Giovanni Calvaruso

Homogeneity on three-dimensional contact metric manifolds

We study ball-homogeneity, curvature homogeneity, natural reductivity, conformal flatness and ϕ-symmetry for three-dimensional contact metric manifolds. Several classification results are given.

Ricci solitons and geometry of non-reductive homogeneous 4-spaces

We study the geometry of non-reductive 4-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.

Special Ball-Homogeneous Spaces

We continue the study of ball-homogeneous Riemannian manifolds, that is, R.iemannian spaces such that the volume of all sufficiently small geodesic balls or spheres only depends on the radius. First, we consider the case of locally reducible spaces. Then we treat the three-dimensional case, in particular for Einstein-like metrics and finally, we study conformally flat ball-homogeneous spaces. Our aim is to provide more partial answers to the question whether a ball-homogeneous space is locally homogeneous or not.

Four-dimensional pseudo-Riemannian homogeneous Ricci solitons

We consider four-dimensional homogeneous pseudo-Riemannian manifolds with non-trivial 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.

Homogeneous geodesics of non-unimodular Lorentzian Lie groups and naturally reductive Lorentzian spaces in dimension three

We determine, for all three-dimensional non-unimodular Lie groups equipped with a Lorentzian metric, the set of homogeneous geodesics through a point. Together with the results of [C] and [CM2], this leads to the full classification of three-dimensional Lorentzian g.o. spaces and naturally reductive spaces.

Ricci solitons in three-dimensional paracontact geometry

Abstract 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.

Einstein-Like Lorentz Metrics and Three-Dimensional Curvature Homogeneity of Order One

We completely classify three-dimensional Lorentz manifolds, curvature homogeneous up to order one, equipped with Einstein-like metrics. New examples arise with respect to both homogeneous examples and three-dimensional Lorentz manifolds admitting a degenerate parallel null line field.

Harmonicity of vector fields on four-dimensional generalized symmetric spaces

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.

On the Ricci operator of locally homogeneous Lorentzian 3-manifolds

We determine the admissible forms for the Ricci operator of three-dimensional locally homogeneous Lorentzian manifolds.

LORENTZIAN SYMMETRIC THREE-SPACES AND THE CLASSIFICATION OF THEIR PARALLEL SURFACES

We describe a global model for Lorentzian symmetric three-spaces admitting a parallel null vector field, and classify completely the surfaces with parallel second fundamental form in all Lorentzian symmetric three-spaces. Interesting differences arise with respect to the Riemannian case studied in [2]. Our results complete the classification of parallel surfaces in all three-dimensional Lorentzian homogeneous spaces.