Autour du théorème de Ferrand–Obata (Concerning the Theorem of Ferrand–Obata)

The aim of this article is to give a new dynamical proof of the Ferrand–Obata theorem when the manifold is compact. This will give us a generalisation of this theorem to transversally conformal foliations ofcodimension greater than three and constant basic functions.     

The Atiyah bundle and connections on a principal bundle

Let M be a C ^∞ manifold and G a Lie a group. Let E _G be a C ^∞ principal G-bundle over M. There is a fiber bundle C(E _G ) over M whose smooth sections correspond to the connections on E _G . The pull back of E _G to C(E _G ) has a tautological connection. We investigate the curvature of this tautological connection.     

The Weyl bundle

Let F be a symplectic vector bundle over a space X. We construct a bundle of elementary C¹-algebras over X, and prove that the Dixmier-Douady invariant of this bundle is zero. The underlying Hilbert bundles, with their associated module structures, determine a characteristic class: we prove that this class is the second Stiefel-Whitney class of F.     

The gunning cohomology bundle and the Torelli group

Kemerovo. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 31, No. 3, pp. 198–203, May–June, 1990.     

The cohomology of the elliptic tangent bundle

In this note we compute the cohomology of the elliptic tangent bundle, a Lie algebroid introduced in Cavalcanti and Gualtieri (2018), Cavalcanti et al. (2020) used to describe singular symplectic forms arising from generalised complex geometry.     

The hypoelliptic Laplacian on the cotangent bundle

In this paper, we construct a new version of Hodge theory, where the corresponding Laplacian acts on the total space of the cotangent bundle. This Laplacian is a hypoelliptic operator, which is in general non-self-adjoint. When properly interpreted, it provides an interpolation between classical Hodge theory and the generator of the geodesic flow. The construction is also done in families in the superconnection formalism of Quillen and extends earlier work by Lott and the author.

     

The geometry of the bundle of connections

A generalized symplectic structure on the bundle of connections of an arbitrary principal G-bundle is defined by means of a -valued differential 2-form on C(P), which is related to the generalized contact structure on . The Hamiltonian properties of are also analyzed. Received August 31, 1999; in final form January 4, 2000 / Published online February 5, 2001     

The Conley conjecture for the cotangent bundle

We prove the Conley conjecture for cotangent bundles of oriented, closed manifolds, and Hamiltonians which are quadratic at infinity, i.e., we show that such Hamiltonians have infinitely many periodic orbits. For the conservative systems, similar results have been proven by Lu and Mazzucchelli using convex Hamiltonians and Lagrangian methods. Our proof uses Floer homological methods from Ginzburg’s proof of the Conley conjecture for closed symplectically aspherical manifolds.     

The oscillator parallelization of the induced scalar bundle

Translated fromSibirskii Matematicheskii Zhurnal, Vol. 36, No. 5, pp. 1122–1129, September–October, 1995.     

Stochastic Calculus of Variations on complex line bundle and construction of unitarizing measures for the Poincaré disk

Holomorphic representation of Lie algebra can be realized through Kählerian symplectic formalism; underlying holomorphic convexity requires then the introduction of elliptic operators with complex coefficients. We construct the Stochastic Calculus of Variations for those elliptic operators; remote past vanishing of projections of the underlying process implies convergence in law; then limit laws lead to the unitarizing measure of the given representation; this general approach is developed in full details on Poincaré disk.     

The cohomology of the cotangent bundle of Heisenberg groups

Given a parabolic subalgebra g₁×n of a semisimple Lie algebra, Kostant (Ann. Math. 1963) and Griffiths (Acta Math. 1963) independently computed the g₁ invariants in the cohomology group of n with exterior adjoint coefficients. By a theorem of Bott (Ann. Math. 1957), this is the cohomology of the associated compact homogeneous space with coefficients in the sheaf of local holomorphic forms. In this paper we determine explicitly the full module structure, over the symplectic group, of the cohomology group of the Heisenberg Lie algebra with exterior adjoint coefficients. This is the cohomology of the cotangent bundle of the Heisenberg group.     

Functions on discrete sets holomorphic in the sense of Ferrand, or monodiffric functions of the second kind

We study the class of functions called monodiffric of the second kind by Isaac.They are discrete analogues of holomorphic functions of one or two complex variables.Discrete analogues of the Cauchy-Riemann operator,of domains of holomorphy in one discrete variable,and of the Hartogs phenomenon in two discrete variables are investigated.Two fundamental solutions to the discrete Cauchy-Riemann equation are studied:one with support in a quadrant,the other with decay at infinity.The first is easy to construct by induction;the second is accessed via its Fourier transform.     

On the Picard bundle

Fix a holomorphic line bundle ξ over a compact connected Riemann surface X of genus g, with g?2, and also fix an integer r such that degree(ξ)>r(2g−1). Let Mξ(r) denote the moduli space of stable vector bundles over X of rank r and determinant ξ. The Fourier-Mukai transform, with respect to a Poincaré line bundle on X×J(X), of any F∈Mξ(r) is a stable vector bundle on J(X). This gives an injective map of Mξ(r) in a moduli space associated to J(X). If g=2, then Mξ(r) becomes a Lagrangian subscheme.     

A bundle view of boundary-value problems: generalizing the Gardner-Jones bundle

Holomorphic families of linear ordinary differential equations on a finite interval with prescribed parameter-dependent boundary conditions are considered from a geometrical viewpoint. The Gardner-Jones bundle, which was introduced for linearized reaction-diffusion equations, is generalized and applied to this abstract class of λ-dependent boundary-value problems, where λ is a complex eigenvalue parameter. The fundamental analytical object of such boundary-value problems (BVPs) is the characteristic determinant, and it is proved that any characteristic determinant on a Jordan curve can be characterized geometrically as the determinant of a transition function associated with the Gardner-Jones bundle. The topology of the bundle, represented by the Chern number, then yields precise information about the number of eigenvalues in a prescribed subset of the complex λ-plane. This result shows that the Gardner-Jones bundle is an intrinsic geometric property of such λ-dependent BVPs. The bundle framework is applied to examples from hydrodynamic stability theory and the linearized complex Ginzburg-Landau equation.     

The variational sequence on finite jet bundle extensions and the Lagrangian formalism

The geometric Lagrangian theory is based on the analysis of some basic mathematical objects such as: the contact ideal, the (exact) variational sequence, the existence of Euler-Lagrange and Helmholtz-Sonin forms, etc. In this paper we give new and much simpler proofs for the whole theory using Fock space methods.     

The Stiefel bundle of a Banach algebra

We introduce the Stiefel bundle associated to a given Banachable algebra and study the properties of this analytic principal fiber bundle over the Grassmannian of equivalence classes of idempotents in the algebra. Our main application concerns the bounded linear operators of a Banach space. In particular, the problem of smooth parametrization of subspaces can then be reduced to one involving the smooth extension of sections.