Some Non-Compactness Results for Locally Homogeneous Contact Metric Manifolds

We exhibit some sufficient conditions ensuring the non-compactness of a locally homogeneous, regular, contact metric manifold, under suitable assumptions on the Jacobi operator of the Reeb vector field.


Introduction
Let (M, η) be a contact manifold of dimension 2n + 1, n ≥ 1, i.e. a smooth manifold endowed with a fixed contact form η, that is, a 1-form satisfying: η ∧ (dη) n = 0 everywhere on M . This condition singles out a distinguished globally defined vector field ξ, called the Reeb vector field, which is transverse to the contact distribution D = Ker(η), and such that η(ξ) = 1 and dη(X, ξ) = 0 for every smooth vector field X on M . Many results are known concerning the Riemannian geometry of associated metrics on (M, η) (see for instance the standard monograph [1], where the reader can find several examples and an extensive bibliography on this subject).
We recall that an associated metric g to η is a Riemannian metric for which there exists a (1, 1) tensor field ϕ : T M → T M such that ϕ 2 = −Id + η ⊗ ξ, η(X) = g(X, ξ), dη(X, Y ) = g(X, ϕY ), Another wide class can be considered in the context of pseudo-Hermitian geometry. Recall that a pseudo-Hermitian manifold is a contact metric manifold whose almost CR-structure (D, J) is formally integrable; it is a strongly pseudoconvex CR manifold. For these manifolds, condition (*) holds automatically under the assumption that the Ricci operator Q commutes with ϕ, i.e., [Q, ϕ] = 0. Indeed, in this case, on the contact subbundle, l and h are related by In particular, this is true for η-Einstein pseudo-Hermitian manifolds, which are characterized by Instead of [Q, ϕ] = 0, one can also consider the case where Q is of the form: All these relevant cases are examined in the last section (see Theorems 4.5, 4.6 and 4.7).
In the above set up, our main result is the following: Theorem 1.1. Let M (ϕ, ξ, η, g) be a locally homogeneous, regular contact metric manifold. Assume that the Jacobi operator l satisfies: for every eigenvalue of h. If, for each positive eigenvalue λ of h, we have c λ = (λ + 1) 2 and, for at least one of them, we have: then M is not compact.
The method of proof consists in constructing a suitable deformation of the original metric g, determining a contact metric structure (ϕ , ξ, η, g ) that satisfies ∇ ξ h = 0 and whose operator h admits at least one eigenvalue λ ≥ 1. But this turns out to be impossible on a compact, regular contact manifold, due to an argument similar to a result about critical metrics in [1, §10.3]. The deformations employed in the proof are given essentially by a rescaling of the original metric by a suitable positive constant factor a λ on each eigenbundle D(λ) (see Definition 3.2).
In Sect. 3, we develop some properties of this kind of deformations for every locally homogeneous contact metric manifold, under our basic assumption (*). In particular, we first discuss when such a deformed structure is a K-contact one (Theorem 3.8). This is also of interest in itself, because it is known that the existence of a K-contact metric is also a strong condition, imposing topological restrictions in the compact case (for instance, a torus cannot carry a K-contact metric structure). In the last section, we apply these properties to prove our main result and some of its applications. In particular, we prove: Theorem 1.2. A locally homogeneous, regular, η-Einstein pseudo-Hermitian manifold of dimension 2n + 1 that satisfies is not compact.

Preliminaries
Given a contact manifold (M, η), we will denote by D the contact distribution, which is the 2n-dimensional distribution defined by ker(η). We can write the tangent bundle of M as T M = D ⊕ Rξ, where ξ is the Reeb vector field of η.
If g is an associated metric to η, it is known (see e.g. [1], [3]) that: where l = R(−, ξ)ξ is the Jacobi operator relative to ξ. Moreover, h is a symmetric operator, h anticommutes with ϕ, and trh = 0 (for example, Lemma 6.2. of [1]). An a consequence, we recall that, if λ is an eigenvalue of h, then −λ is also an eigenvalue. It is also known that hξ = 0. So, whenever convenient, h can also be considered as an endomorphism h : D → D of the contact subbundle.
In the case of pseudo-Hermitian manifolds that satisfy [Q, ϕ] = 0, on the contact subbundle, l and h are related by We end this section by recalling the notion of contact metric (κ, μ)-spaces. These are contact metric manifolds satisfying the equation [2] for every vector fields X, Y on M , where κ and μ are constants. The (κ, μ)spaces obviously include the Sasakian manifolds (κ = 1 and h = 0) and they Applying a D a -homothetic deformation to a (κ, μ)-space yields another (κ , μ )-space; we recall that such a deformation is given by the following change of the structural tensors of M :η where a is a positive constant. Then one has: It is known that the Ricci operator of a (κ, μ)-space can be written as Hence, the η-Einstein ones are characterized by μ = 2(1 − n). The simply connected, complete, non Sasakian (κ, μ)-spaces are all homogeneous and are completely classified (see [5,9,11]) and, considering two such spaces equivalent up to D a -homothetic deformations, they form a oneparameter family parametrized by R. This family contains the tangent sphere bundles T 1 S where S is a Riemannian space form. The classification relies on a result of Boeckx [5], stating that the number I M = 1−μ/2 √ 1−κ completely determines a contact metric (κ, μ)-space M locally up to equivalence and up to D a -homotetic deformations of its contact metric structure. We shall call I M the Boeckx invariant of the (κ, μ)-space.

A Family of Deformations of a Locally Homogeneous Contact Metric Structure
Given a locally homogeneous contact metric manifold M (ϕ, ξ, η, g), the symmetric operator h has constant eigenvalues with constant multiplicity, since h is preserved by all the local automorphisms of the geometric structure ([7, Lemma 10]). We denote by S the set of eigenvalues of h |D , by λ i the positive eigenvalues in S, by D(λ) the eigendistribution of h associated with the eigenvalue λ, which is a vector subbundle of T M, and by [ξ] the one-dimensional subbundle spanned by the vector field ξ. Therefore, we can write T M as: (3.1) The following simple property of the sections of the subbundles D(λ) shall be useful next.
Proof. Since h is symmetric and the eigenvalues are constant, we obtain: thus proving the lemma.
Using the decomposition (3.1), we shall now define a family of deformed structures (ϕ , ξ, η, g ), all of which are associated with the given contact form.

Definition 3.2.
We take ϕ as and the metric g as with a λ a constant attached to each λ in the spectrum of h; these constants are required to satisfy the following constrains: We will now give sufficient conditions for this family of structures to be contact metric. Proposition 3.3. Given a locally homogeneous contact metric manifold (M, ϕ, ξ, η, g), then each structure (ϕ , ξ, η, g ) of the family defined in Definition 3.2 is a contact metric structure.
Proof. If the manifold is K-contact, then the deformation leaves invariant the contact metric structure.
The new metric, g , is compatible with the almost contact structure, i.e. g (ϕ X, ϕ Y ) = g (X, Y )−η(X)η(Y ). This follows by considering the following cases: and Thus our structure is almost contact metric. Finally, we can check that g is associated to the contact form η, i.e. that dη(X, Y ) = g (X, ϕ Y ). This also can be proved case by case. For instance, if X ∈ D(λ) and Y ∈ D(−λ), then In all the remaining cases, the verification is similar and straightforward and we omit it for the sake of brevity.
From now on, we shall restrict our study to locally homogeneous contact metric manifolds whose Jacobi operator l has the simplest behavior with respect to the splitting (3.1). Namely, we shall require that l should preserve the splitting and that, for every eigenvalue λ of h: Under this assumption, we shall determine the operator h = 1 2 L ξ ϕ of a deformed contact metric structure and its covariant derivative ∇ ξ h with respect to the new metric. This information will be used in the next section to prove our main results.
Let us first see a couple of necessary lemmas.
Proof. Given X ∈ D(λ), we have that hX = λX and lX = c λ X. Moreover, since ϕX ∈ D(−λ), then hϕX = −λϕX and lϕX = c −λ ϕX. Substituting these formulas in Eq. (2.3) and using standard contact metric properties give us Proof. Using standard properties of the Levi-Civita connection ∇, Lemma 3.1 and Eq. (2.1), it follows that Applying now Eq. (2.2) and the hypothesis on l, we obtain that concluding the proof.
We can now determine the operator h of the deformed contact metric structures.
for every eigenvalue λ of h |D . Consider a deformation (ϕ , ξ, η, g ) of the contact metric structure according to Definition 3.2. Then the corresponding operator h is determined as follows: Proof. Since the deformed structure is contact metric, h ξ = 0. Therefore, in order to determine h completely, we only have to check h | D . Given X ∈ D(λ) and Y ∈ D(μ), by the definition of h and ϕ , we have that: (3.4) We will now consider two cases: μ = −λ and μ = −λ.
If μ = −λ, we obtain the next equation from the definition of h: Using the definition of ϕ and that ϕ is anti-symmetric, we obtain: It now follows from Lemma 3.5 that In the subcase μ = λ, we know that g(X, Y ) = 0, so g(h X, Y ) = 0. In the subcase μ = λ = 0, we have that thus ending the proof.
Proof. Equation We can give sufficient and necessary conditions for the deformed structures to be K-contact.
(3.6) In this case, such a deformation is unique and it is determined by setting: Proof. First we remark that, if (3.6) holds for all positive eigenvalues, then it actually holds for all non-null eigenvalues. This is a simple consequence of (3.2).
Hence we obtain: for all λ = 0. This is possible provided the right-hand side of this equality is positive for all λ = 0, which is equivalent to the condition (3.6). Vice versa, if condition (3.6) holds true, a deformation of the contact metric structure can be defined by the formula in (3.7), since the argument of each square root is positive. Moreover, if λ = 0 is in the spectrum of h, we have a 0 = 1 and, We can also give sufficient conditions for the locally homogeneous, pseudo-Hermitian manifolds with [Q, ϕ] = 0 to carry a K-contact structure. Proof. From (3.8) we see that, for each positive λ ∈ S, we have λ < n. Indeed, assume by contradiction λ ≥ n for some positive λ. From (2.4), we have Ric(ξ, ξ) = 2n−tr(h 2 ), which implies tr(h 2 ) ≥ 2n 2 and so Ric(ξ, ξ) ≤ 2n−2n 2 , contrary to the hypothesis. On the other hand, since [Q, ϕ] = 0, the Jacobi operator l is determined according to (2.5) and we have at once that for every positive λ ∈ S. Hence, according to Theorem 3.8, we can perform a deformation of the contact metric structure in order to get a K-contact metric structure on M .
Since a torus does not admit a K-contact structure, we obtain the following result. Next we determine the covariant derivative ∇ ξ h of the deformed contact metric structures. Proposition 3.11. Under the same assumptions of Proposition 3.6, for every λ, μ eigenvalues of h and for every X ∈ D(λ) and Y ∈ D(μ), we have: Proof. Observe first that h is symmetric also with respect to the original metric g, owing to Proposition 3.6. Thus, given X ∈ D(λ) and Y ∈ D(μ), by the definition of h we have that: If λ = μ, then ρ λ = ρ μ and thus g((∇ ξ h )X, Y ) = 0.
If λ = μ, then using standard properties of the Levi-Civita connection ∇ and Eq. (2.1) applied to the deformed structure gives: It follows from the definition of ϕ and from Proposition 3.6 that Since λ = μ, we have by Lemma 3.5 that If we substitute this last equation in the previous one, we obtain Now, in the subcase λ = −μ, we know that ϕX ∈ D(−λ) and Y ∈ D(μ) are orthogonal, and therefore g(ϕX, Y ) = 0, so that g((∇ ξ h )X, Y ) = 0.

Main Results
Theorem 4.1. Let M (ϕ, ξ, η, g) be a locally homogeneous, regular contact metric manifold. Assume that the Jacobi operator l satisfies: for every eigenvalue of h. If, for each positive eigenvalue λ of h, we have c λ = (λ + 1) 2 and, for at least one of them, we have: then M is not compact.
Proof. Consider a locally homogeneous, regular contact metric manifold whose operator h behaves as in the statement. We consider first the case where ∇ ξ h = 0. In this case, according to (2.2), we have l = Id − h 2 , so that and, by assumption, h admits at least a positive eigenvalue λ 0 ≥ 1. We shall adopt a reasoning similar to the proof of [1,Theorem 10.12]. Assume by contradiction that M is compact. Hence, by the regularity assumption, M is a principal fibre bundle over a symplectic manifold B with group and fibre S 1 ( [8]). More precisely, the structure group S 1 is generated by a new Reeb vector fieldξ, which is associated to a suitable contact form η = fη, with f = 0 a scalar function.
Fix p ∈ M , and let γ : R → M be the maximal integral curve of ξ passing through p. Then γ is closed ([8, p. 722]) and therefore a periodic geodesic. Fix a unit eigenvector v ∈ T p M of h with eigenvalue λ 0 . Then v can be extended to a parallel vector field X = X(t) along γ, so that X(0) = v. Since ∇ ξ h = 0, for each t ∈ R, X(t) is also an eigenvector of h with the same eigenvalue λ 0 . Since ∇ ξ ϕ = 0, ϕX(t) is also parallel and an eigenvector with eigenvalue −λ 0 , for all t.
On the other hand, v can also be extended to a vector field Y ∈ X(M ), which is a section of the contact distribution D such that [Y, ξ] = 0. Indeed, consider the bundle projection π : (M,η) → B and the vector u = π * (v).
Now consider the functions α and β defined as:

Corollary 4.2. A locally homogeneous, regular contact metric manifold with vanishing Jacobi operator is not compact.
Proof. Indeed, the previous result includes the case l = 0, because, in that case, Eq. (2.2) yields h 2 = Id, so that λ = 1 is the unique positive eigenvalue of h. Therefore, c 1 = 0 and (4.1) is satisfied.
Another consequence of our main result is the following.

Corollary 4.3.
A locally homogeneous, regular contact metric manifold whose ξ-sectional curvatures satisfy: for every tangent vector X orthogonal to ξ, is not compact.
Proof. It follows from (4.5) that for every tangent vector X orthogonal to ξ. Therefore, for tangent vector fields X, Y orthogonal to ξ, which implies that l = Id − h 2 in the contact subbundle. So l |D(λ) = c λ Id, with c λ = 1 − λ 2 , for every eigenvalue of h. Moreover, c λ < 0 since K(ξ, X) = g(lX, X) = c λ g(X, X) < 0 by assumption. Hence λ > 1 and (4.1) holds true for all positive eigenvalues of h, and thus Theorem 4.1 is applicable. Proof. After performing a D a -homothetic deformation of the contact metric structure, which preserves the Boeckx invariant, we may suppose that κ = 0. So l = μh and I M = 1 − μ 2 and it follows that μ ∈ [0, 4). Moreover, κ = 0 means that h has a unique positive eigenvalue λ = 1, so all the hypotheses of Theorem 4.1 hold, yielding the conclusion.
Next we discuss some applications of Theorem 4.1 to pseudo-Hermitian geometry.
Theorem 4.5. Let M be locally homogeneous, regular pseudo-Hermitian manifold of dimension 2n + 1, whose Ricci operator Q satisfies: Then M is not compact, provided that h admits at least one eigenvalue ≥ n.
Proof. Under the above assumptions, (2.5) holds and we see at once that all the hypotheses of Theorem 4.1 are satisfied, yielding that M is not compact.
A similar argument applies also to the following situation: Theorem 4.6. Let M be locally homogeneous, regular pseudo-Hermitian manifold of dimension 2n + 1, whose Ricci operator Q satisfies: Then M is not compact, provided γ−2n 2 is not in the spectrum of h and h admits at least one positive eigenvalue in the interval [− γ−2n 2 , γ−2n 2 ). Proof. Assuming the above expression for Q, one gets the following identity on the contact subbundle: The rest of the proof is similar to that of Theorem 4.5.
Lastly, we prove that the inequality (4.6) is optimal. Proof. Let us distinguish two cases: r = 2n and 2n(1 − n 2 ) < r < 2n. Case r = 2n. It suffices to consider a sphere S 2n+1 endowed with the standard Sasakian structure.
Funding Open access funding provided by Universitá degli Studi di Bari Aldo Moro within the CRUI-CARE Agreement.
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Conflict of interest
The authors declare that they have no conflict of interest.
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