We consider the problem of characterizing Sasakian manifolds of constant ϕ-sectional curvature by using the spectrum 2Spec of the Laplace-Beltrami operator acting on 2-forms. In particular, we show that the sphere S2n+1, equipped with a Berger-Sasakian metric, is characterized by its 2Spec in the class of all compact simply connected Sasakian manifolds

The purpose of this paper is to classify all simply connected homogeneous almost cosymplectic three-manifolds. We show that each such three-manifold is either a Lie group G equipped with a left invariant almost cosymplectic structure or a Riemannian product of type R × N, where N is a Kähler surface of constant curvature. Moreover, we find that the Reeb vector field of any homogeneous almost cosymplectic three-manifold, except one case, defines a harmonic map.

We introduce a systematic study of contact structures with pseudo-Riemannian associated metrics, emphasizing analogies and differences with respect to the Riemannian case. In particular,we classify contact pseudo-metric manifolds of constant sectional curvature, threedimensional locally symmetric contact pseudo-metric manifolds and three-dimensional homogeneous contact Lorentzian manifolds.

We introduce and study H-paracontact metric manifolds, that is, paracontact metric manifolds whose Reeb vector feld xi is harmonic. We prove that they are characterized by the condition that xi is a Ricci eigenvector. We then investigate how harmonicity of the Reeb vector field xi of a paracontact metric manifold is related to some other relevant geometric properties, like infinitesimal harmonic transformations and paracontact Ricci solitons.

We introduce a new family of Riemannian metrics on the three-sphere and study its geometric properties, starting from the description of their curvature. Such metrics, which include the standard metric and Berger metrics as special cases, are called “of Kaluza–Klein type”, because they are induced in a natural way by the corresponding metrics defined on the tangent sphere bundle of the two-sphere.

We characterize H-contact semi-Riemannian manifolds (i.e., contact semi-Riemannian manifolds whose Reeb vector field ξ is harmonic) by the condition that ξ is a Ricci eigenvector. We then investigate how H-contact semi-Riemannian manifolds are related to some relevant geometric properties, like the Reeb vector field being an infinitesimal harmonic transformation or the contact semi-Riemannian structure being a contact Ricci soliton, and we determine to what extent known results for contact Riemannian manifolds remain valid in the contact Lorentzian case.

Let (TM, G) and (T1M, ˜G ) respectively denote the tangent bundle and the unit tangent sphere bundle of a Riemannian manifold (M, g), equipped with arbitraryRiemannian g-natural metrics. After studying the geometry of the canonical projections π : (TM, G) → (M, g) and π1 : (T1M, ˜G) → (M, g), we give necessary and sufficient conditions for π and π1 to be harmonic morphisms. Some relevant classes of Riemannian g-natural metrics will be characterized in terms of harmonicity properties of the canonical projections. Moreover, we study the harmonicity of the canonical projection : (TM −{0}, G) → (T1M, ˜G ) with respect to Riemannian g-natural metrics G, ˜G of Kaluza–Klein type.

Let $(M,g)$ be a Riemannian manifold. When $M$ is compact and the tangent bundle $TM$ is equipped with the Sasaki metric $g^s$, parallel vector fields are the only harmonic maps from $(M,g)$ to $(TM,g^s)$. The Sasaki metric, and other well-known Riemannian metrics on $TM$, are particular examples of $g$-natural metrics. We equip $TM$ with an arbitrary $g$-natural Riemannian metric $G$, and investigate the harmonicity properties of a vector field $V$ of $M$, thought as a map from $(M,g)$ to $(TM,G)$. We then apply this study to the Reeb vector field and, in particular, to Hopf vector fields on odd-dimensional spheres.

The main object in this monograph are harmonic vector fields on Riemannian manifolds. Let (M,g) be an n-dimensional Riemannian manifold, and S(M) be its tangent sphere bundle, where S(M) is endowed with a Riemannian metric G_s (the Sasaki metric) associated to the Riemannian metric g on M. Thus for a smooth unit tangent field X:M→S(M) from (M,g) into (S(M),G_s), we can consider the Dirichlet energy E(X)=1/2∫_M ∥dX∥2 dv_g. A harmonic vector field is a critical point of E(X) for any smooth 1-parameter variation of X through unit tangent vector fields. In terms of the Euler-Lagrange equation, a harmonic vector field is a smooth solution of the nonlinear elliptic PDE system ΔgX=∥∇X∥^2 X, where Δ_g is the rough Laplacian. Harmonic vector fields are not necessarily harmonic maps except when the curvature condition trace_g {R(∇⋅X,X)⋅}=0 is satisfied. The resulting theory of harmonic vector fields is similar in many respects to the more consolidated theory of harmonic maps yet presents new and intriguing aspects captured in a rapidly growing specific literature. The monograph (over 500 pages) is self-contained in the field of harmonic vector fields, and consists of eight chapters, as follows: (1) Geometry of the tangent bundle, (2) Harmonic vector fields, (3) Harmonicity and stability, (4) Harmonicity and contact metric structures, (5) Harmonicity with respect to g-natural metrics, (6) The energy of sections, (7) Harmonic vector fields in CR geometry, and (8) Lorentz geometry and harmonic vector fields.

We introduce and study a new family of pseudo-Riemannian metrics on the anti-de Sitter three-space $H^3_1$. These metrics will be called “of Kaluza-Klein type” , as they are induced in a natural way by the corresponding metrics defined on the tangent sphere bundle $T_1 H_2(κ)$. For any choice of three real parameters λ,μ, ν neq 0, the pseudo-Riemannian manifold $(H^3_1, g_λμν)$ is homogeneous. Moreover, we shall introduce and study some natural almost contact and paracontact structures (ϕ, ξ, η), compatible with $g_λμν$ , such that (ϕ, ξ, η, g_λμν) is a homogeneous almost contact (respectively, paracontact) metric structure. These structures will be then used to show the existence of a three-parameter family of homogeneous metric mixed 3-structures on the anti-de Sitter three-space.

Let (M, g) be a Riemannian manifold and T_1M its unit tangent sphere bundle. Minimality and harmonicity of unit vector fields have been extensively studied considering on T_1M the Sasaki metric G_S. This metric, and other well known Riemannian metrics on T_1M, are particular examples of Riemannian natural metrics. In this paper we equip T_1M with a Riemannian natural metric G and in particular with a natural contact metric structure. Then, we study the minimality for Reeb vector fields of contact metric manifolds and of quasi-umbilical hypersurfaces of a Kaehler manifold. Several explicit examples are given. In particular, the Reeb vector field ξ of a K-contact manifold is minimal for any G belongs to a family depending on two parameters of metrics of Kaluza-Klein type. Next, we show that the Reeb vector field ξ of a K-contact manifold defines a harmonic map ξ : (M, g) → (T_1M, G) for any Riemannian natural metric G. Besides, if the Reeb vector ξ of an almost contact metric manifold is a CR map then the induced almost CR structure on M is strictly pseudoconvex and ξ is a pseudohermitian map; if in addition ξ is geodesic then ξ : (M, g) → (T_1M, G) is a harmonic map. Moreover, the Reeb vector field ξ of a contact metric manifold is a CR map iff ξ is Killing and G is a special metric of Kaluza-Klein type. Finally, in the last Section, we get that there is a family of strictly pseudoconvex CR structure on T_1S^{2n+1} depending on two-parameter for which a Hopf vector field ξ determines a pseudoharmonic map (in the sense of Barletta-Dragomir-Urakawa) from S^{2n+1} to T_1S^{2n+1}.