arXiv subjects

João Nuno Mestre

Publications and source records attributed to João Nuno Mestre.

10 recordsLinked to original sources

arXiv2026

Jacobi Hamiltonian Integrators

We develop a method of constructing structure-preserving integrators for Hamiltonian systems in Jacobi manifolds. Hamiltonian mechanics, rooted in symplectic and Poisson geometry, has long provided a foundation for modeling conservative systems in classical physics. Jacobi manifolds, generalizing both contact and Poisson manifolds, extend this theory and are suitable for incorporating time-dependent, dissipative and thermodynamic phenomena. Building on recent advances in geometric integrators - specifically Poisson Hamiltonian Integrators (PHI), which preserve key features of Poisson systems - we propose a construction of Jacobi Hamiltonian Integrators. Our approach explores the correspondence between Jacobi and homogeneous Poisson manifolds, with the aim of extending the PHI techniques while ensuring preservation of the homogeneity structure. This work develops the theoretical tools required for this generalization and outlines a numerical integration technique compatible with Jacobi dynamics. { By focusing on the homogeneous Poisson perspective instead of direct contact realizations, we establish a clear pathway for constructing structure-preserving integrators for time-dependent and dissipative systems that are embedded in the Jacobi framework.

math.DG↗

arXiv2026

Sub-Riemannian structures and non-transitive Cartan geometries via Lie groupoids

In this paper we discuss how to associate a suitable non-transitive version of a Cartan connection to sub-Riemannian manifolds of corank 1 (including contact and quasi-contact sub-Riemannian manifolds) with non-necessarily constant sub-Riemannian symbols. In particular, we recast the variation of the sub-Riemannian symbols into a suitable "type" map, which is constant if and only if the symbols are constant. We then consider the (non-transitive) groupoid of sub-Riemannian symmetries and investigate its smoothness, properness, regularity, and other properties in relation with the type map. Last, we describe how to build a "non-transitive" analogue of a Cartan connection on top of such (Lie) groupoid, obtained as the sum of a tautological form with a multiplicative Ehresmann connection. We conclude by illustrating our results on concrete examples in dimension 5.

math.DG↗

arXiv2026

Jacobi Hamiltonian Integrators: construction and applications

We propose a systematic framework for constructing geometric integrators for Hamiltonian systems on Jacobi manifolds. By combining Poissonization of Jacobi structures with homogeneous symplectic bi-realizations, Jacobi dynamics are lifted to homogeneous Poisson Hamiltonian systems, enabling the construction of structure-preserving Jacobi Hamiltonian integrators. The resulting schemes are constructed explicitly and applied to a range of examples, including contact Hamiltonian systems and classical models. Numerical experiments highlight their qualitative advantages over standard integrators, including better preservation of geometric structure and improved long-time behavior.

math.NA↗

arXiv2025

Equivariant deformation problems and homotopy operators

We use homotopy operators for the $L_\infty$-algebra associated with an equivariant deformation problem in order to describe a smooth parametrization of the space of structures around a given one. Along the way we give new algebraic and explicit proofs of rigidity and unobstructedness theorems.

math.DG↗

arXiv2021

Deformations of symplectic groupoids

We describe the deformation cohomology of a symplectic groupoid, and use it to study deformations via Moser path methods, proving a symplectic groupoid version of the Moser Theorem. Our construction uses the deformation cohomologies of Lie groupoids and of multiplicative forms, and we prove that in the symplectic case, deformation cohomology of both the underlying groupoid and of the symplectic groupoid have de Rham models in terms of differential forms. We use the de Rham model, which is intimately connected to the Bott-Shulman-Stasheff double complex, to compute deformation cohomology in several examples. We compute it for proper symplectic groupoids using vanishing results; alternatively, for groupoids satisfying homological 2-connectedness conditions we compute it using a simple spectral sequence. Finally, without making assumptions on the topology, we constructed a map relating differentiable and deformation cohomology of the underlying Lie groupoid of a symplectic groupoid, and related it to its Lie algebroid counterpart via van Est maps.

math.DG↗

arXiv2020

Deformations of Lie groupoids

We study deformations of Lie groupoids by means of the cohomology which controls them. This cohomology turns out to provide an intrinsic model for the cohomology of a Lie groupoid with values in its adjoint representation. We prove several fundamental properties of the deformation cohomology including Morita invariance, a van Est theorem, and a vanishing result in the proper case. Combined with Moser's deformation arguments for groupoids, we obtain several rigidity and normal form results.

math.DG↗

arXiv2020

Suborbifolds and groupoid embeddings

Given the notion of suborbifold of the second author (based on ideas of Borzellino/Brunsden) and the classical correspondence (up to certain equivalences) between (effective) orbifolds via atlases and effective orbifold groupoids, we analyze which groupoid embeddings correspond to suborbifolds and give classes of suborbifolds naturally leading to groupoid embeddings.

math.GT↗

arXiv2020

The transverse density bundle and modular classes of Lie groupoids

In this note we revisit the notions of transverse density bundle and of modular classes of Lie algebroids and Lie groupoids; in particular, we point out that one should use the transverse density bundle $\mathcal{D}_{A}^{\textrm{tr}}$ instead of $Q_A$, which is the representation that is commonly used when talking about modular classes. One of the reasons for this is that, as we will see, $Q_A$ is not really an object associated to the stack presented by a Lie groupoid (in general, it is not a representation of the groupoid!).

math.DG↗

arXiv2020

Measures on differentiable stacks

We introduce and study measures and densities (= geometric measures) on differentiable stacks, using a rather straightforward generalization of Haefliger's approach to leaf spaces and to transverse measures for foliations. In general we prove Morita invariance, a Stokes formula which provides reinterpretations in terms of (Ruelle-Sullivan type) algebroid currents, and a Van Est isomorphism. In the proper case we reduce the theory to classical (Radon) measures on the underlying space, we provide explicit (Weyl-type) formulas that shed light on Weinstein's notion of volumes of differentiable stacks, in particular, in the symplectic case, we prove the conjecture left open in Weinstein's "The volume of a differentiable stack". We also revisit the notion of Haar systems (and the existence of cut-off functions). Our original motivation comes from the study of Poisson manifolds of compact types, which provide two important examples of such measures: the affine and the Duistermaat-Heckman measures.

math.DG↗

arXiv2017

Orbispaces as differentiable stratified spaces

We present some features of the smooth structure, and of the canonical stratification on the orbit space of a proper Lie groupoid. One of the main features is that of Morita invariance of these structures - it allows us to talk about the canonical structure of differentiable stratified space on the orbispace (an object analogous to a separated stack in algebraic geometry) presented by the proper Lie groupoid. The canonical smooth structure on an orbispace is studied mainly via Spallek's framework of differentiable spaces, and two alternative frameworks are then presented. For the canonical stratification on an orbispace, we extend the similar theory coming from proper Lie group actions. We make no claim to originality. The goal of these notes is simply to give a complementary exposition to those available, and to clarify some subtle points where the literature can sometimes be confusing, even in the classical case of proper Lie group actions.

math.DG↗