Symplectic coordinates found on a Hitchin component for a hyperbolic surface.
problem Parametrizing the PSL3(R)-Hitchin component with canonical coordinates. method Proved global Darboux coordinates with half canonical Goldman coordinates.
result Global Darboux coordinates exist for the PSL3(R)-Hitchin component. We introduce the concepts of a multisymplectic structure and a polysymplectic structure on a general fiber bundle over a general base manifold, define the concept of the symbol of a multisymplectic form, which is a polysymplectic form representing its leading order contribution, and prove Darboux theorems for the exist…
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.
Paper desingularizes bm-symplectic structures for even m.
problem Understanding bm-symplectic structures and their properties. method Desingularization procedure converting bm-symplectic forms to symplectic forms ωε. result Desingularized forms converge to bm-symplectic forms as ε tends to zero. Researchers found new spectral coordinates for the Calogero-Moser system.
problem The Rational Calogero-Moser system.
method Bi-Hamiltonian geometry and canonical spectral coordinates.
result Explicit construction of spectral canonical coordinates.
We introduce non-smooth symplectic forms on manifolds and describe corresponding Poisson structures on the algebra of Colombeau generalized functions. This is achieved by establishing an extension of the classical map of smooth functions to Hamiltonian vector fields to the setting of non-smooth geometry. For mildly sin…
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…
The Darboux-Weinstein theorem is extended to LCS manifolds.
problem Extending classical results to locally conformally symplectic manifolds.
method Presentation of a version of the Darboux-Weinstein theorem in the LCS setting.
result Application to Lagrangian submanifolds.
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.
New example shows multisymplectic forms without Darboux charts.
problem Existence of multisymplectic forms without Darboux charts.
method Construction of multisymplectic three-forms on R6. result Provided an explicit counterexample to Darboux theorem in multisymplectic framework.
Characterizes density-valued symplectic forms on multisymplectic manifolds.
problem Understanding density-valued symplectic forms on multisymplectic manifolds.
method Intrinsic characterization and Darboux-type theorems.
result Proves Darboux-type theorems for density-valued symplectic forms.
The paper proves a theorem for non-integrable projective distributions of degree one.
problem Classifying non-integrable projective distributions of degree one.
method Proving a singular Darboux type theorem for homogeneous polynomial closed 2-forms of degree one.
result Classification of non-integrable codimension one distributions of degree one.
New method linearizes Darboux transformations of discrete curves.
problem Linearizing Darboux transformations of discrete curves.
method Expressing Darboux transformations as parallel sections of discrete connections in quaternionic formalism.
result Closed-form discrete parametrisations of all Darboux transforms and bicycle correspondences.
The paper studies symplectic forms on projective limits of Banach bundles and their Darboux Theorem.
problem Conditions for weak symplectic forms on projective limits of Banach bundles.
method Analyzing projective sequences of Banach bundles and applying Darboux Theorem.
result Necessary and sufficient conditions for the Darboux Theorem on projective limits of Banach manifolds.
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.
The article interprets Darboux's surfaces using triality in differential geometry.
problem Understanding Darboux's surfaces through triality.
method Geometric interpretation of Darboux's surfaces using triality.
result Darboux's surfaces are related to totally isotropic surfaces in a 6D projective quadric.
The paper extends classical Darboux theorems to various geometric structures in field theories.
problem Extending classical Darboux theorems to new geometric structures.
method Exploring flat connections and polarizations for geometric structures.
result New Darboux theorems for various geometric structures in field theories.
The notion of a generalized harmonic inverse mean curvature surface in the Euclidean four-space is introduced. A backward Bäcklund transform of a generalized harmonic inverse mean curvature surface is defined. A Darboux transform of a generalized harmonic inverse mean curvature surface is constructed by a backward Bäck…
New theorem for convex 1-forms on manifolds.
problem Convexity requirement for Pfaff-Darboux theorem in economic models.
method Generalized convex Pfaff-Darboux theorem for Legendrian foliations.
result Characterization of necessary and sufficient conditions for convex local normal forms.
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 RP2-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 RP2-structures. result Global Darboux coordinates for a new family of symplectic forms ωf are found. 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.
For a finite dimensional symplectic manifold (M,ω) with a symplectic form ω, corresponding loop space (LM=C∞(S1,M)) admits a weak symplectic form Ωω. We prove that the loop space over $\mbr^n$ admits Darboux chart for the weak symplectic structure Ωω. Further, we show that inclusion map from the symp…
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.
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 discrete models for constant mean curvature surfaces and tori.
problem Creating discrete models for constant mean curvature surfaces and tori.
method Integrable theory of discrete polarised curves and Darboux transforms.
result Closed-form discrete parametrisations of discrete isothermic cylinders, discrete constant mean curvature cylinders, and discrete isothermic tori.
Introduces generalized moment maps for almost Hermitian settings.
problem Extending classical moment map theory to almost Hermitian settings.
method Introduces momentumly closed forms and proves a variant of the Darboux-Weinstein theorem.
result Establishes convexity property and constructs reduction space for generalized moment maps.
Classifies different types of Darboux transformations for multidimensional operators.
problem Classifying Darboux transformations for multidimensional operators.
method Analyzes all known types of Darboux transformations and introduces new types.
result Full classification of first-order Darboux transformations and a description of higher-order transformations.
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.
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.
Investigates Darboux rectifying curves on smooth surfaces.
problem Characterizing Darboux rectifying curves on smooth surfaces.
method Analyzes the position vector under isometry and finds conformal invariance conditions.
result Identifies sufficient conditions for conformal invariance of Darboux rectifying curves.
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…
Introduces Darboux-Lie derivative for fiber bundles.
problem None explicitly stated; focuses on introducing a new derivative.
method Study of Darboux-Lie derivative for fiber-bundle maps.
result Properties of Darboux-Lie derivative for fiber bundles.
Minimal surfaces proof via Darboux transformations.
problem Minimal surface properties and transformations.
method Christoffel, Goursat, and Darboux transformations of isothermic surfaces.
result Simple proof of Darboux pairs relation.
The paper explores unique properties of Darboux transformations of spacelike curves in the Lorentz-Minkowski plane.
problem Exploring Darboux transformations of spacelike curves in the Lorentz-Minkowski plane.
method Using the Penrose diagram for conformal compactification, the paper investigates unique properties of Darboux transformations of spacelike curves.
result The paper identifies unique properties of Darboux transformations of spacelike curves in the Lorentz-Minkowski plane, especially regarding singularities and blowup.
The paper extends a theorem for complex structures on Lie groups to Courant algebroids.
problem Characterizing integrable generalized complex structures on transitive Courant algebroids.
method Analyzing skew-symmetric fields of endomorphisms and their closure under the Dorfman bracket.
result Local form of integrable generalized complex structures is determined under certain conditions.
Darboux inverses explain Kepler orbits on curved surfaces.
problem Understanding orbits on curved surfaces.
method Analyzing the Darboux inverses of the Kepler problem.
result Kepler orbits are periodic on open sets of phase space.
In this study, we introduce Darboux slant ruled surfaces in the Euclidean 3-space which is defined by the property that the Darboux vector of orthonormel frame of ruled surface makes a constant angle with a fixed, non-zero direction. We obtain the characterizations of Darboux slant ruled surfaces regarding the conical …
New integrable matrix PDEs derived from Frölicher-Nijenhuis brackets.
problem Developing integrable systems from tensor field properties.
method Using Frölicher-Nijenhuis brackets to generate bi-differential graded algebras and PDE systems.
result New integrable nonlinear matrix PDEs and systems are derived.
The study offers conditions for Darboux charts on specific types of manifolds.
problem Existence of Darboux charts on weakly symplectic manifolds.
method Using Moser's trick to find sufficient conditions.
result Sufficient conditions for Darboux charts on weakly symplectic manifolds.
The basic theory on the conformal geometry of timelike surfaces in pseudo-Riemannian space forms is introduced, which is parallel to the classical framework of Burstall etc. for spacelike surfaces. Then we provide a discussion on the transforms of timelike (±)−isothermic surfaces (or real isothermic, complex isothe…
We give a full description of Darboux transformations of any order for arbitrary (nondegenerate) differential operators on the superline. We show that every Darboux transformation of such operators factorizes into elementary Darboux transformations of order one. Similar statement holds for operators on the ordinary lin…
The study of multisymplectic structures using Spencer cohomology.
problem Integrability of multisymplectic structures.
method Applying Spencer cohomology to identify multisymplectic structures as G-structures and giving conditions for integrability. result Conditions for a multisymplectic form to admit a chart with constant coefficients.
We define a transformation on harmonic maps from a Riemann surface into the 2-sphere which depends on a complex parameter, the so-called mu-Darboux transformation. In the case when the harmonic map N is the Gauss map of a constant mean curvature surface f and the parameter is real, the mu-Darboux transformation of -N i…
In this study, we give definitions and characterizations of eikonal slant helices, eikonal Darboux helices and non-normed eikonal Darboux helices in 3-dimensional pseudo- Riemannian manifold M . We show that every eikonal slant helix is also an eikonal Darboux helix for timelike and spacelike curves. Furthermore, we ob…
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.
Study periodic solutions using spectral data and Darboux coordinates.
problem Elliptic sinh-Gordon equation periodic solutions.
method Spectral data and Darboux coordinates.
result Existence of Darboux coordinates.
In this study, we give definitions and characterizations of eikonal slant helix curves, eikonal Darboux helices and non-normed eikonal Darboux helices in three dimensional Riemannian manifold 3 M . We show that every eikonal slant helix is also an eikonal Darboux helix. Furthermore, we obtain that if the curve a is a n…