Geometric structures encoded in the lie structure of an Atiyah algebroid

We investigate Atiyah algebroids, i.e. the infinitesimal objects of principal bundles, from the viewpoint of the Lie algebraic approach to space. First we show that if the Lie algebras of smooth sections of two Atiyah algebroids are isomorphic, then the corresponding base manifolds are necessarily diffeomorphic. Further, we give two characterizations of the isomorphisms of the Lie algebras of sections for Atiyah algebroids associated to principal bundles with semisimple structure groups. For instance we prove that in the semisimple case the Lie algebras of sections are isomorphic if and only if the corresponding Lie algebroids are, or, as well, if and only if the integrating principal bundles are locally isomorphic. Finally, we apply these results to describe the isomorphisms of sections in the case of reductive structure groups—surprisingly enough they are no longer determined by vector bundle isomorphisms and involve dive rgences on the base manifolds.     

Local structure of Koszul-Vinberg and of Lie algebroids

In this short note we continue our study of Koszul-Vinberg algebroids which form a subcategory of the category of Lie algebroids, and which appear naturally in the study of affine structures, affine and transversally affine foliations [N. Nguiffo Boyom, R. Wolak, J. Geom. Phys. 42 (2002) 307-317]. We prove a local decomposition theorem for KV-algebroids. Using the notion of KV-algebroids we introduce a new class of singular foliations: affine singular foliations. In the last section we study the holonomy of these foliations and prove a stability theorem.     

Lie algebroid structures on double vector bundles and representation theory of Lie algebroids

A VB-algebroid is essentially defined as a Lie algebroid object in the category of vector bundles. There is a one-to-one correspondence between VB-algebroids and certain flat Lie algebroid superconnections, up to a natural notion of equivalence. In this setting, we are able to construct characteristic classes, which in special cases reproduce characteristic classes constructed by Crainic and Fernandes. We give a complete classification of regular VB-algebroids, and in the process we obtain another characteristic class of Lie algebroids that does not appear in the ordinary representation theory of Lie algebroids.     

Lie Brackets on Affine Bundles

Natural affine analogs of Lie brackets on affine bundles are studied.In particular, a close relation to Lie algebroids and a duality withcertain affine analog of Poisson structure is established as well asaffine versions of complete lifts and Cartan exterior calculi.     

Banach–Lie groupoids and generalized inversion

We study a few basic properties of Banach–Lie groupoids and algebroids, adapting some classical results on finite dimensional Lie groupoids. As an illustration of the general theory, we show that the notion of locally transitive Banach–Lie groupoid sheds fresh light on earlier research on some infinite-dimensional manifolds associated with Banach algebras.     

On characterization of Poisson and Jacobi structures

We characterize Poisson and Jacobi structures by means of complete lifts of the corresponding tensors: the lifts have to be related to canonical structures by morphisms of corresponding vector bundles. Similar results hold for generalized Poisson and Jacobi structures (canonical structures) associated with Lie algebroids and Jacobi algebroids.     

On Almost Complex Lie Algebroids

The almost complex Lie algebroids over smooth manifolds are considered in the paper. In the first part, we give some examples and we extend some basic results from almost complex manifolds to almost complex Lie algebroids. Next the almost Hermitian Lie algebroids and some related structures on the associated complex Lie algebroid are studied.     

Geometric structures as deformed infinitesimal symmetries

A general model for geometric structures on differentiable manifolds is obtained by deforming infinitesimal symmetries. Specifically, this model consists of a Lie algebroid, equipped with an affine connection compatible with the Lie algebroid structure. The curvature of this connection vanishes precisely when the structure is locally symmetric.

This model generalizes Cartan geometries, a substantial class, to the intransitive case. Simple examples are surveyed and corresponding local obstructions to symmetry are identified. These examples include foliations, Riemannian structures, infinitesimal -structures, symplectic and Poisson structures.

     

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.     

具有殆仿切触结构流形的新例子

本文证明了(2n+1)维连通实Lie群具有殆仿切触结构,并给出它们是可积的充要条件,从而给出这类流形一批新的例子。它们与过去已知的这类流形大都是满足某些条件的仿复流形上的主圆丛不同。     

Locally Vacant Double Lie Groupoids and the Integration of Matched Pairs of Lie Algebroids

We prove a general integrability result for matched pairs of Lie algebroids. (Matched pairs of Lie algebras are also known as double Lie algebras or twilled extensions of Lie algebras.) The method used is an extension of a method introduced by Lu and Weinstein in the case of Poisson Lie groups, and yields double groupoids which satisfy an étale form of the vacancy condition.     

Lie and Courant algebroids on foliated manifolds

This is an exposition of the subject, which was developed in the author’s papers [19, 20]. Various results from the theory of foliations (cohomology, characteristic classes, deformations, etc.) are extended to subalgebroids of Lie algebroids that generalize the tangent integrable distributions. We also suggest a definition of foliated Courant algebroids and give some corresponding results and constructions.     

Reduction of Jacobi manifolds via Dirac structures theory

We first recall some basic definitions and facts about Jacobi manifolds, generalized Lie bialgebroids, generalized Courant algebroids and Dirac structures. We establish an one-one correspondence between reducible Dirac structures of the generalized Lie bialgebroid of a Jacobi manifold (M,Λ,E) for which 1 is an admissible function and Jacobi quotient manifolds of M. We study Jacobi reductions from the point of view of Dirac structures theory and we present some examples and applications.     

Rigid and Finite Type Geometric Structures

Rigid geometric structures on manifolds, introduced by Gromov, are characterized by the fact that their infinitesimal automorphisms are determined by their jets of a fixed order. Important examples of such structures are those given by an H-reduction of the first order frame bundle of a manifold, where the Lie algebra of H is of finite type; in fact, for structures given by reductions to closed subgroups of first order frame bundles, finite type implies rigidity. The goal of this paper is to generalize this to geometric structures defined by reductions of frame bundles of arbitrary order, and to give an algebraic characterization of the property of being rigid in terms of a suitable notion of finite type.     

The Euler-Poincaré-Hopf theorem for flat connections in some transitive Lie algebroids

This paper is a continuation of [19], [21], [22]. We study flat connections with isolated singularities in some transitive Lie algebroids for which either ℝ or sl(2, ℝ) or so(3) are isotropy Lie algebras. Under the assumption that the dimension of the isotropy Lie algebra is equal to n + 1, where n is the dimension of the base manifold, we assign to any such isolated singularity a real number called an index. For ℝ-Lie algebroids, this index cannot be an integer. We prove the index theorem (the Euler-Poincaré-Hopf theorem for flat connections) saying that the index sum is independent of the choice of a connection. Multiplying this index sum by the orientation class of M, we get the Euler class of this Lie algebroid. Some integral formulae for indices are given.     

Homological sections of Lie algebroids

We examine Lie (super)algebroids equipped with a homological section, i.e., an odd section that ‘self-commutes’, we refer to such Lie algebroids as inner Q-algebroids: these provide natural examples of suitably “superised” Q-algebroids in the sense of Mehta. Such Lie algebroids are a natural generalisation of Q-manifolds and Lie superalgebras equipped with a homological element. Amongst other results, we show that, via the derived bracket formalism, the space of sections of an inner Q-algebroid comes equipped with an odd Loday–Leibniz bracket.     

Lie algebroid fibrations

A degree 1 non-negative graded super manifold equipped with a degree 1 vector field Q satisfying [Q,Q]=1, namely a so-called NQ-1 manifold is, in plain differential geometry language, a Lie algebroid. We introduce a notion of fibration for such super manifolds, that essentially involves a complete Ehresmann connection. As it is the case for Lie algebras, such fibrations turn out not to be just locally trivial products. We also define homotopy groups and prove the expected long exact sequence associated to a fibration. In particular, Crainic and Fernandes's obstruction to the integrability of Lie algebroids is interpreted as the image of a transgression map in this long exact sequence.     

关于李群胚和泊松作用的讨论

定义了纤维丛的相配群胚的概念,从作用的角度研究了李群胚与主丛的关系;给出了一个泊松群胚在泊松流形上的作用是泊松作用的充要条件;文末得到了一些关于泊松流形上Casimir函数的结果.     

Complete involutive algebras of functions on cotangent bundles of homogeneous spaces

Homogeneous spaces of all compact Lie groups admit Riemannian metrics with completely integrable geodesic flows by means of C –smooth integrals [9, 10]. The purpose of this paper is to give some constructions of complete involutive algebras of analytic functions, polynomial in velocities, on the (co)tangent bundles of homogeneous spaces of compact Lie groups. This allows us to obtain new integrable Riemannian and sub-Riemannian geodesic flows on various homogeneous spaces, such as Stiefel manifolds, flag manifolds and orbits of the adjoint actions of compact Lie groups. Mathematics Subject Classification (2000): 70H06, 37J35, 53D17, 53D25