Contact Metric Spaces and pseudo-Hermitian Symmetry

We prove that a contact strongly pseudo-convex CR (Cauchy–Riemann) manifold M2n+1, n ≥ 2, is locally pseudo-Hermitian symmetric and satisfies ∇ξ h = μhφ, μ ∈ R, if and only if M is either a Sasakian locally φ-symmetric space or a non-Sasakian (k, μ)-space. When n = 1, we prove a classification theorem of contact strongly pseudo-convex CR manifolds with pseudo-Hermitian symmetry.


Introduction
For a contact manifold (M 2n+1 ; η), we have the two fundamental structures associated with the contact form η: the Riemannian metric g and the Levi form related with an endomorphism J on D(=Ker of η) such that J 2 = −I. Between the two associated structures we have a one-to-one correspondence by the equation where L denotes the (0,2)-tensor field on M which naturally extends the Levi form. When the integrability of the almost complex structure J on D is assumed, we call it an (integrable) CR structure. A Sasakian manifold is a contact Riemannian manifold (M; η, g) satisfying a normality condition (see, Section 2.1). Then, we may describe the Sasakian structure as a contact strongly pseudo-convex CR (Cauchy-Riemann) structure whose characteristic vector field is a Killing vector field for its associated Riemannian metric.
In studying a contact manifold from the Riemannian view point, the question of contact metric manifolds (M; η, g) satisfying (Cartan's) local symmetry, that is, whose Riemannian curvature tensor R satisfies ∇R = 0, where ∇ denotes the Levi-Civita connection of g, has a long history (cf. Section 7.5 in Reference [1]). (For Sasakian manifolds, the problem was settled already by M. Okumura [2].) At last, E. Boeckx and the present author [3] have proved that a locally symmetric contact metric space is locally isometric to either the unit sphere or the unit tangent sphere bundle of the Euclidean space with its standard contact metric structure. This result says that the local symmetry is too strong a condition to impose on contact manifolds. In this context, T. Takahashi [4] introduced the notion of Sasakian locally φ-symmetric spaces as the analogue of locally Hermitian symmetric spaces. A Sasakian manifold is said to be locally φ-symmetric if the Riemannian curvature tensor R satisfies for all vector fields X, Y, Z, V and U orthogonal to ξ. Making a generalization of this definition, in Reference [5], the authors call a contact metric manifold locally φ-symmetric if it satisfies the same curvature condition ( * ) as in the Sasakian case.
Taking a look at contact manifolds from the point of view of its pseudo-Hermitian structure, then we have a canonical affine connection, distinct from the Levi-Civita connection of an associated metric g. Indeed, the generalized Tanaka-Webster connection∇ on a strongly pseudo-convex almost CR manifold is invariant under pseudo-homothetic transformations (which preserve φ). In previous works [6][7][8][9][10][11], the generalized Tanaka-Webster connection has been playing an important part when we studied the interplay between the contact Riemannian structure and the contact strongly pseudo-convex almost CR structure. In Reference [10] (for the CR-integrable case in Reference [6]), we defined locally pseudo-Hermitian symmetric spaces to be strongly pseudo-convex almost CR manifolds whose pseudo-Hermitian curvature tensorR satisfies for all vector fields X, Y, Z, V and U orthogonal to ξ. Then, in Section 3 of the present paper, we prove that a contact strongly pseudo-convex CR manifold M 2n+1 , n ≥ 2, is locally pseudo-Hermitian symmetric and satisfies ∇ ξ h = µhφ, µ ∈ R, if and only if M is either a Sasakian locally φ-symmetric space or a non-Sasakian (k, µ)-space (Theorem 2). In Section 4, we treat the three-dimensional case. In particular, we prove a classification theorem of three-dimensional contact strongly pseudo-convex CR manifolds with pseudo-Hermitian symmetry (Theorem 4). Moreover, we give non-homogeneous examples of locally pseudo-Hermitian symmetric contact strongly pseudo-convex CR manifolds.

Contact Riemannian Structures
First of all, we recall some basic notions and formulas in contact Riemannian geometry. All manifolds treated in this paper are assumed to be connected and of class C ∞ .
For a contact form η, we have a unique vector field ξ, which is called the characteristic vector field or the Reeb vector field, satisfying η(ξ) = 1 and i ξ dη = 0. Here, i ξ denotes the interior product operator by ξ. Then we have also a Riemannian metric g and a (1, 1)-tensor field φ such that for all X, Y ∈ X(M), where X(M) denotes the Lie algebra of all smooth vector fields on M. From (1), we have easily that A manifold M equipped with structure tensors (η, ξ, φ, g) satisfying (1) is said to be a contact Riemannian manifold or contact metric manifold and is denoted by M = (M; η, g). We define a (1, 1)-tensor field h on M by 2h = L ξ φ, where L ξ denotes Lie differentiation with respect to ξ. The operator h is self-adjoint and satisfies where ∇ is the Levi-Civita connection. From (3) and (4) we see that any flow line of ξ is a geodesic. Furthermore, we know that ∇ ξ φ = 0 in general (cf. p. 67 in Reference [1]). From the second equation of (3) it follows also that The Riemannian curvature tensor R is defined by for all X, Y, Z ∈ X(M). The characteristic Jacobi operator is the symmetric (1, 1)-tensor defined by (X) = R(X, ξ)ξ. Then, from the definition of R, by using (4) we have Using (6), together with (3) and (5), then we have A contact metric manifold is called a K-contact manifold if ξ is a Killing vector field, or equivalently, h vanishes. Given M = (M; η, g), an almost complex structureJ on M × R is naturally defined bỹ If the almost complex structureJ is integrable, M is said to be normal or Sasakian. Note that any 3-dimensional K-contact manifold is already Sasakian, but in higher dimension it does not hold any more. A Sasakian structure is characterized by the following equation: or eqiuvalently for all X, Y ∈ X(M). We refer to Blair's book [1] for more details.

Strongly Pseudo-Convex almost CR Structures
In this subsection, we begin by a brief reviewing of pseudo-Hermitian CR structure.  If the Levi form is non-degenerate (positive or negative definite, resp.), then (η, L) is called a non-degenerate (strongly pseudo-convex, resp.) pseudo-Hermitian CR structure. Then there exists a unique globally defined nowhere vanishing vector field ξ such that η(ξ) = 1 and i ξ dη = 0. In particular, for a strongly pseudo-convex pseudo-Hermitian CR structure (η, L), we can extend the Levi form canonically to a Riemannian metric on M defined by where i ξ L = 0. It is called the Webster metric (cf. Reference [13]).
Let's return to a contact Riemannian manifold M = (M; η, g). At each point p ∈ M, the subspace D p (= Ker of η p ) of the tangent space T p M of M gives the decomposition T p M = D p ⊕ {ξ} p (direct sum). Then D : p → D p defines a 2n-dimensional distribution orthogonal to ξ, which is called the contact distribution or the contact subbundle. The associated pseudo-Hermitian structure is naturally provided by the subbundle for all X, Y ∈ Γ(D). Such (η, J) is called an pseudo-Hermitian almost CR structure. It should be notable that the CR-integrability does not hold in general for a contact metric manifold. In terms of the structure tensors, CR-integrability condition is equivalent to the condition for all X, Y ∈ X(M) (see Reference [14], [Proposition 2.1]). From (8) and (11), we see that the associated pseudo-Hermitian structure of a Sasakian manifold is strongly pseudo-convex and CR-integrable. The same is true for all three-dimensional contact metric spaces [6].

The pseudo-Hermitian Symmetric Spaces
In this section, we study on the pseudo-Hermitian curvature tensor and symmetric properties.
for all X, Y ∈ X(M).
Using (4), we may rewrite∇ as∇ Here, A is (1, 2)-tensor field defined by Then∇ has the torsionT In particular, for the K-contact case we get Proposition 1 ([14]). The generalized Tanaka-Webster connection∇ on a contact Riemannian manifold M = (M; η, g) is the unique linear connection satisfying the following conditions: In the case Ω = 0, that is, CR-integrability holds,∇ reduces to the Tanaka-Webster connection, which is defined on a non-degenerate integrable pseudo-Hermitian manifold [16,17]. So, the above definition is useful especially for the non-integrable case. The generalized Tanaka-Webster curvature tensorR is defined by: We call sometimesR the pseudo-Hermitian curvature tensor. Then we have Corollary 1 ([6]). The pseudo-Hermitian curvature tensorR and its covariant derivative∇R are pseudo-homothetically invariant.

Proposition 5 ([6]). A Sasakian manifold is locally pseudo-Hermitian symmetric if and only if it is locally φ-symmetric.
Since g((∇ XR )(ξ, Y)Z, ξ) = 0, we find that where {e i , e 2n+1 = ξ} (i = 1, · · · , 2n) is an adapted orthonormal frame. Now, we introduce a very useful class of contact metric manifolds when we study their pseudo-Hermitian geometry. This is the so-called (k, µ)-spaces [18] defined by the condition: where k, µ ∈ R. It includes Sasakian spaces for k = 1 and h = 0. Popular examples of non-Sasakian (k, µ)-spaces are provided by the unit tangent sphere bundles of spaces of constant curvature c (k = c(2 − c) and µ = −2c) when c = 1. (Due to Tashiro's result [19], we know that the standard contact metric structure is Sasakian only when c = 1.) Very recently, the present author [13] proved a complete classification theorem of non-Sasakian (k, µ)-spaces which are realized as real hypersurfaces in the complex quadric Q n+1 , its dual Q * n+1 , or the complex Euclidean space C n+1 . Other remarkable properties of (k, µ)-spaces [18] are as follows: • such a class is invariant under pseudo-homothetic transformations, • the associated pseudo-Hermitian structure is CR-integrable.
From the second property, we obtain a formula of ∇φ by (11). Also for ∇h, we have an explicit expression: for all X, Y ∈ X(M) [18]. From (21), we have that a (k, µ)-space is an η-parallel contact metric space, which means that it satisfies g((∇ X h)Y, Z) = 0 for all vector fields X, Y, Z ⊥ ξ. E. Boeckx and the present author proved that the converse holds also: 20]). An η-parallel contact metric space is either a K-contact space or a (k, µ)-space.
Now, we prove the following theorem.
From these equations, we calculate the pseudo-Hermitian curvature tensor: Then we have the holomorphic sectional curvatureĤ for∇ and the pseudo-Hermitian scalar curvaturer: Thus we have the following theorem.

Theorem 3.
A three-dimensional contact strongly pseudo-convex CR manifold is a locally pseudo-Hermitian symmetric space if and only if it is of constant holomorphic sectional curvatureĤ.

Proof of Theorem 3.
Suppose that M is locally pseudo-Hermitian symmetric. Then, since h = 0 on U 2 , we see that it has Sasakian structure. Then by Proposition 5 it is locally φ-symmetric, moreover, due to a result in Reference [25] we have that it has a constant holomorphic sectional curvatureĤ(= H + 3) on U 2 , where H denotes the holomorphic sectional curvature for ∇. Next, from (32) and (33) we compute on U 1 : Since M is locally pseudo-Hermitian symmetric,Ĥ satisfies e(Ĥ) = φe(Ĥ) = 0.
We also have, using the second condition of (1): These computations imply thatĤ is constant on U 1 . Since U 1 ∪ U 2 is dense and M is connected,Ĥ is constant on the whole M. In addition, we can easily show that the converse also holds. After all, we conclude the proof of Theorem 3.

Corollary 4.
A three-dimensional Jacobi (k, µ)-contact manifold is locally pseudo-Hermitian symmetric if and only if the scalar curvature is constant.
In a previous paper, we have proved the following fact.  Proof of Theorem 4. Sppose that M is a locally pseudo-Hermitian symmetric space with constant scalar curvature r. We divide our proof into two cases: U 1 is empty or not. If U 1 is empty, then M = U 2 and M is a Sasakian space form by Corollary 3. Now, we treat the case that U 1 is not empty, and we deliver our argument on U 1 . Then, from Proposition 4 we haver is a constant. But, from (18), r = r − ρ(ξ, ξ) + 4. Hence, we have that ρ(ξ, ξ) = 2 − 2λ 2 is a constant, which gives that λ(> 0) is a constant. From this, we have M = U 1 . We compute Since a vertical Ricci curvature σ(X) for X ∈ Γ(D) is constant, from (36) we have that p and q are constants, and from (34) we find that u is also a constant. Then, since p and q are constants, using again (30) we have (u + λ − 1)q = 0 and (−u + λ + 1)p = 0. Since λ = 0, we have pq = 0. We divide our arguments into three cases: Then, owing to Perrone's result [26] (or Milnor's result [27]) for the above two cases M is locally isometric to a non-unimodular Lie group with left-invariant contact metric structure.
Using Perrone's result again, M is locally isometric to a unimodular Lie group with left-invariant contact metric structure. Conversely, by Propositions 5 we know that any Sasakian space form is locally pseudo-Hermitian symmetric space and by Proposition 7 we also know that a Lie group is locally pseudo-Hermitian symmetric space. Therefore, we have completed the proof.
We have non-homogeneous locally pseudo-Hermitian symmetric spaces.
For any smooth function f (y, z) of variables y, z, we define a global frame field {e 1 , e 2 , e 3 } by Define a Riemannian metric g such that {e 1 , e 2 , e 3 = ξ} is orthonormal with respect to it. As usual, the endomorphism field φ is defined by φe 1 = e 2 , φe 2 = −e 1 and φe 3 = 0. Then we see that M f = (R 3 ; η, g) is a Jacobi (0, 0)-contact space, but its vertical Ricci curvature σ(φe) = 2 f are not constant, in general. Then we have locally pseudo-Hermitan symmetric spaces of constant holomorphic sectional curvature (for∇)Ĥ = c for all c ∈ R by the following examples: • f = z: M f has a constant holomorphic sectional curvatureĤ = 3. It is not locally φ-symmetric in the sense of Reference [5]. • f = α (const.): M f has a constant holomorphic sectional curvatureĤ = 2 + α 2 . It is curvature homogeneous. A locally φ-symmetric space occurs only when α = 0.