_{4}, e_{2}) + 8λg(∇_{e_{4}}e_{1}, e_{2})

Then, using b), we obtain −e_{3}(λ^{2}) = 4(1 − λ)γ_{1} and

e_{3}(λ) = −2\frac{1 − λ}{λ}γ_{1}. (8)

Now, condition d) implies γ_{1} ≠ 0 and applying the first Bianchi identity to e_{1}, e_{3}, ξ and using R_{e_{3}ξ} = 0 we obtain: −2γ_{1} + e_{3}(λ) + (1 + λ)(β′_{1} − γ′_{2}) = 0 (9) Again, using R_{e_{3}ξ} = 0 and c) we find:

γ′_{2} = −\frac{1 − λ}{1 + λ}β′_{1} (10)

and substituing (8) , (10) in (9) , we get

β′_{1} = \frac{1}{λ}γ_{1}.

Finally, by direct computation, we have

g(R_{e_{1}e_{2}}e_{1}, e_{2}) = −(γ′_{1})^{2} − \frac{(1 − λ)^{2}}{λ^{2}} (γ_{1})^{2} + 1 − λ^{2}. (11)

Now, we suppose that M^{5} = H^{2}(k_{1}) × H^{2}(k_{2}) × R and recall that a) holds. Obviously, ξ has non zero component tangent to H^{2}(k_{1})×H^{2}(k_{2}) , otherwise we have R_{XY} ξ = 0 for all X, Y and λ = 1. Moreover, since the foliation spanned by {e_{3}, e_{4}, ξ} induces foliations by geodesics on each H^{2}(k_{i}), we can consider (f_{1}, f_{2}) orthonormal vectors tangent to H^{2}(k_{1}) , and (f_{3}, f_{4}) orthonormal vectors tangent to H^{2}(k_{2}) such that {f_{2}, f_{4}, f_{5}} span the distribution [+1] ⨁[−1] ⨁[ξ]. It follows that e_{1} and e_{2} belong to the span{f_{1}, f_{3}} and, since the sectional curvature K({f_{1}, f_{3}}) = 0, we have K({e_{1}, e_{2}}) = 0 and (11) implies

1 − λ^{2} = (γ′_{1})^{2} + \frac{(1 − λ)^{2}}{λ^{2}} (γ_{1})^{2} > 0. (12)

On the other hand, writing ξ = af_{2} + bf_{4} + cf_{5} and using (4) we obtain R_{f_{1}ξ}ξ = (1 − λ^{2})f_{1} whereas using the sectional curvature, we get R_{f_{1}ξ}ξ = a^{2}k_{1} so that 1 − λ^{2} = a^{2}k_{1}.

We conclude that 1 − λ^{2} < 0, contradicting (12).

3 Some results in the higher dimensional case

Let M^{2n+1} be a locally symmetric contact metric manifold and suppose that h ≠ 0. Arguing as at the beginning of section 2, we consider the set

{0, +1, −1, λ_{1}, −λ_{1}, …, λ_{r}, −λ_{r}}

of the distinct eigenvalues of h such that dim[0] = p + 1, dim[+1] = m, dim[λ_{1}] = m_{1}, …, dim[λ_{r}] = m_{r} and 2n + 1 = p + 1 + 2m + 2m_{1} + … + 2m_{r}. Here [λ] denotes the eigenspace corresponding to the eigenvalue λ. Theorem 2 Let M^{2n+1}, 2n + 1 > 5, be a locally symmetric contact metric manifold and suppose that the spectrum of h is given by the set {0, +1, −1} with +1 and −1 as eigenvalues of multiplicity n and 0 as simple eigenvalue. Then M^{2n+1} is locally isometric to the Riemannian product E^{n+1} × S^{n}(4). Proof. By means of (4), we get R_{Xξ}ξ = 0 for any eigenvector X ∈ [±1]. Consequently, the sectional curvatures of the tangent 2-planes containing ξ vanish.

If M^{2n+1} is irreducible, it is Einstein with Ric(ξ, ξ) = 2n − tr(h^{2}) = 0 and consequently it is Ricci-flat and then flat, contradicting the result of Olszak in [6]. Hence, M^{2n+1} is reducible and the vanishing of the ξ-curvatures implies that ξ has to be tangent to a flat factor. It follows that R_{XY} ξ = 0 for all tangent vectors X, Y and theorem B applies.

Now,we suppose m < n, we put [0] = [ξ]⊕V_{0} (orthogonal sum), and H = [ξ]⊕[±1]. To prove that the distribution H is integrable we need some lemma.

Lemma 1. For any X ∈ H we have [ξ, X] ∈ H.

Proof. Clearly, for X ∈ H we have:

X ∈ [+1] ⇒ (∇_{X}ξ = −2φX ∈ [−1], ∇_{ξ}X ∈ [+1])
X ∈ [−1] ⇒ (∇_{X}ξ = 0, ∇_{ξ}X ∈ [−1])

94 A.M.Pastore Finally, ∇_{ξ}ξ = 0 and [X, ξ] ∈ [±1] ⊂ H follows. Lemma 2. For any X, Y belonging to [+1] we have ∇_{φY} X ∈ [±1] ⊂ H. Proof. We use the following formula stated as formula (5) in [3]

\begin{aligned}R_{Y X}ξ + R_{ξX}Y − R_{hY X}ξ − R_{ξX}hY & = g(X, Y)ξ − 2η(Y)X + η(X)Y \\ & −g(X, hY)ξ + 2η(Y)h^{2}X \\ & −η(X)hY + (∇_{φY} h^{2})(X). \end{aligned} (13)

obtaining (∇_{φY} h^{2})(X) = 0, i.e., ∇_{φY} X − h^{2}(∇_{φY} X) = 0 (14) and this implies ∇_{φY} X ∈ [±1]. Namely, we decompose ∇_{φY} X with respect to the direct sum of the eigenspaces: ∇_{φY} X = A_{0} + A_{+1} + A_{−1} + A_{λ_{1}} + A_{−λ_{1}} + … + A_{λ_{r}} + A_{−λ_{r}} (15) Then, we have

h^{2}(∇_{φY} X) = A_{+1} + A_{−1} + λ_{1}^{2}A_{λ_{1}} + λ_{1}^{2}A_{−λ_{1}} + … + λ_{r}^{2}A_{λ_{r}} + λ_{r}^{2}A_{−λ_{r}}. Using (14) and (15), we get A_{0} = 0, A_{λ_{1}} = 0, A_{−λ_{1}} = 0, …, A_{λ_{r}} = 0, A_{−λ_{r}} = 0, λ_{1}, …, λ_{r} being different from +1, −1. Finally, from (15) we obtain ∇_{φY} X = A_{+1} + A_{−1} ∈ [±1] ⊂ H. Corollary 1. For any X ∈ [−1] and Y ∈ [+1] we have ∇_{X}Y ∈ [±1]. Proof. Apply Lemma 2 to φX and Y. Lemma 3. For any Y ∈ [+1] and X ∈ [−1], we have ∇_{φY} X ∈ [±1]. Proof. From (13), since g(X, Y) = 0, we obtain (∇_{φY} h^{2})(X) = 0 and we continue as in the proof of Lemma 2. Corollary 2. We have: a) (X ∈ [−1], Y ∈ [−1]) ⇒ ∇_{X}Y ∈ [±1]

b) X, Y ∈ [−1] ⇒ [X, Y] ∈ [±1] Lemma 4. For any X ∈ [−1] and Y ∈ [−1], we have ∇_{φY} X ∈ H. Proof. Using (13) we have:

2R_{XY} ξ + 2R_{ξX}Y = 2g(X, Y)ξ + (∇_{φY} h^{2})(X).

Lemma 1 and Corollary 2 easily imply that R_{Y X}ξ ∈ [±1] and R_{ξX}Y ∈ [±1]. It follows B = 2g(X, Y)ξ + ∇_{φY} X − h^{2}(∇_{φY} X) ∈ [±1] (16) On the other hand, decomposing ∇_{φY} X as in (15) and computing h^{2}(∇_{φY} X), we get

\begin{aligned}B & = 2g(X, Y)ξ + A_{0} + (1 − λ_{1}^{2})A_{λ_{1}} + (1 − λ_{1}^{2})A_{−λ_{1}} + … \\ & +(1 − λ_{r}^{2})A_{λ_{r}} + (1 − λ_{r}^{2})A_{−λ_{r}} \end{aligned} (17)

Comparing (16) and (17) we conclude

A_{0} = −2g(X, Y)ξ, A_{λ_{1}} = 0, A_{−λ_{1}} = 0, …, A_{λ_{r}} = 0, A_{−λ_{r}} = 0

so that

∇_{φY} X = −2g(X, Y)ξ + A_{+1} + A_{−1} ∈ H.

Corollary 3. (X ∈ [+1], Y ∈ [−1]) ⇒ (∇_{X}Y ∈ H, [X, Y] ∈ H).

Lemma 5. For any Y ∈ [−1] and X ∈ [+1] we have ∇_{φY} X ∈ [±1].

Proof. Using (13), since g(X, Y) = 0, we get

2R_{Y X}ξ + 2R_{ξX}Y = (∇_{φY} h^{2})(X)

Now, Lemma 1 and the previous corollaries easily imply that R_{Y X}ξ ∈ [±1] and R_{ξX}Y ∈ H, so that

∇_{φY} X = h^{2}(∇_{φY} X) ∈ H. (18)

Again, decomposing ∇_{φY} X with respect to the direct sum of eigenspaces, (18) implies A_{0} = aξ, A_{λ_{1}} = 0 …, A_{−λ_{r}} = 0 ,so that we have

∇_{φY} X = aξ + A_{+1} + A_{−1}

Now, since φY ∈ [+1], we get g(∇_{φY} X, ξ) = −g(X, ∇_{φY} ξ) = −2g(x, φ^{2}Y) = 2g(X, Y) = 0 and ∇_{φY} X ∈ [±1].

Corollary 4. (X ∈ [+1], Y ∈ [+1]) ⇒ (∇_{X}Y ∈ [±1].[X, Y] ∈ [±1].

Proposition 4.1. The distribution H = [ξ] ⊕ [±1] is integrable with totally geodesic leaves.

Proof. The previous lemma and corollaries imply that [X, Y] ∈ H for any X ∈ H and Y ∈ H. Thus the distribution H is involutive and integrable. Let N be a maximal integral submanifold. Since ∇_{X}Y is tangent to N for any vector fields X, Y tangent to N , the second fundamental form vanishes and N is totally geodesic. Proposition 4.2. The integral manifolds of the distribution H are locally isometric to the Riemannian product E^{m+1} × S^{m}(4). Proof. Let N be an integral manifold of H, A local frame for T N is given by ξ and the eigenvectors {e_{i}, φe_{i}}, i ∈ {1, …, m} corresponding to the eigenvalues +1, −1, and N has a canonically induced contact metric structure (ξ, φ′, g) where φ′ is the restriction of φ to N. Moreover, N turns out to be locally symmetric since it is totally geodesic in the locally symmetric manifold M^{2n+1}. It is easy to verify that h′ = \frac{1}{2}L_{ξ}φ′ is the restriction of h to N. Now, h′ has eigenvalues +1, −1 with multiplicity m and 0 as a simple eigenvalue. Theorem 2 insures that N is locally isometric to E^{m+1}×S^{m}(4). Hence, we can conclude with the following theorem: Theorem 3 Let M^{2n+1} be a locally symmetric contact metric manifold. Then M^{2n+1} admits a foliation whose leaves are totally geodesic and locally isometric to the Riemannian product E^{m+1} × S^{m}(4). The integer m is the multiplicity of the eigenvalue +1 of the operator \frac{1}{2}L_{ξ}φ.

References

[1] D.E.Blair, Contact manifolds in Riemannian geometry, Lecture Notes in Math., 509 , 1976.

[2] D.E.Blair, Two remarks on contact metric structures, Tôhoku Math. J. 29 (1977), 319-324.

[3] D.E.Blair, When is the tangent sphere bundle locally symmetric?, in Geometry and Topology, World Scientific, Singapore (1989), 15-30.

[4] D.E.Blair and R.Sharma, Three dimensional locally symmetric contact metric manifolds, B.U.M.I (7), 4-A, (1990) , 385-390.

[5] D.E.Blair and J.M.Sierra, 5-dimensional locally symmetric contact metric manifolds, B.U.M.I. (7), 7-A, (1993), 299-311.

[6] Z.Olszak, On contact metric manifolds, Tôhoku Math.J., 31, (1979), 247-253.

[7] S.Tanno, Locally symmetric K-contact Riemannian manifolds, Proc. Japan Acad. 43, (1967), 581-583.

Dipartimento Interuniversitario di Matematica Università di Bari, via Orabona, 4 70125 Bari, Italy e-mail Pastore@pascal.dm.uniba.it