arXiv subjects

Eva Miranda

Publications and source records attributed to Eva Miranda.

72 records · Page 4Linked to original sources

arXiv2015

Examples of integrable and non-integrable systems on singular symplectic manifolds

We present a collection of examples borrowed from celestial mechanics and projective dynamics. In these examples symplectic structures with singularities arise naturally from regularization transformations, Appell's transformation or classical changes like McGehee coordinates, which end up blowing up the symplectic structure or lowering its rank at certain points. The resulting geometrical structures that model these examples are no longer symplectic but symplectic with singularities which are mainly of two types: $b^m$-symplectic and $m$-folded symplectic structures. These examples comprise the three body problem as non-integrable exponent and some integrable reincarnations such as the two fixed-center problem. Given that the geometrical and dynamical properties of $b^m$-symplectic manifolds and folded symplectic manifolds are well-understood [GMP, GMP2, GMPS, KMS, Ma, CGP, GL,GLPR, MO, S, GMW], we envisage that this new point of view in this collection of examples can shed some light on classical long-standing problems concerning the study of dynamical properties of these systems seen from the Poisson viewpoint.

math.SG↗

arXiv2015

A note on symplectic topology of $b$-manifolds

A Poisson manifold $(M^{2n},\p)$ is $b$-symplectic if $\bigwedge^n\p$ is transverse to the zero section. In this paper we apply techniques native to Symplectic Topology to address questions pertaining to $b$-symplectic manifolds. We provide constructions of $b$-symplectic structures on open manifolds by Gromov's $h$-principle, and of $b$-symplectic manifolds with a prescribed singular locus, by means of surgeries.

math.SG↗

arXiv2015

Rigidity of infinitesimal momentum maps

In this paper we prove rigidity theorems for Poisson Lie group actions on Poisson manifolds. In particular, we prove that close infinitesimal momentum maps associated to Poisson Lie group actions are equivalent using a normal form theorem for SCI spaces. When the Poisson structure of the acted manifold is integrable, this yields rigidity also for lifted actions to the symplectic groupoid.

math-ph↗

arXiv2015

A note on symplectic and Poisson linearization of semisimple Lie algebra actions

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 Hamiltonian action with semisimple linear part. The smooth analogue only holds if the semisimple Lie algebra is of compact type. An analytic equivariant b-Darboux theorem for b-Poisson manifolds and an analytic equivariant Weinstein splitting theorem for general Poisson manifolds are also obtained in the Poisson setting.

math.SG↗

arXiv2015

Action-angle variables and a KAM theorem for b-Poisson manifolds

In this article we prove an action-angle theorem for b-integrable systems on b-Poisson manifolds improving the action-angle theorem contained in [LMV11] for general Poisson manifolds in this setting. As an application, we prove a KAM-type theorem for b-Poisson manifolds.

math.SG↗

arXiv2014

Toric actions on b-symplectic manifolds

We study Hamiltonian actions on $b$-symplectic manifolds with a focus on the effective case of half the dimension of the manifold. In particular, we prove a Delzant-type theorem that classifies these manifolds using polytopes that reside in a certain enlarged and decorated version of the dual of the Lie algebra of the torus.

math.SG↗

arXiv2014

Symplectic and Poisson geometry on b-manifolds

Let $M^{2n}$ be a Poisson manifold with Poisson bivector field $Π$. We say that $M$ is b-Poisson if the map $Π^n:M\toΛ^{2n}(TM)$ intersects the zero section transversally on a codimension one submanifold $Z\subset M$. This paper will be a systematic investigation of such Poisson manifolds. In particular, we will study in detail the structure of $(M,Π)$ in the neighbourhood of $Z$ and using symplectic techniques define topological invariants which determine the structure up to isomorphism. We also investigate a variant of de Rham theory for these manifolds and its connection with Poisson cohomology.

math.SG↗

arXiv2014

A Poincare lemma in Geometric Quantisation

This article presents a Poincare lemma for the Kostant complex, used to compute geometric quantisation, when the polarisation is given by a Lagrangian foliation defined by an integrable system with nondegenerate singularities.

math.SG↗

arXiv2013

Geometric Quantization of real polarizations via sheaves

In this article we develop tools to compute the Geometric Quantization of a symplectic manifold with respect to a regular Lagrangian foliation via sheaf cohomology and obtain important new applications in the case of real polarizations. The starting point is the definition of representation spaces due to Kostant. Besides the classical examples of Gelfand-Cetlin systems due to Guillemin and Sternberg very few examples of explicit computations of real polarizations are known. The computation of Geometric Quantization for Gelfand-Cetlin systems is based on a theorem due to Śniatycki for fibrations which identifies the representation space with the set of Bohr-Sommerfeld leaves determined by the integral action coordinates. In this article we check that the associated sheaf cohomology apparatus of Geometric Quantization satisfies Mayer-Vietoris and Künneth formulae. As a consequence, a new short proof of this classical result for fibrations due to Śniatycki is obtained. We also compute Geometric Quantization with respect to any generic regular Lagrangian foliation on a 2-torus and the case of the irrational flow. In the way, we recover some classical results in the computation of foliated cohomology of these polarizations.

math.SG↗

arXiv2013

On a Poincare lemma for foliations

In this paper we revisit a Poincare lemma for foliated forms, with respect to a regular foliation, and compute the foliated cohomology for local models of integrable systems with singularities of nondegenerate type. A key point in this computation is the use of some analytical tools for integrable systems with nondegenerate singularities, including a Poincare lemma for the deformation complex associated to this singular foliation.

math.SG↗

arXiv2013

Coupling symmetries with Poisson structures

We study local normal forms for completely integrable systems on Poisson manifolds in the presence of additional symmetries. The symmetries that we consider are encoded in actions of compact Lie groups. The existence of Weinstein's splitting theorem for the integrable system is also studied giving some examples in which such a splitting does not exist, i.e. when the integrable system is not, locally, a product of an integrable system on the symplectic leaf and an integrable system on a transversal. The problem of splitting for integrable systems with additional symmetries is also considered.

math.SG↗

arXiv2011

Rigidity of Hamiltonian actions on Poisson manifolds

This paper is about the rigidity of compact group actions in the Poisson context. The main resut is that Hamiltonian actions of compact semisimple type are rigid. We prove it via a Nash-Moser normal form theorem for closed subgroups of SCI-type. This Nash-Moser normal form has other applications to stability results that we will explore in a future paper. We also review some classical rigidity results for differentiable actions of compact Lie groups and export it to the case of symplectic actions of compact Lie groups on symplectic manifolds.

math.SG↗

arXiv2011

Codimension one symplectic foliations and regular Poisson structures

In this short note we give a complete characterization of a certain class of compact corank one Poisson manifolds, those equipped with a closed one-form defining the symplectic foliation and a closed two-form extending the symplectic form on each leaf. If such a manifold has a compact leaf, then all the leaves are compact, and furthermore the manifold is a mapping torus of a compact leaf. These manifolds and their regular Poisson structures admit an extension as the critical hypersurface of a Poisson b-manifold as we consider in a later paper.

math.SG↗

arXiv2009

Geometric quantization of integrable systems with hyperbolic singularities

We construct the geometric quantization of a compact surface using a singular real polarization coming from an integrable system. Such a polarization always has singularities, which we assume to be of nondegenerate type. In particular, we compute the effect of hyperbolic singularities, which make an infinite-dimensional contribution to the quantization, thus showing that this quantization depends strongly on polarization.

math.SG↗

arXiv2008

Action-angle coordinates for integrable systems on Poisson manifolds

We prove the action-angle theorem in the general, and most natural, context of integrable systems on Poisson manifolds, thereby generalizing the classical proof, which is given in the context of symplectic manifolds. The topological part of the proof parallels the proof of the symplectic case, but the rest of the proof is quite different, since we are naturally led to using the calculus of polyvector fields, rather than differential forms; in particular, we use in the end a Poisson version of the classical Caratheodory-Jacobi-Lie theorem, which we also prove. At the end of the article, we generalize the action-angle theorem to the setting of non-commutative integrable systems on Poisson manifolds.

math.SG↗

arXiv2006

A note on equivariant normal forms of Poisson structures

We prove an equivariant version of the local splitting theorem for tame Poisson structures and Poisson actions of compact Lie groups. As a consequence, we obtain an equivariant linearization result for Poisson structures whose transverse structure has semisimple linear part of compact type.

math.SG↗

arXiv2004

Equivariant normal form for nondegenerate singular orbits of integrable Hamiltonian systems

We consider an integrable Hamiltonian system with n-degrees of freedom whose first integrals are invariant under the symplectic action of a compact Lie group G. We prove that the singular Lagrangian foliation associated to this Hamiltonian system is symplectically equivalent, in a G-equivariant way, to the linearized foliation in a neighborhood of a compact singular non-degenerate orbit. We also show that the non-degeneracy condition is not equivalent to the non-resonance condition for smooth systems.

math.SG↗

arXiv2004

A singular Poincare lemma

We prove a Poincare lemma for a set of r smooth functions on a 2n-dimensional smooth manifold satisfying a commutation relation determined by r singular vector fields associated to a Cartan subalgebra of $\frak{sp}(2r,\mathbb R)$. This result has a natural interpretation in terms of the cohomology associated to the infinitesimal deformation of a completely integrable system.

math.SG↗