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Rui Loja Fernandes

Publications and source records attributed to Rui Loja Fernandes.

At least 37 records · Page 2Linked to original sources
arXiv2015

Riemannian metrics on Lie groupoids

We introduce a notion of metric on a Lie groupoid, compatible with multiplication, and we study its properties. We show that many families of Lie groupoids admit such metrics, including the important class of proper Lie groupoids. The exponential map of these metrics allow us to establish a Linearization Theorem for Riemannian groupoids, obtaining both a simpler proof and a stronger version of the Weinstein-Zung Linearization Theorem for proper Lie groupoids. This new notion of metric has a simplicial nature which will be explored in future papers of this series.

math.DG↗
arXiv2015

Normal Forms and Lie Groupoid Theory

In these lectures notes I discuss the Linearization Theorem for Lie groupoids, and its relation to the various classical linearization theorems for submersions, foliations and group actions. In particular, I explain in some detail the recent metric approach to this problem.

math.DG↗
arXiv2013

Integrability and Reduction of Hamiltonian Actions on Dirac Manifolds

For a Hamiltonian, proper and free action of a Lie group $G$ on a Dirac manifold $(M,L)$, with a regular moment map $μ:M\to \mathfrak{g}^*$, the manifolds $M/G$, $μ^{-1}(0)$ and $μ^{-1}(0)/G$ all have natural induced Dirac structures. If $(M,L)$ is an integrable Dirac structure, we show that $M/G$ is always integrable, but $μ^{-1}(0)$ and $μ^{-1}(0)/G$ may fail to be integrable, and we describe the obstructions to their integrability.

math.SG↗
arXiv2012

The modular class of a Poisson map

We introduce the modular class of a Poisson map. We look at several examples and we use the modular classes of Poisson maps to study the behavior of the modular class of a Poisson manifold under different kinds of reduction. We also discuss their symplectic groupoid version, which lives in groupoid cohomology.

math.DG↗
arXiv2011

A h-principle for symplectic foliations

We show that a classical result of Gromov in symplectic geometry extends to the context of symplectic foliations, which we regard as a $h$-principle for (regular) Poisson geometry. Namely, we formulate a sufficient cohomological criterion for a regular bivector to be homotopic to a regular Poisson structure, in the spirit of Haefliger's criterion for homotoping a distribution to a foliation. We give an example to show that this criterion is not too unsharp.

math.SG↗
arXiv2010

Stability of symplectic leaves

We find computable criteria for stability of symplectic leaves of Poisson manifolds. Using Poisson geometry as an inspiration, we also give a general criterion for stability of leaves of Lie algebroids, including singular ones. This not only extends but also provides a new approach (and proofs) to the classical stability results for foliations and group actions.

math.DG↗
arXiv2009

Integrability of Poisson-Lie group actions

We establish a 1:1 correspondence between Poisson-Lie group actions on integrable Poisson manifolds and twisted multiplicative hamiltonian actions on source 1-connected symplectic groupoids. For an action of a Poisson-Lie group $G$ on a Poisson manifold $M$, we find an explicit description of the lifted hamiltonian action on the symplectic groupoid $Σ(M)$. We give applications of these results to the integration of Poisson quotients $M/G$, Lu-Weinstein quotients $μ^{-1}(e)/G$ and Poisson homogeneous spaces $G/H$.

math.DG↗
arXiv2009

The momentum map in Poisson geometry

Every action on a Poisson manifold by Poisson diffeomorphisms lifts to a Hamiltonian action on its symplectic groupoid which has a canonically defined momentum map. We study various properties of this momentum map as well as its use in reduction.

math.SG↗
arXiv2008

A geometric approach to Conn's linearization theorem

We give a soft geometric proof of the classical result due to Conn stating that a Poisson structure is linearizable around a singular point (zero) at which the isotropy Lie algebra is compact and semisimple.

math.SG↗
arXiv2008

Lie Algebroids and Classification Problems in Geometry

We show how one can associate to a given class of finite type G-structures a classifying Lie algebroid. The corresponding Lie groupoid gives models for the different geometries that one can find in the class, and encodes also the different types of symmetry groups.

math.DG↗
arXiv2006

The Symplectization Functor

We describe a symplectization functor from the Poisson category to the symplectic "category" and we study some of its properties.

math.DG↗
arXiv2006

Integrable hierarchies and the modular class

We observe that the modular class of a Poisson-Nijhenhuis manifold has a canonical representative and that, under a cohomological assumption, this vector field is bi-hamiltonian. In many examples the associated hierarchy of flows reproduces classical integrable hierarchies.

math.DG↗
arXiv2005

A Note on Proper Poisson Actions

We show that the fixed point set of a proper action of a Lie group $G$ on a Poisson manifold $M$ by Poisson automorphisms has a natural induced Poisson structure and we give several applications.

math.DG↗