Related papers: Non-Integrable Pfaffian Systems
The system of a closed vortex filament is an integrable Hamiltonian one, namely, a Hamiltonian system with an infinite sequense of constants of motion in involution. An algebraic framework is given for the aim of describing differential…
Invariant manifolds of unstable periodic orbits organize the dynamics of chaotic orbits in phase space. They provide insight into the mechanisms of transport and chaotic advection and have important applications in physical situations…
Multicomponent KdV-systems are defined in terms of a set of structure constants and, as shown by Svinolupov, if these define a Jordan algebra the corresponding equations may be said to be integrable, at least in the sense of having…
Representing graphs as sets of node embeddings in certain curved Riemannian manifolds has recently gained momentum in machine learning due to their desirable geometric inductive biases, e.g., hierarchical structures benefit from hyperbolic…
We introduce a class of maps from an affine flat into a Riemannian manifold that solve an elliptic system defined by the natural second order elliptic operator of the affine structure and the nonlinear Riemann geometry of the target. These…
We consider the proximal gradient method on Riemannian manifolds for functions that are possibly not geodesically convex. Starting from the forward-backward-splitting, we define an intrinsic variant of the proximal gradient method that uses…
We introduce a geometric invariant that we call the index of symmetry, which measures how far is a Riemannian manifold from being a symmetric space. We compute, in a geometric way, the index of symmetry of compact naturally reductive…
Integral formulae for foliated Riemannian manifolds provide obstructions for existence of foliations or compact leaves of them with given geometric properties. This paper continues our recent study and presents new integral formulae and…
The symmetry frame formalism is an effective tool for computing the symmetries of a Riemann-Cartan geometry and, in particular, in metric teleparallel geometries. In the case of non-vanishing torsion in a four dimensional Riemann-Cartan…
We introduce an invariant linked to some foundational questions in geometric measure theory and provide bounds on this invariant by decomposing an arbitrary cycle into uniformly rectifiable pieces. Our invariant measures the difficulty of…
We review the results of several of our papers about the procedure of extension of Hamiltonians, allowing the construction of families of superintegrable systems with non-trivial polynomial first integrals (or symmetry operators) of…
We study codimension $q \geq 2$ holomorphic foliations defined in a neighborhood of a point $P$ of a complex manifold that are completely integrable, i.e. with $q$ independent meromorphic first integrals. We show that either $P$ is a…
Integrals of the Pfaffian form over the nonsingular part of a projective variety compute information closely related to the Mather-Chern class of the variety and to other invariants such as the local Euler obstruction along strata of its…
Building on the interplay between geometry and integrability, we show that F-manifolds with compatible connection $(\nabla,\circ,e)$ are the geometric counterpart of integrable systems of quasilinear first order evolutionary PDEs. We…
The superintegrability of a rational harmonic oscillator (non-central harmonic oscillator with rational ratio of frequencies) with non-linear "centrifugal" terms is studied. In the first part, the system is directly studied in the Euclidean…
We show that a large class of non-degenerate second-order (maximally) superintegrable systems gives rise to Hessian structures, which admit natural (Hessian) coordinates adapted to the superintegrable system. In particular, abundant…
This article discusses the geometric application of the method of multiplier ideal sheaves. It first briefly describes its application to effective problems in algebraic geometry and then presents and explains its application to the…
We carry out a detailed quantitative analysis on the geometry of invariant manifolds for smooth dissipative systems in dimension two. We begin by quantifying the regularity of any orbit (finite or infinite) in the phase space with a set of…
In this paper we deal with Seifert fibre spaces, which are compact 3-manifolds admitting a foliation by circles. We give a combinatorial description for these manifolds in all the possible cases: orientable, non-orientable, closed, with…
In this article, we propose a way to consider processes indexed by a collection $\mathcal{A}$ of subsets of a general set $\mathcal{T}$. A large class of vector spaces, manifolds and continuous $\mathbb{R}$-trees are particular cases.…