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Integrable systems in low dimensions, constructed through the symmetry reduction method, are studied using phase portrait and variable separation techniques. In particular, invariant quantities and explicit periodic solutions are…
We extend Ahlbrandt and Ziegler's reconstruction results to the metric setting: we show that separably categorical metric structures are determined, up to bi-interpretability, by their automorphism groups.
This work deals with the structure of the isometry group of pseudo-Riemannian 2-step nilmanifolds. We study the action by isometries of several groups and we construct examples showing substantial differences with the Riemannain situation;…
We discuss some specializations of the frames of flat orthonormal frame bundles over geometries of indefinite signature, and the resulting symmetries of families of embedded Riemannian or pseudo-Riemannian geometries. The specializations…
A survey of problems, conjectures, and theorems about quasi-isometric classification and rigidity for finitely generated solvable groups.
This paper proposes an original Riemmanian geometry for low-rank structured elliptical models, i.e., when samples are elliptically distributed with a covariance matrix that has a low-rank plus identity structure. The considered geometry is…
Methods were developed in Ref. [1] for constructing reference metrics (and from them differentiable structures) on three-dimensional manifolds with topologies specified by suitable triangulations. This note generalizes those methods by…
In image set classification, a considerable advance has been made by modeling the original image sets by second order statistics or linear subspace, which typically lie on the Riemannian manifold. Specifically, they are Symmetric Positive…
In this paper we consider symplectic and Hamiltonian structures of systems generated by actions of sigma-model type and show that these systems are naturally connected with specific symplectic geometry on loop spaces of Riemannian and…
Starting from the formulation of pseudo-Riemannian generalisation of real spectral triples we develop the data of geometries over finite-dimensional algebras with indefinite metric and their Riemannian parts. We then discuss the Standard…
The main results on the theory of conformal and almost Grassmann structures are presented. The common properties of these structures and also the differences between them are outlined. In particular, the structure groups of these structures…
In this note we discuss the geometry of Riemannian surfaces having a discrete set of singular points. We assume the conformal structure extends through the singularities and the curvature is integrable. Such points are called \emph{simple…
Actions of Lie groups on presymplectic manifolds are analyzed, introducing the suitable comomentum and momentum maps. The subsequent theory of reduction of presymplectic dynamical systems with symmetry is studied. In this way, we give a…
We give a construction of a Poisson transform mapping density valued differential forms on generalized flag manifolds to differential forms on the corresponding Riemannian symmetric spaces, which can be described entirely in terms of finite…
The compatibility between the conformal symmetry and the closure of conformal algebras is discussed on the nonlinear sigma model. The present approach, above the basis of field redefinition employed in the Hamiltonian scheme, attempts the…
This report concerns the problem of dimensionality reduction through information geometric methods on statistical manifolds. While there has been considerable work recently presented regarding dimensionality reduction for the purposes of…
Parametric Cartan theory of exterior differential systems, and explicit cohomology of projective manifolds reveal united rationality features of differential algebraic geometry.
The question whether a Riemannian manifold is geodesically connected can be studied from geometrical as well as variational methods, and accurate results can be obtained by using the associated distance and related properties of the…
By virtue of the well-known theorem, a structure Lie group K of a principal bundle $P$ is reducible to its closed subgroup H iff there exists a global section of the quotient bundle P/K. In gauge theory, such sections are treated as Higgs…
The paper surveys highlights of the ongoing program to classify discrete polyhedral structures in Euclidean 3-space by distinguished transitivity properties of their symmetry groups, focussing in particular on various aspects of the…