Some Non-Compactness Results for Locally Homogeneous Contact Metric Manifolds

Results Math (2022) 77:150 c 2022 The Author(s) 1422-6383/22/040001-18 published online June 20, 2022 https://doi.org/10.1007/s00025-022-01699-0 Results in Mathematics Some Non-Compactness Results for Locally Homogeneous Contact Metric Manifolds Antonio Lotta and Ver´onica Mart´ın-Molina Abstract. 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. Mathematics Subject Classification. 53C25, 53C30. Keywords. Locally homogeneous, contact metric manifold, regular contact manifold, Jacobi operator. 1. 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)=1anddη(X, ξ)=0for every smooth vector field Xon 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 gto ηis a Riemannian metric for which there exists a (1,1) tensor field ϕ:TM →TM such that ϕ2=−Id+η⊗ξ, η(X)=g(X,ξ),dη(X,Y )=g(X, ϕY ), V. Mart´ın-Molina was partially supported by the group FQM-226 of the Junta de Andaluc´ıa (Spain) and by the grant VIPPIT-IV.4 of the Universidad de Sevilla (Spain). 150 Page 2 of 18 A. Lotta and V. Mart´ın-Molina Results Math for every X,Y vector fields on M. The tensor field ϕis uniquely determined by g, and the tensors (ϕ, ξ, η, g) make up a contact metric structure on M. On every contact manifold, associated metrics abound; indeed, by the polarization technique, one can construct them starting from the non-degenerate (symplectic) two-form dη on the contact subbundle D. A choice of such a metric endows the contact manifold of an almost CR-structure (D,J), where the complex structure Jis the restriction ϕ|D. The most studied class of associated metrics is that of Sasakian ones, strictly related to K¨ahler geometry, characterized by the additional requirements that (D,J) should be a (formally integrable) CR-structure and moreover ξshould be a Killing vector field. For any associated metric g, this last condition is also equivalent to Lξϕ=0, and, when it holds, gis usually called a K-contact metric. In the general case, the above Lie derivative is a non-null symmetric (1,1) tensor field; in the literature, it is customary to adopt the notation h:= 1 2Lξϕ. The operator hhas a key role in the development of the theory of general contact metric structures. Another important symmetric operator is the Jacobi operator associated to ξ,l:= R(−,ξ)ξ, where Ris the curvature tensor. Observe that hand lboth annihilate ξ, so they can be thought of as endomorphisms of the contact subbundle. The behavior of these two operators, which appear in several key formulas, greatly influence the geometry of the underlying contact metric manifold. The aim of this paper is to discuss an instance of this fact, exhibiting sufficient conditions ensuring the non-compactness of the contact manifold. We consider the class of locally homogeneous contact metric manifolds, whose contact form is regular, meaning essentially that the orbit space M/ξ determined by the flow of the Reeb vector field is smooth and the canonical projection π:M→M/ξ is a submersion (for more details, see for instance [1, Chapter 3]). We remark that this class contains all the homogeneous contact metric manifolds, due to a general result of Boothy-Wang [8]. We shall restrict to the case where land hare most nicely related: namely, our basic assumption will be: l|D(λ)=cλId (*) for every eigenvalue λof h:D→D, with eigenbundle D(λ), where cλis a constant. Many relevant examples of (locally) homogeneous, non K-contact, contact metric manifolds satisfy this condition. We cite here the (κ, μ)-spaces, studied by several authors (see [1,§7.3], [2], [5]), and tangent sphere bundles over symmetric spaces of rank one (see [6]). Vol. 77 (2022) Some Non-Compactness Results for Locally Page 3 of 18 150 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 Qcommutes with ϕ, i.e., [Q, ϕ] = 0. Indeed, in this case, on the contact subbundle, land hare related by l=Id−h2+ 2(1 −n)h. In particular, this is true for η-Einstein pseudo-Hermitian manifolds, which are characterized by Q=αId+βη ⊗ξ, α,β ∈R. Instead of [Q, ϕ] = 0, one can also consider the case where Qis of the form: Q=αId+βη ⊗ξ+γh, α,β,γ ∈R. 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 lsatisfies: l|D(λ)=cλId, for every eigenvalue of h. If, for each positive eigenvalue λof h, we have cλ=(λ+1) 2and, for at least one of them, we have: cλ∈[(λ+1) 2−4λ2,(λ+1) 2), then Mis 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 ∇ ξh= 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 aλon 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, 150 Page 4 of 18 A. Lotta and V. Mart´ın-Molina Results Math 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+1that satisfies Ric(ξ,ξ)≤2n(1 −n2) (1.1) is not compact. We remark that the inequality (1.1) is optimal; this is showed by giving suitable examples of compact homogeneous, η-Einstein pseudo-Hermitian manifolds that satisfy Ric(ξ,ξ)>2n(1 −n2) (see Proposition 4.8). 2. Preliminaries Given a contact manifold (M,η), we will denote by Dthe contact distribution, which is the 2n-dimensional distribution defined by ker(η). We can write the tangent bundle of Mas TM =D⊕Rξ, where ξis the Reeb vector field of η. If gis an associated metric to η, it is known (see e.g. [1], [3]) that: ∇ξ=−ϕ−ϕh, (2.1) (∇ξh)=ϕ(Id−h2−l),(2.2) ϕlϕ −l=2(h2+ϕ2),(2.3) Ric(ξ,ξ)=2n−tr(h2),(2.4) where l=R(−,ξ)ξis the Jacobi operator relative to ξ.Moreover,his a symmetric operator, hanticommutes with ϕ,andtrh= 0 (for example, Lemma 6.2. of [1]). An a consequence, we recall that, if λis an eigenvalue of h, then −λ is also an eigenvalue. It is also known that hξ = 0. So, whenever convenient, hcan also be considered as an endomorphism h:D→Dof the contact subbundle. In the case of pseudo-Hermitian manifolds that satisfy [Q, ϕ] = 0, on the contact subbundle, land hare related by l=Id−h2+ 2(1 −n)h, (2.5) because of the following identity [1, Prop. 10.2]: Qϕ −ϕQ =lϕ −ϕl +4(n−1)hϕ −η⊗ϕQξ +η◦Qϕ ⊗ξ. We end this section by recalling the notion of contact metric (κ, μ)-spaces. These are contact metric manifolds satisfying the equation [2] R(X,Y )ξ=κ(η(Y)X−η(X)Y)+μ(η(Y)hX −η(X)hY ), for every vector fields X,Y on M, where κand μare constants. The (κ, μ)- spaces obviously include the Sasakian manifolds (κ=1andh= 0) and they Vol. 77 (2022) Some Non-Compactness Results for Locally Page 5 of 18 150 are non-Sasakian, locally homogeneous, strongly pseudoconvex CR manifolds when κ<1. In this case, the spectrum of his {0,−√1−κ, √1−κ}. Applying aDa-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:= aη, ¯ ξ:= 1 aξ, ¯g=ag +a(a−1)η⊗η, where ais a positive constant. Then one has: κ=κ+a2−1 a2,μ =μ+2a−2 a.(2.6) It is known that the Ricci operator of a (κ, μ)-space can be written as Q= (2(n−1) −nμ)Id+ (2(1 −n)+n(2κ+μ))η⊗ξ+ (2(n−1) + μ)h. (2.7) 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 Da-homothetic deformations, they form a oneparameter family parametrized by R. This family contains the tangent sphere bundles T1Swhere Sis a Riemannian space form. The classification relies on a result of Boeckx [5], stating that the number IM=1−μ/2 √1−κcompletely determines a contact metric (κ, μ)-space Mlocally up to equivalence and up to Da-homotetic deformations of its contact metric structure. We shall call IM the Boeckx invariant of the (κ, μ)-space. 3. 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 hhas constant eigenvalues with constant multiplicity, since his preserved by all the local automorphisms of the geometric structure ([7, Lemma 10]). We denote by Sthe set of eigenvalues of h|D,byλithe positive eigenvalues in S,byD(λ) the eigendistribution of hassociated with the eigenvalue λ, which is a vector subbundle of TM, and by [ξ] the one-dimensional subbundle spanned by the vector field ξ. Therefore, we can write TM as: TM =[ξ]⊕D(0) ⊕D(λ1)⊕···D(λm)⊕D(−λ1)⊕···⊕D(−λm),if 0 ∈S, [ξ]⊕D(λ1)⊕···D(λm)⊕D(−λ1)⊕···⊕D(−λm),if 0 /∈S. (3.1) The following simple property of the sections of the subbundles D(λ) shall be useful next. 150 Page 6 of 18 A. Lotta and V. Mart´ın-Molina Results Math Lemma 3.1. Let M(ϕ, ξ, η, g)be a locally homogeneous contact metric manifold. Then, for every X∈D(λ)and Y∈D(μ), we have: g((∇ξh)X,Y )=(λ−μ)g(∇ξX,Y ). Proof. Since his symmetric and the eigenvalues are constant, we obtain: g((∇ξh)X,Y )=λg(∇ξX,Y )−g(∇ξX,hY )=(λ−μ)g(∇ξX,Y ), 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 ϕξ=0,ϕ =aλϕon D(λ), and the metric gas g(X,Y )=⎧ ⎪ ⎨ ⎪ ⎩ 1,if X=Y=ξ, aλg(X,Y ),if X, Y ∈D(λ), 0,otherwise, with aλa constant attached to each λin the spectrum of h; these constants are required to satisfy the following constrains: aλ⎧ ⎪ ⎨ ⎪ ⎩ =1 ifλ=0, >0ifλ=0, =1 a−λ. 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. If the manifold is not K-contact, we start by verifying ϕ2=−Id+η⊗ξ. Indeed, taking into account the splitting (3.1), when computing ϕ2(X), we just have to consider two possible cases: – Case X=ξ. Then ϕξ=ϕξ =0,thusϕ2ξ= 0. On the other hand, −ξ+η(ξ)ξ=−ξ+ξ=0. – Case X∈D(λ). Then ϕX ∈D(−λ), so that ϕ2X=aλϕ(ϕX)=aλ1 a−λϕ2X=ϕ2X=−X+η(X)ξ. Vol. 77 (2022) Some Non-Compactness Results for Locally Page 7 of 18 150 Thus the structure is almost contact. 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: – Case X,Y =ξ. Then g(ϕξ,ϕξ)=0andg(ξ,ξ)−η(ξ)η(ξ)=1−1=0. – Case X=ξ,Y∈D(λ). Then g(ϕξ,ϕY)=0andg(ξ,Y )−η(ξ)η(Y)= 0−1·0=0. – Case X,Y ∈D(λ). Then g(ϕX,ϕY)=a2 λg(ϕX, ϕY )=a2 λ(a−λ)g(ϕX, ϕY )=a2 λ1 aλg(ϕX, ϕY ) =aλg(ϕX, ϕY )=aλ(g(X,Y )−η(X)η(Y)) = aλg(X,Y ) and g(X,Y )−η(X)η(Y)=aλg(X, Y ). – Case X∈D(λ), Y∈D(μ) with λ=μ. Then g(ϕX,ϕY)=aλaμg(ϕX, ϕY )=0 and g(X,Y )−η(X)η(Y)=0+0. 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)andY∈D(−λ), then g(X,ϕY)=a−λg(X,ϕY )=a−λaλg(X,ϕY ) =1 aλaλg(X,ϕY )=g(X,ϕY ). 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 lhas the simplest behavior with respect to the splitting (3.1). Namely, we shall require that lshould preserve the splitting and that, for every eigenvalue λof h: l|D(λ)=cλId, where cλis a constant. Under this assumption, we shall determine the operator h=1 2Lξϕof a deformed contact metric structure and its covariant derivative ∇ ξh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. 150 Page 8 of 18 A. Lotta and V. Mart´ın-Molina Results Math Lemma 3.4. Let M(ϕ, ξ, η, g)be a locally homogeneous contact metric manifold. Assume that l|D(λ)=cλId for every eigenvalue λof h|D.Then, c−λ=−cλ−2λ2+2,(3.2) In particular, if λ=0, then cλ=1. Proof. Given X∈D(λ), we have that hX =λX and lX =cλX.Moreover, since ϕX ∈D(−λ), then hϕX =−λϕX and lϕX =c−λϕX. Substituting these formulas in Eq. (2.3) and using standard contact metric properties give us −(c−λ+cλ)X=2(λ2−1)X, from which Eq. (3.2) follows. When λ= 0, we have that −(c0+c0)=−2, so c0=1.  Lemma 3.5. Let M(ϕ, ξ, η, g)be a locally homogeneous contact metric manifold. Assume that l|D(λ)=cλId for every eigenvalue λof h|D. Then, for every X∈D(λ)and Y∈D(μ) satisfying λ+μ=0, we have: g([ξ,X],ϕY)=1−λ2−cλ λ+μ+λ+1 g(X,Y ). Proof. Using standard properties of the Levi-Civita connection ∇, Lemma 3.1 and Eq. (2.1), it follows that g([ξ,X],ϕY)=g(∇ξX, ϕY )−g(∇Xξ,ϕY ) =1 λ+μg((∇ξh)X,ϕY )+(λ+1)g(X, Y ). Applying now Eq. (2.2) and the hypothesis on l, we obtain that g([ξ,X],ϕY)= 1 λ+μg(X−h2X−lX,Y )+(λ+1)g(X, Y ) =1 λ+μg(X−λ2X−cλX,Y )+(λ+1)g(X,Y ) =1−λ2−cλ λ+μ+λ+1 g(X,Y ), concluding the proof.  We can now determine the operator hof the deformed contact metric structures. Proposition 3.6. Let M(ϕ, ξ, η, g)be a locally homogeneous contact metric manifold. Assume that l|D(λ)=cλId, Vol. 77 (2022) Some Non-Compactness Results for Locally Page 9 of 18 150 for every eigenvalue λof 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: h|D(λ)=ρλId, where ρλ=(−(λ+1)2+4λ2+cλ)a2 λ+(λ+1)2−cλ 4λaλfor λ=0, 0for λ=0.(3.3) 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)andY∈D(μ), by the definition of hand ϕ,wehave that: 2g(hX,Y )=g([ξ,ϕX],Y)−g(ϕ[ξ,X],Y)=aλg([ξ,ϕX],Y)−g(ϕ[ξ,X],Y). (3.4) We will now consider two cases: μ=−λand μ=−λ. If μ=−λ, we obtain the next equation from the definition of h: 2g(hX,Y )=aλg([ξ,ϕX],Y)−aλg(ϕ[ξ,X],Y)=2aλg(hX, Y )=2aλλg(X,Y ). Now, in the subcase λ= 0 this gives directly g(hX, Y ) = 0. Assuming λ=0, then g(X,Y ) = 0 since eigenvectors belonging to different eigendistributions are orthogonal, and hence we have g(hX,Y ) = 0 again. If μ=−λ, then substituting [ξ,ϕX]=2hX +ϕ[ξ,X]=2λX +ϕ[ξ,X] in Eq. (3.4) gives us: 2g(hX,Y )=2aλλg(X, Y )+aλg(ϕ[ξ,X],Y)−g(ϕ[ξ,X],Y). Using the definition of ϕand that ϕis anti-symmetric, we obtain: 2g(hX,Y )=2aλλg(X, Y )+(aλ−a−μ)g(ϕ[ξ,X],Y) =2aλλg(X,Y )−(aλ−a−μ)g([ξ,X],ϕY). It now follows from Lemma 3.5 that 2g(hX,Y )=2aλλ−(aλ−a−μ)1−λ2−cλ λ+μ+λ+1 g(X, Y ). In the subcase μ=λ, we know that g(X,Y )=0,sog(hX, Y )=0. In the subcase μ=λ=0,wehavethat g(hX,Y )=1 22aλλ−(aλ−a−λ)1−λ2−cλ 2λ+λ+1 g(X, Y ) =(4λ2−(λ+1) 2+cλ)a2 λ+(λ+1) 2−cλ 4λaλ g(X,Y ), thus ending the proof.  150 Page 16 of 18 A. Lotta and V. Mart´ın-Molina Results Math Proof. After performing a Da-homothetic deformation of the contact metric structure, which preserves the Boeckx invariant, we may suppose that κ=0. So l=μh and IM=1−μ 2and it follows that μ∈[0,4). Moreover, κ=0 means that hhas 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 Mbe locally homogeneous, regular pseudo-Hermitian manifold of dimension 2n+1, whose Ricci operator Qsatisfies: [Q, ϕ]=0. Then Mis not compact, provided that hadmits 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 Mis not compact.  A similar argument applies also to the following situation: Theorem 4.6. Let Mbe locally homogeneous, regular pseudo-Hermitian manifold of dimension 2n+1, whose Ricci operator Qsatisfies: Q=αId+βη ⊗ξ+γh, α,β,γ ∈R. Then Mis not compact, provided γ−2n 2is not in the spectrum of hand 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: l=Id−h2+(γ+ 2(1 −n))h. The rest of the proof is similar to that of Theorem 4.5. Theorem 4.7. A locally homogeneous, regular, η-Einstein, pseudo-Hermitian manifold of dimension 2n+1that satisfies Ric(ξ,ξ)≤2n(1 −n2) (4.6) is not compact. Proof. This follows from Theorem 4.5. Indeed, the hypothesis [Q, ϕ] = 0 holds true because the manifold is η-Einstein. There must be an eigenvalue of h greater than or equal to nbecause (2.4) and the hypothesis mean that Ric(ξ,ξ)=2n−tr(h2)≤2n(1 −n2) and thus tr(h2)≥2n3. Lastly, we prove that the inequality (4.6) is optimal. Vol. 77 (2022) Some Non-Compactness Results for Locally Page 17 of 18 150 Proposition 4.8. For any n≥1and rwith 2n(1 −n2)<r≤2n, there exists a homogeneous, η-Einstein, pseudo-Hermitian manifold of dimension 2n+1 that satisfies Ric(ξ,ξ)=rand is compact. Proof. Let us distinguish two cases: r=2nand 2n(1 −n2)<r<2n. Case r=2n. It suffices to consider a sphere S2n+1 endowed with the standard Sasakian structure. Case 2n(1−n2)<r<2n. If we define c:= (√2n3−√2n−r)2 2n3−(2n−r), then 0 <c<1 and we can take a sphere S=Sn+1 with constant curvature c. Then its tangent sphere bundle M=T1S, with the standard contact metric structure, is a compact homogeneous (κ, μ)-space M2n+1 with κ=c(2 −c)<1and μ=−2c<0([2]). If we now perform a Da-homothetic deformation with constant a=c+1 n>0, then it follows from (2.6) that the deformed manifold is a non-Sasakian (κ,μ )-space with κ=r 2n<1andμ= 2(1 −n) and thus it is η-Einstein. Since Qξ=2nκξ=rξ, it also satisfies that Ric(ξ,ξ) =r. Funding Open access funding provided by Universit´a degli Studi di Bari Aldo Moro within the CRUI-CARE Agreement. Data availability statement Data sharing not applicable to this article as no datasets were generated or analysed during the current study. Declarations Conflict of interest The authors declare that they have no conflict of interest. Open Access. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons. org/licenses/by/4.0/. References [1] Blair, D.E.: Riemannian geometry of contact and symplectic manifolds, 2nd edn. Progress in Mathematics 203. Birkh¨auser, Boston (2010) [2] Blair, D.E., Koufogiorgos, T., Papantoniou, B.J.: Contact metric manifolds satisfying a nullity condition. Israel J. Math. 91, 189–214 (1995) 150 Page 18 of 18 A. Lotta and V. Mart´ın-Molina Results Math [3] Blair, D.E., Perrone, D.: Conformally Anosov flows in contact metric geometry. Balkan J. Geom. Appl. 3, 33–46 (1998) [4] Boeckx, E.: A class of locally ϕ-symmetric contact metric spaces. Arch. Math. 72, 466–472 (1999) [5] Boeckx, E.: A full classification of contact metric (κ, μ)-spaces. Illinois J. 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Orabona 4 70125 Bari Italy e-mail: [email protected] Ver´onica Mart´ın-Molina Departamento de Did´actica de las Matem´aticas, Facultad de Ciencias de la Educaci´on, c/ Pirotecnia s/n Universidad de Sevilla Sevilla 41013 Spain e-mail: [email protected] Received: June 25, 2021. Accepted: May 25, 2022. Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.