arXiv research

Proves local bi-integrability of bi-Hamiltonian systems via bi-Poisson reduction.

Let $X$ be a manifold with a bi-Poisson structure $\{η^t\}$ generated by a pair of $G$-invariant symplectic structures $ω_1$ and $ω_2$, where the Lie group $G$ acts properly on $X$. Let $H$ be some isotropy subgroup for this action representing the principle orbit type and $X^r_\mathfrak{h}$ be the submanifold of $X$ c…

We propose a method of constructing completely integrable systems based on reduction of bihamiltonian structures. More precisely, we give an easily checkable necessary and sufficient conditions for the micro-kroneckerity of the reduction (performed with respect to a special type action of a Lie group) of micro-Jordan b…

We give a characterisation of Atiyah's and Hitchin's transverse Hilbert schemes of points on a symplectic surface in terms of bi-Poisson structures. Furthermore, we describe the geometry of hyperkähler manifolds arising from the transverse Hilbert scheme construction, with particular attention paid to the monopole modu…

We study the relation between the centro-affine geometry of star-shaped planar curves and the projective geometry of parametrized maps into $\RP^1$. We show that projectivization induces a map between differential invariants and a bi-Poisson map between Hamiltonian structures. We also show that a Hamiltonian evolution …

This paper identifies knot projections with reductivity two.

This paper classifies instantons with closed reductions and provides examples of non-closed reductions.

We consider locally conformal Kaehler geometry as an equivariant (homothetic) Kaehler geometry: a locally conformal Kaehler manifold is, up to equivalence, a pair (K,Γ) where K is a Kaehler manifold and Γa discrete Lie group of biholomorphic homotheties acting freely and properly discontinuously. We define a new invari…

In this paper we describe Routhian reduction as a special case of standard symplectic reduction, also called Marsden-Weinstein reduction. We use this correspondence to present a generalization of Routhian reduction for quasi-invariant Lagrangians, i.e. Lagrangians that are invariant up to a total time derivative. We sh…

Two reduction schemes for symplectic manifolds are shown equivalent.

Study extends Kobayashi's method to non-reductive subgroups for homogeneous spaces.

The purpose of this paper is to generalize the regular Optimal Reduction Theorem to general proper Dirac actions, formulated both in terms of point and orbit reduction. A comparison to general standard singular Dirac reduction is given emphasizing the desingularization role played by optimal reduction.

We show that the contact reduction can be specialized to Sasakian manifolds. We link this Sasakian reduction to Kähler reduction by considering the Kähler cone over a Sasakian manifold. We present examples of Sasakian manifolds obtained by $S^1$ reduction of standard Sasakian spheres.

Study characterizes naturally reductive metrics on homogeneous manifolds.

Abstract: Generalized reduction methods for symmetries in graded geometry.

In this note we give conditions which ensure the reduction of a symplectic connection in the process of a Marsden-Weinstein reduction and of the reduction of a presymplectic manifold.

This work introduces a unified approach to the reduction of Poisson manifolds using their description by graded symplectic manifolds. This yields a generalization of the classical Poisson reduction by distributions (Marsden-Ratiu reduction). Further it allows one to construct actions of strict Lie 2-groups and to descr…

The reduction of nonholonomic systems is formulated in terms of Dirac reduction. An optimal reduction method for a class of nonholonomic systems is formulated. Several examples are studied in detail.

This paper extends symplectic reduction to cosymplectic groupoids and explores their properties.

Let $E_G$ be a stable principal $G$--bundle over a compact connected Kaehler manifold, where $G$ is a connected reductive linear algebraic group defined over the complex numbers. Let $H\subset G$ be a complex reductive subgroup which is not necessarily connected, and let $E_H\subset E_G$ be a holomorphic reduction of s…

A new method for classifying naturally reductive spaces is presented. This method relies on the structure theory of naturally reductive spaces developed in \cite{Storm2018a} and the new construction of naturally reductive spaces in \cite{Storm2018}. We obtain the classification of all naturally reductive spaces in dime…

New definition of naturally reductive Finsler manifolds using geodesic graphs.

Survey of Lagrangian reduction for discrete mechanical systems.

New method corrects missing data bias in dimension reduction.

Paper compares Lagrangian reduction methods for rigid body systems.

The paper extends Marsden-Weinstein reduction to mechanical presymplectic structures for time-dependent Hamiltonian systems.

The paper simplifies symmetries in complex geometric structures.

A new construction of naturally reductive spaces is presented. This construction gives a large amount of new families of naturally reductive spaces. First the infinitesimal models of the new naturally reductive spaces are constructed. A concrete transitive group of isometries is given for the new spaces and also the na…

The paper explores polysymplectic structures and their reductions in field theories.

We complete the reduction scheme in the whole LP category, introduced in [7] to perform Lagrangian reduction by stages. We answer affirmatively the open question of whether reduction can be done in the whole category and analyze the Noether theorem on LP-bundles, the relationship with Hamiltonian reduction by stages an…

Develops Marsden-Meyer-Weinstein reduction for $k$-contact field theories.

Paper shows spectra can't distinguish naturally reductive manifolds.

The paper explores reductions of self-dual conformal structure equations.

In the present paper we study naturally reductive homogeneous $(α,β)$-metric spaces. Under some conditions, we give some necessary and sufficient conditions for a homogeneous $(α,β)$-metric space to be naturally reductive. Then we show that for such spaces the two definitions of naturally reductive homogeneous Finsler …

Abstract reviews symmetry and reduction in dynamical systems.

We discuss the use of Dirac structures to obtain a better understanding of the geometry of a class of optimal control problems and their reduction by symmetries. In particular we will show how to extend the reduction of Dirac structures recently proposed by Yoshimura and Marsden [Yo09] to describe the reduction of a cl…

This work extends reduction processes for nonholonomic discrete mechanical systems.

We define reduction of locally conformal Kaehler manifolds, considered as conformal Hermitian manifolds, and we show its equivalence with an unpublished construction given by Biquard and Gauduchon. We show the compatibility between this reduction and Kaehler reduction of the universal cover. By a recent result of Kamis…

We propose a reduction procedure for symplectic connections with symmetry. This is applied to coadjoint orbits whose isotropy is reductive.

Adapts DR objectives for both sample and feature size reduction.

The reductivity of a spherical curve represents how reduced the spherical curve is. It is unknown if there exists a spherical curve whose reductivity is four. In this paper we give an unavoidable set for spherical curves with reductivity four by considering 4-gons.

Survey of recent developments in symmetric reductions and controls for Hamiltonian systems.

In this paper, we first study the Poisson reductions of controlled Hamiltonian (CH) system and symmetric CH system by controllability distributions. These reductions are the extension of Poisson reductions by distribution for Poisson manifolds to that for phase spaces of CH systems with external force and control. We g…

The not-quite-Hamiltonian theory of singular reduction and reconstruction is described. This includes the notions of both regular and collective Hamiltonian reduction and reconstruction.

Extends symplectic reduction and theorem to Lie algebroids.

We extend the theorems concerning the equivariant symplectic reduction of the cotangent bundle to contact geometry. The role of the cotangent bundle is taken by the cosphere bundle. We use Albert's method for reduction at zero and Willett's method for non-zero reduction. We provide examples for both cases.

Consensus dimension reduction combines multiple visualizations to identify shared patterns.

The version of Marsden-Ratiu reduction theorem for Nambu-Poisson manifolds by a regular distribution has been studied by Ib$\acute{\text{a}}\tilde{\text{n}}$ez et al. In this paper we show that the reduction is always ensured unless the distribution is zero. Next we extend the more general Falceto-Zambon Poisson reduct…