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Giovanni Calvaruso

Publications and source records attributed to Giovanni Calvaruso.

16 recordsLinked to original sources
arXiv2025

Ricci solitons of special Lorentzian Lie groups with a four-dimensional isometry group

In the framework of the study of homogeneous Lorentzian three-manifolds, we consider here the only class of examples which admit a four-dimensional group of isometries but are neither Lorentzian Bianchi-Cartan-Vranceanu spaces nor plane waves. We obtain an explicit description in global coordinates of these special homogeneous Lorentzian manifolds. We then prove that all such examples are non-gradient expanding Ricci solitons.

math.DG↗
arXiv2023

Helix surfaces for Berger-like metrics on the anti-de Sitter space

We consider the Anti-de Sitter space $\mathbb{H}^3_1$ equipped with Berger-like metrics, that deform the standard metric of $\mathbb{H}^3_1$ in the direction of the hyperbolic Hopf vector field. Helix surfaces are the ones forming a constant angle with such vector field. After proving that these surfaces have (any) constant Gaussian curvature, we achieve their explicit local description in terms of a one-parameter family of isometries of the space and some suitable curves. These curves turn out to be general helices, which meet at a constant angle the fibers of the hyperbolic Hopf fibration.

math.DG↗
arXiv2016

Ricci solitons on low-dimensional generalized symmetric spaces

We consider three- and four-dimensional pseudo-Riemannian generalized symmetric spaces, whose invariant metrics were explicitly described in [15]. 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, we prove 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 [21]. We also investigate three-dimensional generalized symmetric spaces of any signature and prove that they are Ricci solitons.

math.DG↗
arXiv2016

Symmetries of Lorentzian Three-Manifolds with Recurrent Curvature

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.

math.DG↗
arXiv2016

From almost (para)-complex structures to affine structures on Lie groups

Let $G=H\ltimes K$ denote a semidirect product Lie group with Lie algebra $\mathfrak g=\mathfrak h \oplus \mathfrak k$, where $\mathfrak k$ is an ideal and $\mathfrak h$ is a subalgebra of the same dimension as $\mathfrak k$. There exist some natural split isomorphisms $S$ with $S^2=\pm \,Id$ on $\mathfrak g$: given any linear isomorphism $j:\mathfrak h \to \mathfrak k$, we have the almost complex structure $J(x,v)=(-j^{-1}v, jx)$ and the almost paracomplex structure $E(x,v)=(j^{-1}v, jx)$. In this work we show that the integrability of the structures $J$ and $E$ above is equivalent to the existence of a left-invariant torsion-free connection $\nabla$ on $G$ such that $\nabla J=0=\nabla E$ and also to the existence of an affine structure on $H$. Applications include complex, paracomplex and symplectic geometries.

math.DG↗
arXiv2016

Cosymplectic and $α$-cosymplectic Lie algebras

We completely characterize cosymplectic and $α$-cosymplectic Lie algebras in terms of corresponding symplectic Lie algebras and suitable derivations on them. Several examples are given and classification results are obtained in dimension five for cosymplectic, $K$-cosymplectic and coKähler Lie algebras.

math.DG↗
arXiv2016

Three-dimensional homogeneous generalized Ricci solitons

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.

math.DG↗
arXiv2015

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.

math.DG↗
arXiv2014

Ricci solitons in three-dimensional paracontact geometry

We completely describe paracontact metric three-manifolds whose Reeb vector field satisfies the Ricci soliton equation. While contact Riemannian (or Lorentz\-ian) 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 of [AGAG-D-13-00189], concerning three-dimensional normal paracontact Ricci solitons.

math.DG↗
arXiv2013

Paracontact metric structures on the unit tangent sphere bundle

Starting from $g$-natural pseudo-Riemannian metrics of suitable signature on the unit tangent sphere bundle $T_1 M$ of a Riemannian manifold $(M,\langle,\rangle)$, we construct a family of paracontact metric structures. We prove that this class of paracontact metric structures is invariant under $\mathcal D$-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 $T_1 M$.

math.DG↗
arXiv2013

Geometry of $H$-paracontact metric manifolds

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

math.DG↗
arXiv2010

Algebraic Properties of Curvature Operators in Lorentzian Manifolds with Large Isometry Groups

Together with spaces of constant sectional curvature and products of a real line with a manifold of constant curvature, the socalled Egorov spaces and $\varepsilon$-spaces exhaust the class of $n$-dimensional Lorentzian manifolds admitting a group of isometries of dimension at least ${1/2} n(n-1)+1$, for almost all values of $n$ [Patrangenaru V., Geom. Dedicata 102 (2003), 25-33]. We shall prove that the curvature tensor of these spaces satisfy several interesting algebraic properties. In particular, we will show that Egorov spaces are Ivanov-Petrova manifolds, curvature-Ricci commuting (indeed, semi-symmetric) and $\mathcal P$-spaces, and that $\varepsilon$-spaces are Ivanov-Petrova and curvature-curvature commuting manifolds.

math.DG↗