Improves Marsden-Weinstein reduction theory for k-polysymplectic manifolds.
problem Improving Marsden-Weinstein reduction theory for k-polysymplectic manifolds.
method Developed a theory of affine Lie group actions for k-polysymplectic momentum maps, removing technical conditions.
result Devise a k-polycosymplectic Marsden-Weinstein reduction theory.
Abstract: Linking field theory to Floer theory via regularization.
problem Finding periodic solutions of Hamilton's equation.
method Regularization scheme for polysymplectic formalism linking Euclidean field theory to hyperkähler Floer theory.
result Proved a cuplength estimate.
The paper explores polysymplectic structures and their reductions in field theories.
problem Invariance of Lagrangian and Hamiltonian field theories under symmetry groups.
method Application of polysymplectic reduction theorem for both Lagrangian and Hamiltonian field equations.
result Identification and relation of polysymplectic structures through Routhian function and Legendre transformation.
Novel method for solving ODEs on k-polysymplectic manifolds.
problem Solving ordinary differential equations on k-polysymplectic manifolds.
method k-polysymplectic energy-momentum method.
result Novel stability analysis techniques applied to Hamiltonian systems.
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…
We adapt the framework of geometric quantization to the polysymplectic setting. Considering prequantization as the extension of symmetries from an underlying polysymplectic manifold to the space of sections of a Hermitian vector bundle, a natural definition of prequantum vector bundle is obtained which incorporates in …
A polysymplectic structure is a vector-valued symplectic form, that is, a closed nondegenerate 2-form with values in a vector space. We first outline the polysymplectic Hamiltonian formalism with coefficients in a vector space V, then apply this framework to show that the moduli space M(P) of flat connect…
The paper simplifies symmetries in complex geometric structures.
problem Redundancy in conditions for symmetry reduction in polysymplectic and polycosymplectic structures.
method Exploring and proving necessary and sufficient conditions for polycosymplectic reduction.
result A one-to-one relationship between polycosymplectic reduction and the reduction of a larger polysymplectic manifold.
In this paper we introduce poly-Poisson structures as a higher-order extension of Poisson structures. It is shown that any poly-Poisson structure is endowed with a polysymplectic foliation. It is also proved that if a Lie group acts polysymplectically on a polysymplectic manifold then, under certain regularity conditio…
The polysymplectic (n+1)-form is introduced as an analogue of the symplectic form for the De Donder-Weyl polymomentum Hamiltonian formulation of field theory. The corresponding Poisson brackets on differential forms are constructed. The analogues of the Poisson algebra are shown to be generalized (non-commutative and…
In the first part of this paper we begin the study of polysymplectic manifolds, and of their relationship with PDE's. This notion provides a generalization of symplectic manifolds which is very well suited for the geometric study of PDE's with values in a smooth manifold. Some of the standard tools of analytical mechan…
Paper introduces Floer theory for field theories, proving periodic solutions for particle-field systems.
problem Defining Hamiltonian Floer theory for covariant field theories, especially those with degenerate action functionals.
method Regularization procedure to handle degeneracy, leading to Floer curves that converge to periodic solutions.
result Existence of Floer curves and space-time periodic solutions for coupled particle-field systems.
Abstract: Generalizes multisymplectic forms to vector-valued versions.
problem Generalizing multisymplectic forms to vector-valued versions.
method Obtained a standard local presentation and proved an entropy inequality for partial compositions.
result Vector-valued multisymplectic forms form a non-unital operad.
Reduces field theories on principal bundles by a subgroup, deriving reduced equations.
problem Hamiltonian field theories on principal G-bundles with invariant densities.
method Lie-Poisson reduction using covariant bracket formulation.
result Derives reduced observables, brackets, and equations of motion for field theories.
Canonical structure of the space-time symmetric analogue of the Hamiltonian formalism in field theory based on the De Donder-Weyl (DW) theory is studied. In n space-time dimensions the set of n polymomenta is associated to the space-time derivatives of field variables. The polysymplectic (n+1)-form generalizes th…
Study surfaces with constant ratio of principal curvatures in Euclidean and isotropic geometries.
problem Characterize surfaces with constant ratio of principal curvatures in different geometries.
method Differential geometry, line geometry, Lie sphere geometry, ordinary differential equations, algebraic geometry.
result Characterized various types of surfaces like rotational, channel, ruled, helical, and translational.
It is shown that Electromagnetism creates geometry different from Riemannian geometry. General geometry including Riemannian geometry as a special case is constructed. It is proven that the most simplest special case of General Geometry is geometry underlying Electromagnetism. Action for electromagnetic field and Maxwe…
We define (p,q) hermitian geometry as the target space geometry of the two dimensional (p,q) supersymmetric sigma model. This includes generalised Kähler geometry for (2,2), generalised hyperkähler geometry for (4,2), strong Kähler with torsion geometry for (2,1) and strong hyperkähler with torsion geometry f…
Non-lorentzian geometry reviewed, including Lie algebras and Klein geometries.
problem Understanding non-lorentzian spacetimes.
method Classification and characterization of kinematical Lie algebras and their geometries.
result Characterization of Cartan geometries based on intrinsic torsion.
Simpler method derived for path geometries on surfaces, characterizing projective path geometries.
problem Characterizing projective path geometries on surfaces.
method Solving the equivalence problem of sub-Riemannian geometry of signature (1,1) on a contact 3-manifold.
result Characterization of projective path geometries in terms of their chains.
New definition of Born geometry connects to known geometries.
problem Defining and understanding Born geometries.
method Using Künneth structures and recursion operators.
result Born connection derived from Künneth connection for integrable geometries.
We give an introduction to the theory of varieties of minimal rational tangents, emphasizing its aspect as a fusion of algebraic geometry and differential geometry, more specifically, a fusion of Mori geometry of minimal rational curves and Cartan geometry of cone structures.
The paper extends group constructions to coset geometries, creating new ways to combine geometries.
problem Combining and gluing incidence geometries in a general framework.
method Extending classical group-theoretic constructions to coset geometries.
result Provides a general framework for combining or gluing incidence geometries.
New symmetries found in Riemann-Cartan geometries.
problem Investigating symmetries in geometries with curvature and torsion.
method Mathematical tools to determine symmetries and subclasses of geometries.
result Determined all static and stationary spherically symmetric Riemann-Cartan geometries and subclasses with specific symmetries.
Survey explores interactions between convex and complex geometry.
problem Understanding intersections between convex and complex geometry.
method Survey and review of existing literature.
result Demonstrates fascinating interactions between convex and complex geometry.
Lecture notes on geodesics in differential geometry.
problem Understanding geodesics in differential geometry.
method Expository lecture notes with exercises.
result Explains the geometry of geodesics.
Lecture notes on Finslerian geometry.
problem No specific problem stated; covers Finslerian geometry.
method Lecture notes.
result No specific key result mentioned.
Spin(7) geometry linked to multisymplectic geometry.
problem Understanding Spin(7) structures through multisymplectic geometry.
method Utilized Spin(7) identities to prove non-degeneracy of Cayley four-form in multisymplectic context.
result Spin(7) geometry is a special case of multisymplectic geometry.
The extension functors between categories of Cartan geometries can be used to define different categories of Cartan geometries with additional morphisms. The Cartan geometries modeled on skeletons can be used for the description of such categories of Cartan geometries and therefore we develop the theory of Cartan geome…
A geometric transition is a continuous path of geometric structures that changes type, meaning that the model geometry, i.e. the homogeneous space on which the structures are modeled, abruptly changes. In order to rigorously study transitions, one must define a notion of geometric limit at the level of homogeneous spac…
Paper introduces a generalized Bures-Wasserstein geometry for SPD matrices.
problem Understanding the geometry of SPD matrices for machine learning.
method Proposes a generalized Bures-Wasserstein geometry parameterized by a symmetric positive definite matrix.
result The GBW geometry outperforms the BW geometry in machine learning applications.
We show that a car, viewed as a nonholonomic system, provides an example of a flat parabolic geometry of type (SO(2,3),P12), where P12 is a Borel parabolic subgroup in SO(2,3). We discuss the relations of this geometry of a car with the geometry of circles in the plane (a low dimensional Lie sph…
The study sets limits on the complexity of Klein geometries.
problem Understanding the complexity of Klein geometries.
method Simple upper and lower bounds for the order of Klein geometries.
result Established upper and lower bounds for the order of Klein geometries.
Surveying probabilistic real algebraic geometry.
problem Classical problems in real algebraic geometry.
method Probabilistic perspective on classical topics.
result Modern approach to Hilbert's Sixteenth Problem.
Develops Weyl structures for path geometries, simplifying their study.
problem Complexity in studying path geometries using traditional differential geometry methods.
method Defines distinguished connections and Schouten tensor, proving their dependence on line bundle sections.
result Shows a smaller subclass of Weyl structures for path geometries, with interesting connections to BGG sequences.
Ray-marching method visualizes 8 Thurston geometries in real-time.
problem Accurately rendering and visualizing Thurston geometries in real-time.
method Ray-marching algorithms with theoretical framework for non-Euclidean geometries.
result Accurate interactive real-time views of Thurston geometries achieved.
The paper extends Ruh-Vilms theorem to hypersurfaces in Weitzenböck geometry.
problem Extending Ruh-Vilms theorem to hypersurfaces in Weitzenböck geometry.
method Extending the Ruh-Vilms theorem to Weitzenböck geometry, considering the flat geometry with torsion.
result The Laplacian of the Gauss map for hypersurfaces in Weitzenböck geometry is related to the mean curvature vector field.
New algebraic-geometry method for Ribaucour transformations.
problem Classical differential geometry problems.
method Algebraic-geometry approach to constructing orthogonal nets.
result Obtains smooth orthogonal nets as Ribaucour transformations.
Paper develops formulas and theorems in Hermitian geometry.
problem None explicitly stated in the abstract.
method Develops second variational formulas and index forms in Hermitian geometry.
result Establishes results analogous to classical theorems in Riemannian geometry.
Introduces a new geometry based on difference angles, showing unique properties.
problem Defining angles independently of circles or rotations.
method Axiomatic system for difference angles, defining new geometric constructs.
result Explicit confirmation of the concurrency of the parabolic Miquel configuration.
The study of special metrics in various parabolic geometries.
problem Finding special metrics in parabolic geometries.
method Investigating invariant conditions for special metrics in different parabolic geometries.
result Characterization of special metrics in hypersurface CR and contact Legendrean cases.
We introduce the notion of manifolds of amalgamation geometry and its generalization, split geometry. We show that the limit set of any surface group of split geometry is locally connected, by constructing a natural Cannon-Thurston map.
KT-geometry is the geometry of a Hermitian connection whose torsion is a 3-form. HKT-geometry is the geometry of a hyper-Hermitian connection whose torsion is a 3-form. We identify non-trivial conditions for a reduction theory for these types of geometry.
We classify the 5-dimensional homogeneous geometries in the sense of Thurston. The present paper (part 3 of 3) classifies those in which the linear isotropy representation is nontrivial but reducible. Most of the resulting geometries are products. Some interesting examples include a countably infinite family of inequiv…
New framework for noncommutative Carrollian geometry using Lie-Rinehart pairs.
problem Developing a geometric framework for ultra-relativistic physics in noncommutative settings.
method Using ρ-Lie-Rinehart pairs to generalize Carrollian Lie algebroids to almost commutative geometry.
result Foundational principles of Carrollian geometry hold in almost commutative geometry.
Authors discuss complex and non-Archimedean geometry, proving a conjecture.
problem Proving a version of the Yau--Tian--Donaldson conjecture for Kähler metrics.
method Relation between complex, analytic, and non-Archimedean geometry.
result Sketch of proof for Yau--Tian--Donaldson conjecture.
The target space geometry of abelian vector multiplets in N=2 theories in four and five space-time dimensions is called special geometry. It can be elegantly formulated in terms of Hessian geometry. In this review, we introduce Hessian geometry, focussing on aspects that are relevant for the special geometrie…
New geometry based on Siegel upper half-space with volume formula.
problem Developing a new 3D geometry based on Siegel upper half-space.
method Constructing a geometry fibered over Siegel upper half-space and providing a volume formula.
result Volume of Siegel-Seifert closed manifolds is the fiber circle length times base manifold's Euler characteristic.