We present a compared analysis of some properties of 3-Sasakian and 3-cosymplectic manifolds. We construct a canonical connection on an almost 3-contact metric manifold which generalises the Tanaka-Webster connection of a contact metric manifold and we use this connection to show that a 3-Sasakian manifold does not adm…
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48 results for “Darboux coordinate system”
In this paper we analyze the tangential symmetries of Darboux integrable decomposable exterior differential systems. The decomposable systems generalize the notion of a hyperbolic exterior differential system and include the classic notion of Darboux integrability for first order systems and second order scalar equatio…
Bernhard Riemann 1861 revisited: existence of flat coordinates for an arbitrary bilinear formmath.DG
Generalizes Riemann's results on flat coordinates for non-symmetric bilinear forms.
problem Finding flat coordinates for non-symmetric bilinear forms.
method Provides explicit necessary and sufficient conditions for a tensor field of type (0,2) to be flat.
result Explicit conditions for a tensor field to have constant entries in local coordinates.
The paper finds global Darboux coordinates for a new family of symplectic forms on the deformation space of -structures.
problem Finding global Darboux coordinates for a specific family of symplectic forms.
method Study of complete Lagrangian fibrations and application to the deformation space of -structures.
result Global Darboux coordinates for a new family of symplectic forms are found.
Bi-Hamiltonian geometry and canonical spectral coordinates for the Rational Calogero-Moser systemmath-ph
We reconsider the (rational) Calogero-Moser system from the point of view of bi-Hamiltonian geometry. By using geometrical tools of the latter, we explicitly construct set(s) of spectral canonical coordinates, that is, complete sets of Darboux coordinates defined by the eigenvalues and the eigenvectors of the Lax matri…
The Goldman symplectic form is trivialized for a class of surfaces.
problem Trivializing the Goldman symplectic form on specific surfaces.
method Using ideal triangulations and compatible bridge systems, the authors prove the existence of a symplectic trivialization.
result A global Darboux coordinate system is established for the PGL(V)-Hitchin component.
Symplectic coordinates found on a Hitchin component for a hyperbolic surface.
problem Parametrizing the -Hitchin component with canonical coordinates.
method Proved global Darboux coordinates with half canonical Goldman coordinates.
result Global Darboux coordinates exist for the -Hitchin component.
The paper extends a proposal for effective twisted superpotentials to higher rank.
problem Describing effective twisted superpotentials from class S theories geometrically.
method Introducing higher rank analogues of spectral networks and spectral coordinates, and finding generating functions.
result The generating functions of the effective twisted superpotentials agree with known results.
We study the space of periodic solutions of the elliptic -Gordon equation by means of spectral data consisting of a Riemann surface and a divisor and prove the existence of certain Darboux coordinates.
Symplectic coordinates on the deformation spaces of convex projective structures on 2-orbifoldsmath.GT
Symplectic coordinates found on projective structures on orbifolds.
problem Symplectic structure on deformation spaces of convex projective structures.
method Global Darboux coordinates system construction and symplectic space decomposition.
result Symplectic form on deformation space of convex projective structures.
Let G be a split semi-simple algebraic group over Q. Let S be a decorated surface, that is a topological oriented surface with a finite set of marked points on the boundary, considered modulo isotopy. We introduce a moduli space D(G,S) and define a collection of special rational coordinate systems on it. The moduli spa…
Geometrically revisits Dupin cyclidic systems using evolving circles and cyclides.
problem Understanding the geometric properties and evolution of Dupin cyclidic systems.
method Evolving initial circles or Dupin cyclides to generate Lamé families of Dupin cyclidic systems in various space forms.
result Lamé families are parallel surfaces in different space forms.
Identifies spectral curves for SU(3) coadjoint orbits.
problem Understanding the geometry of coadjoint orbits in SU(3).
method Using Hitchin pairs and spectral curves, identifies a Hamiltonian circle action and finds Darboux coordinates.
result Identifies a differential equation for the Hamiltonian.
On a discretization of confocal quadrics. II. A geometric approach to general parametrizationsmath.DG
A geometric approach discretizes confocal quadrics, leading to novel discrete nets.
problem Discretization of confocal quadrics.
method Geometric characterization and factorizable orthogonal coordinate systems.
result Explicit computation of coordinate functions for discrete confocal quadrics.
Confocal quadrics lie at the heart of the system of confocal coordinates (also called elliptic coordinates, after Jacobi). We suggest a discretization which respects two crucial properties of confocal coordinates: separability and all two-dimensional coordinate subnets being isothermic surfaces (that is, allowing a con…
Study of meromorphic connections and their spectral duals in .
problem Exploring -deformed meromorphic connections and their spectral duals.
method Using apparent singularities and their dual partners as Darboux coordinates, the Hamiltonian evolutions and reductions are derived.
result Spectral duality extends to Hamiltonian evolutions, tau-functions, and Hermitian matrix models on both sides.
We explain that spectral networks are a unifying framework that incorporates both shear (Fock-Goncharov) and length-twist (Fenchel-Nielsen) coordinate systems on moduli spaces of flat SL(2,C) connections, in the following sense. Given a spectral network W on a punctured Riemann surface C, we explain the process of "abe…
Classical approaches to isometric embedding simplified.
problem Isometric embedding of Riemannian surfaces in Euclidean 3-space.
method Coordinate-based and moving-frames approaches, focusing on integrability of PDEs.
result The integrability of the PDE is surprisingly easy and related to moving frames approach.
New flows defined on Hitchin components for PGL(V).
problem Understanding dynamics on Hitchin components.
method Define and analyze new flows on Hitchin components.
result Construct a global coordinate system on the Hitchin component.
The paper connects isomonodromic and isospectral deformations for connections.
problem Connecting isomonodromic and isospectral deformations for connections.
method Explicitly constructing Lax pairs and Darboux coordinates to bridge isomonodromic and isospectral deformations.
result Explicit change of Darboux coordinates to match spectral invariants, solving an open issue.
The paper proves conditions for Darboux integrability in diagonal hydrodynamic systems.
problem Conditions for Darboux integrability in diagonal hydrodynamic systems.
method Proof of conditions using Laplace transformation sequences and geometric interpretations.
result Diagonal systems of hydrodynamic type are Darboux integrable if and only if the corresponding systems for commuting flows are Darboux integrable.
Classifies homogeneous Pfaffian forms on graded manifolds.
problem Local classification of homogeneous Pfaffian forms on graded manifolds.
method Darboux coordinates and characteristic distribution analysis.
result Darboux-type normal forms for homogeneous Pfaffian forms.
Derives a new first order differential equation for smooth surfaces.
problem Finding new equations to describe smooth surfaces.
method Derives a linear differential equation of the first order.
result Proves the maximum principle for Darboux rotation fields.
Resurgence of Joyce structures gauged to a standard form using gauge transformations.
problem Resurgence of Joyce structures in complex hyperkähler geometry.
method Gauge transformations and Borel transforms to show resurgent behavior.
result Established the resurgent behavior of infinitesimal gauge transformations for Joyce structures.
New hierarchies and equations derived from Poisson structures.
problem Development of new hierarchies and equations from Poisson structures.
method Introduction and classification of multicomponent Poisson structures, derivation of hierarchies and equations.
result New Harry Dym and Hunter-Saxton equations derived for arbitrary number of components.
The study explores autonomous systems and their connections to contact geometry and Frobenius manifolds.
problem Understanding the connections between autonomous systems and geometric structures.
method Investigation of the Darboux-Halphen-Ramanujan system, contact geometry, and Frobenius manifolds.
result Highlighting the role of contact geometry in autonomous systems.
Explains properties of multisymplectic manifolds.
problem None explicitly stated; focuses on defining and characterizing multisymplectic manifolds.
method Review and introduction of multisymplectic geometry concepts.
result Characterization of multisymplectic manifolds and their Hamiltonian structures.
We consider BPS states in a large class of d=4, N=2 field theories, obtained by reducing six-dimensional (2,0) superconformal field theories on Riemann surfaces, with defect operators inserted at points of the Riemann surface. Further dimensional reduction on S^1 yields sigma models, whose target spaces are moduli spac…
The Darboux-Halphen system arises in various geometric and physical contexts.
problem Finding solutions to differential equations and understanding geometric structures.
method Review of different problems and their connections to the Darboux-Halphen system.
result The Darboux-Halphen system is a unifying concept across diverse mathematical and physical problems.
New hyperkähler metrics constructed via Riemann-Hilbert problems.
problem Constructing hyperkähler metrics on complex integrable systems with singular fibers.
method Solving Riemann-Hilbert problems and isomonodromic deformations to obtain Darboux coordinates.
result Holomorphic symplectic form constructed, leading to hyperkähler metric.
Privileged Coordinates and Nilpotent Approximation for Carnot Manifolds, II. Carnot Coordinatesmath.DG
The paper introduces Carnot coordinates for Carnot manifolds, simplifying nilpotent approximation.
problem Nilpotent approximation of Carnot manifolds.
method Identification of Carnot coordinates as privileged coordinates with specific properties.
result Carnot coordinates provide a precise and effective nilpotent approximation of Carnot manifolds.
Elliptic systems are characterized by Darboux integrability.
problem Characterizing elliptic differential systems with holomorphic solutions.
method Using a complex manifold and associated holomorphic Pfaffian system.
result Elliptic systems are Darboux integrable under generic conditions.
We study an analogue of the classical Bianchi-Darboux transformation for L-isothermic surfaces in Laguerre geometry, the Bianchi-Darboux transformation. We show how to construct the Bianchi-Darboux transforms of an L-isothermic surface by solving an integrable linear differential system. We then establish a permutabili…
The paper discusses a new method for constructing two-step Darboux transforms of isothermic surfaces.
problem Constructing two-step Darboux transforms of isothermic surfaces.
method Sym-type construction using parallel sections of the associated family.
result All two-step Darboux transforms of an isothermic surface are given without further integration.
Bi-Hamiltonian structures are of great importance in the theory of integrable Hamiltonian systems. The notion of compatibility of symplectic structures is a key aspect of bi-Hamiltonian systems. Because of this, a few different notions of compatibility have been introduced. In this paper we show that, under some additi…
Conservation laws, heirarchies, scattering theory and Bäcklund transformations are known to be the building blocks of integrable partial differential equations. We identify these as facets of a theory of Poisson group actions, and apply the theory to the ZS-AKNS nxn heirarchy (which includes the non-linear Schrödinger …
We give a new mechanism for constructing Backlund transformations by using symmetry reduction of differential systems. We then characterize a family of Backlund transformations between Darboux integrable systems where the Backlund transformation can be constructed by the proposed symmetry reduction method.
Theorems adapted for prequantum systems, showing symplectomorphism and gauge transformation.
problem Adapting classical theorems to prequantum systems.
method Establishing analogs of the Darboux, Moser, and Weinstein theorems.
result Prequantum systems with vanishing first cohomology are equivalent up to symplectomorphism and gauge transformation.
In this paper we present a far-reaching generalization of E. Vessiot's analysis of the Darboux integrable partial differential equations in one dependent and two independent variables. Our approach provides new insights into this classical method, uncovers the fundamental geometric invariants of Darboux integrable syst…
The Darboux-Egoroff system of PDEs with any number of independent variables plays an essential role in the problems of describing -dimensional flat diagonal metrics of Egoroff type and Frobenius manifolds. We construct a recursion operator and its inverse for symmetries of the Darboux-Egoroff system and des…
We prove that second-order hyperbolic Monge-Ampere equations for one function of two variables are connected to the wave equation by a Backlund transformation if and only if they are integrable by the method of Darboux at second order. One direction of proof, proving Darboux integrability, follows the implications of t…
In this note we prove that an analytic symplectic action of a semisimple Lie algebra can be locally linearized in Darboux coordinates. This result yields simultaneous analytic linearization for Hamiltonian vector fields in a neighbourhood of a common zero. We also provide an example of smooth non-linearizable Hamiltoni…
We study the geometric properties of Darboux transforms of constant mean curvature (CMC) surfaces and use these transforms to obtain an algebro-geometric representation of constant mean curvature tori. We find that the space of all Darboux transforms of a CMC torus has a natural subset which is an algebraic curve (call…
The first, second and fourth Painlevé equations are studied by means of dynamical systems theory and three dimensional weighted projective spaces $\C P^3(p,q,r,s)$ with suitable weights determined by the Newton diagrams of the equations or the versal deformations of vector fields. Singular normal forms of t…
To every Darboux integrable system there is an associated Lie group which is a fundamental invariant of the system and which we call the Vessiot group. This article shows that solving the Cauchy problem for a Darboux integrable partial differential equation can be reduced to solving an equation of Lie type for the …
New methods create full discretized isothermic tori in Euclidean spaces.
problem Creating full discretized isothermic tori in Euclidean spaces.
method Using Darboux transformations and periodic curvature line systems.
result Discrete and semi-discrete k-dimensional isothermic tori in n-dimensional Euclidean space.
New kinematic model for a spin-rolling sphere using Darboux frame.
problem Control and planning of an underactuated sphere on a plane.
method Darboux frame transformation to fully-actuated model.
result Underactuated sphere model transformed into fully-actuated model.
Deroin and Tholozan's representations are mapped to complex projective space via action-angle coordinates.
problem Mapping representations of a punctured sphere into to a simpler geometric space.
method Polygonal model and chains of triangles to extract action-angle coordinates.
result Action-angle coordinates give an explicit isomorphism and almost global Darboux coordinates.