Homogeneous structures on three-dimensional Lorentzian manifolds

* Basic Algebraic Notions* Basic Geometrical Notions* Walker Structures* Three-Dimensional Lorentzian Walker Manifolds* Four-Dimensional Walker Manifolds* The Spectral Geometry of the Curvature Tensor* Hermitian Geometry* Special Walker Manifolds

The Geometry of Walker Manifolds

Contact symmetries and Hamiltonian thermodynamics

Based on the construction of Poisson-Lie T -dual σ-models from a common parent action we study a candidate for the non-abelian respectively Poisson-Lie T -duality group. This group generalises the well-known abelian T -duality group O(d, d) and we explore some of its subgroups, namely factorised dualities, B- and β-shifts. The corresponding duality transformed σ-models are constructed and interpreted as generalised (non-geometric) flux backgrounds. We also comment on generalisations of results and techniques known from abelian T -duality. This includes the Lie algebra cohomology interpretation of the corresponding non-geometric flux backgrounds, remarks on a double field theory based on non-abelian T -duality and an application to the investigation of Yang-Baxter deformations. This will show that homogeneously Yang-Baxter deformed σ-models are exactly the non-abelian T -duality β-shifts when applied to principal chiral models.

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Generalised fluxes, Yang-Baxter deformations and the O(d,d) structure of non-abelian T -duality

We completely classify three-dimensional homogeneous Lorentzian manifolds, equipped with Einstein-like metrics. Similarly to the Riemannian case (E. Abbena et al., Simon Stevin Quart J Pure Appl Math 66:173–182, 1992), if (M, g) is a three-dimensional homogeneous Lorentzian manifold, the Ricci tensor of (M, g) being cyclic-parallel (respectively, a Codazzi tensor) is related to natural reductivity (respectively, symmetry) of (M, g). However, some exceptional examples arise.

Einstein-like metrics on three-dimensional homogeneous Lorentzian manifolds

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.

Three-dimensional lorentzian homogeneous ricci solitons

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.

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Homogeneous paracontact metric three-manifolds

We introduce and study $H$-paracontact metric manifolds, that is, paracontact metric manifolds whose Reeb vector field $\xi$ 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.

Giovanni Calvaruso

Papers

Homogeneous structures on three-dimensional Lorentzian manifolds

Einstein-like metrics on three-dimensional homogeneous Lorentzian manifolds

Three-dimensional lorentzian homogeneous ricci solitons

Homogeneous paracontact metric three-manifolds

Geometry of $H$-paracontact metric manifolds