Using an explicit description in global coordinates of invariant metrics of four-dimensional homogeneous pseudo-Riemannian manifolds, we completely classify all examples of Ricci solitons among these metrics. Yamabe solitons are also completely classified.

We study left-invariant almost paracontact metric structures on arbitrary three-dimensional Lorentzian Lie groups.We obtain a complete classification and description under a natural assumption, which includes relevant classes as normal and almost para-cosymplectic structures, and we investigate geometric properties of these structures.

We completely classify invariant Hermitian and K¨ahler structures, together with their paracomplex analogues, on four-dimensional homogeneous pseudo-Riemannian manifolds with nontrivial isotropy subalgebra.

We obtain a complete classification of four-dimensional conformally flat homogeneous pseudo-Riemannian manifolds.

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 consider Lie groups equipped with a left-invariant cyclic Lorentzian metric.As in the Riemannian case, in terms of homogeneous structures, such metrics can be considered as different as possible from bi-invariant metrics. We show that several results concerning cyclic Riemannian metrics do not extend to their Lorentzian analogues, and obtain a full classification of three- and four-dimensional cyclic Lorentzian metrics.

We introduce a general approach to the study of left-invariant K-contact structures on Lie groups and we obtain a full classification in dimension five

We consider odd-dimensional Lie algebras gequipped with a paracontact metric structure. In the case of non-trivial center, paracontact Lie algebras are obtained as central extensions of almost paraKähler Lie algebras. As such, they are necessarily K-paracontact, and a complete classification is given in dimension five, also specifying the paraSasakian examples. Thus, paracontact, not K-paracontact structures can only occur among Lie algebras with trivial center. Starting from the classification of five-dimensional contact Lie algebras given in [17], examples with trivial center, both K-paracontact and not, are explicitly discussed and classified.

We describe four-dimensional Lie groups equipped with a left-invariant Lorentzian metric, obtaining a complete classification of the Einstein and Ricci-parallel examples.

Four-dimensional locally homogeneous Riemannian manifolds are either locally symmetric or locally isometric to Riemannian Lie groups. We determine how and to what extent this result holds in the Lorentzian case.

We consider four-dimensional homogeneous pseudo-Riemannian manifolds with nontrivial isotropy and completely classify the cases giving rise to non-trivial homogeneous Ricci solitons. In particular, we show the existence of non-compact homogeneous (and also invariant) pseudo-Riemannian Ricci solitons which are not isometric to solvmanifolds, and of conformally flat homogeneous pseudo-Riemannian Ricci solitons which are not symmetric.

A paraK¨ahler Lie algebra is an even-dimensional Lie algebra g endowed with a pair $(J, g)$, where $J$ is a paracomplex structure and $g$ a pseudo-Riemannian metric, such that the fundamental 2-form $Omega(X, Y) = g(X, JY)$ is symplectic. We completely classify left-invariant paraK¨ahler structures on four-dimensional Lie algebras and then study the geometry of their paraK¨ahler metric

We classify Walker structures, self-dual and anti-self-dual metrics among the invariant metrics of four-dimensional generalized symmetric spaces.

We investigate the geometric properties of four-dimensional non-reductive pseudo-Riemannian manifolds admitting an invariant metric of signature $(2,2)$. In particular, we obtain the classification of Walker structures, self-dual and anti-self-dual metrics and para-Hermitian structures for all the invariant metrics of these manifolds. For the examples admitting an invariant parallel null plane distribution, we shall also obtain an explicit description of the invariant metrics in canonical Walker coordinates.

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.

Let (M = G/H; g) denote a four-dimensional pseudo-Riemannian generalized symmetric space and g = m + h the corresponding decomposition of the Lie algebra g of G. We completely determine the harmonicity properties of vector fields belonging to m. In some cases, all these vector fields are critical points for the energy functional restricted to vector fields. Vector fields defining harmonic maps are also classified, and the energy of these vector fields is explicitly calculated.

Let (M, g) be a pseudo-Riemannian manifold. If M is compact, g is Riemannian and thetangent bundle TM is equipped with the Sasaki metric gs, parallel vector fields are the only harmonic maps from (M, g) to (TM, gs). On the other hand, if g is Lorentzian, then vector fields satisfying some harmonicity properties need not be parallel. We investigate such properties for left-invariant vector fields onthree-dimensional Lorentzian Lie groups, obtaining several classification results and new examples of critical points of energy functionals.

We obtain the full classification of invariant contact metric structu res on five-dimensional Riemannian generalized symmetric spaces. Different classes of examples of these spaces show different behaviours.

After obtaining an explicit description in global coordinates of invariant metrics on four-dimensional non-reductive homogeneous pseudo-Riemannian manifolds, for each of these spaces we completely describe Killing and geodesic vector elds and homogeneous geodesics through a point.

The complete classification of three-dimensional homogeneous paracontact metric manifolds is obtained. In the symmetric case, such a manifold is either flat or of constant sectional curvature −1. In the non-symmetric case, it is a Lie group equipped with a left-invariant paracontact metric structure.

We explicitly determine invariant Ricci collineations on four-dimensional non-reductive homogeneous pseudo-Riemannian manifolds, and invariant matter collineations for the Lorentzian examples.

For an arbitrary three-dimensional normal paracontact metric structure equipped with a Killing characteristic vector field, we obtain a complete classification of the magnetic curves of the corresponding magnetic field. In particular, this yields to a complete description of magnetic curves for the characteristic vector field of three-dimensional paraSasakian and paracosymplectic manifolds. Explicit examples are described for the hyperbolic Heisenberg group and a paracosymplectic model.

We characterize three-dimensional manifolds admitting an almost contact metric 3-structure and completely classify left-invariant hypercontact structures on threedimensional Lie groups.

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.

We consider a natural condition determining a large class of almost contact metric structures. We study their geometry, emphasizing that this class shares several properties with contact metric manifolds. We then give a complete classification of left-invariant examples on three-dimensional Lie groups, and show that any simply connected homogeneous Riemannian three-manifold $(M, g)$ admits a natural almost contact structure having $g$ as a compatible metric. Moreover, we investigate left-invariant CR structures corresponding to natural almost contact metric structures.

We consider four-dimensional Lie groups equipped with left-invariant met- rics of signature (2; 2). After describing the general structure of four-dimensional Lie algebras with a neutral inner product, a complete classication of left-invariant Einstein and Ricci-parallel metrics of neutral signature is obtained.

We consider the four-dimensional oscillator group, equipped with a well known one-parameter family of left-invariant Lorentzian metrics. We obtain a full classification of its Ricci (curvature, Weyl) collineations and matter collineations, and also point out the left-invariant collineations.

We consider the four-dimensional oscillator group, equipped with a well-known one parameter family of left-invariant Lorentzian metrics, which includes the bi-invariant one (Gadea & Oubina, 1999). In a suitable system of global coordinates, the Ricci soliton equation for these metrics translates into a system of partial differential equations. Solving such system, we prove that all these metrics are Ricci solitons. In particular, the bi-invariant metric on the oscillator group gives rise to infinitely many Ricci solitons (and so, also to Yamabe solitons).

Starting from g-natural pseudo-Riemannian metrics of suitable signature on the unit tangent sphere bundle T1M of a Riemannian manifold (M,g), we construct a family of paracontact metric structures. We prove that this class of paracontact metric structures is invariant under D-homothetic deformations, and classify paraSasakian and paracontact (κ,μ)-spaces inside this class. We also present a way to build paracontact (κ,μ)-spaces from corresponding contact metric structures on T1M.

We completely classify non-degenerate surfaces with parallel second fundamental form in three-dimensional Riemannian and Lorentzian reducible spaces.

We study the geometry of non-reductive four-dimensional homogeneous spaces. In particular, after describing their Levi-Civita connection and curvature properties, we classify homogeneous Ricci solitons on these spaces, proving the existence of shrinking, expanding, and steady examples. For all the non-trivial examples we find, the Ricci operator is diagonalizable.

We completely describe paracontact metric three-manifolds whose Reeb vector field satisfies the Ricci soliton equation. While contact Riemannian (or Lorentzian) Ricci solitons are necessarily trivial, that is, K-contact and Einstein, the paracontact metric case allows nontrivial examples. Both homogeneous and inhomogeneous nontrivial three-dimensional examples are explicitly described. Finally, we correct the main result in [1], concerning three-dimensional normal paracontact Ricci solitons

We consider three- and four-dimensional pseudo-Riemannian generalized symmetric spaces, whose invariant metrics were explicitly described in Černý and Kowalski (1982). While four-dimensional pseudo-Riemannian generalized symmetric spaces of types A, C and D are algebraic Ricci solitons, the ones of type B are not so. The Ricci soliton equation for their metrics yields a system of partial differential equations. Solving such system,weprove that almost all the four-dimensional pseudo-Riemannian generalized symmetric spaces of type B are Ricci solitons. These examples show some deep differences arising for the Ricci soliton equation between the Riemannian and the pseudo-Riemannian cases, as any homogeneous Riemannian Ricci soliton is algebraic Jablonski (2015). We also investigate three-dimensional generalized symmetric spaces of any signature and prove that they are Ricci solitons.

Locally homogeneous Lorentzian three-manifolds with recurrect curvature are special examples of Walker manifolds, that is, they admit a parallel null vector field. We obtain a full classification of the symmetries of these spaces, with particular regard to symmetries related to their curvature: Ricci and matter collineations, curvature and Weyl collineations. Several results are given for the broader class of three-dimensional Walker manifolds.

We obtain the full classification of invariant symplectic, (almost) complex and Kähler structures, together with their paracomplex analogues, on four-dimensional pseudo-Riemannian generalized symmetric spaces. We also apply these results to build some new examples of five-dimensional homogeneous K-contact, Sasakian, K-paracontact and para-Sasakian manifolds.

We describe some recent results concerning the inverse curvature problem, that is, the existence and description of metrics with prescribed curvature, focusing on the low-dimensional homogeneous cases.

We consider left-invariant almost contact metric structures on three-dimensional Lie groups, satisfying a quite natural and mild condition. We prove that any three-dimensional Riemannian Lie group admits one of such structures. Moreover, our study leads to the complete classification of three-dimensional left-invariant normal almost contact metric structures, as well as all cases where the one-form η is contact. We then study almost contact metric properties of these examples and harmonicity properties of their characteristic vector fields.

We study three-dimensional generalized Ricci solitons, both in Riemannian and Lorentzian settings. We shall determine their homogeneous models, classifying left-invariant generalized Ricci solitons on three-dimensional Lie groups.

We study three-dimensional Lorentzian homogeneous Ricci solitons, proving the existence of shrinking, expanding and steady Ricci solitons. For all the non-trivial examples, the Ricci operator is not diagonalizable and has three equal eigenvalues.

We obtain the complete classification and explicitly describe totally geodesic and parallel hypersurfaces of four-dimensional oscillator groups, equipped with a one-parameter family of left-invariant Lorentzian metrics.