Wyniki wyszukiwania

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Calculus on Lie algebroids, Lie groupoids and Poisson manifolds

100%

Marle C.-M.

Instytut Matematyczny Polskiej Akademii Nauk

EN

We begin with a short presentation of the basic concepts related to Lie groupoids and Lie algebroids, but the main part of this paper deals with Lie algebroids. A Lie algebroid over a manifold is a vector bundle over that manifold whose properties are very similar to those of a tangent bundle. Its dual bundle has properties very similar to those of a cotangent bundle: in the graded algebra of sections of its exterior powers, one can define an operator $d_{E}$ similar to the exterior derivative. We present the theory of Lie derivatives, Schouten-Nijenhuis brackets and exterior derivatives in the general setting of a Lie algebroid, its dual bundle and their exterior powers. All the results (which, for the most part, are already known) are given with detailed proofs. In the final sections, the results are applied to Poisson manifolds, whose links with Lie algebroids are very close.

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The works of Charles Ehresmann on connections: from Cartan connections to connections on fibre bundles

96%

Marle C.-M.

Banach Center Publications

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2007

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tom 76

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nr 1

65-86

EN

Around 1923, Élie Cartan introduced affine connections on manifolds and defined the main related concepts: torsion, curvature, holonomy groups. He discussed applications of these concepts in Classical and Relativistic Mechanics; in particular he explained how parallel transport with respect to a connection can be related to the principle of inertia in Galilean Mechanics and, more generally, can be used to model the motion of a particle in a gravitational field. In subsequent papers, Élie Cartan extended these concepts for other types of connections on a manifold: Euclidean, Galilean and Minkowskian connections which can be considered as special types of affine connections, the group of affine transformations of the affine tangent space being replaced by a suitable subgroup; and more generally, conformal and projective connections, associated to a group which is no more a subgroup of the affine group. Around 1950, Charles Ehresmann introduced connections on a fibre bundle and, when the bundle has a Lie group as structure group, connection forms on the associated principal bundle, with values in the Lie algebra of the structure group. He called Cartan connections the various types of connections on a manifold previously introduced by É. Cartan, and explained how they can be considered as special cases of connections on a fibre bundle with a Lie group G as structure group: the standard fibre of the bundle is then an homogeneous space G/G'; its dimension is equal to that of the base manifold; a Cartan connection determines an isomorphism of the vector bundle tangent to the the base manifold onto the vector bundle of vertical vectors tangent to the fibres of the bundle along a global section. These works are reviewed and some applications of the theory of connections are sketched.

3

On symmetries and constants of motion in Hamiltonian systems with nonholonomic constraints

96%

Marle C.-M.

Banach Center Publications

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2003

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tom 59

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nr 1

223-242

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On submanifolds and quotients of Poisson and Jacobi manifolds

96%

Marle C.-M.

Banach Center Publications

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2000

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tom 51

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nr 1

197-209

EN

We obtain conditions under which a submanifold of a Poisson manifold has an induced Poisson structure, which encompass both the Poisson submanifolds of A. Weinstein [21] and the Poisson structures on the phase space of a mechanical system with kinematic constraints of Van der Schaft and Maschke [20]. Generalizations of these results for submanifolds of a Jacobi manifold are briefly sketched.

Ograniczanie wyników

3 Banach Center Publications

4 Marle C.-M.

1 2008

1 2007

1 2003

1 2000

Calculus on Lie algebroids, Lie groupoids and Poisson manifolds

The works of Charles Ehresmann on connections: from Cartan connections to connections on fibre bundles

On symmetries and constants of motion in Hamiltonian systems with nonholonomic constraints

On submanifolds and quotients of Poisson and Jacobi manifolds