Constraint Vector Bundles and Reduction of Lie (Bi-)Algebroids

TL;DR

This paper develops a comprehensive framework for reducing complex geometric structures like Lie algebroids and Dirac manifolds using constraint vector bundles, extending classical reduction methods.

Contribution

It introduces constraint vector bundles, proves a constraint Serre-Swan theorem, and develops a Cartan calculus compatible with reduction for advanced geometric structures.

Findings

Constraint vector bundles are identified with finitely generated projective modules.

A Cartan calculus for constraint forms and multivector fields is established.

A reduction procedure for Lie (bi-)algebroids and Dirac manifolds is developed.

Abstract

We present a framework for the reduction of various geometric structures extending the classical coisotropic Poisson reduction. For this we introduce constraint manifolds and constraint vector bundles. A constraint Serre-Swan theorem is proven, identifying constraint vector bundles with certain finitely generated projective modules, and a Cartan calculus for constraint differentiable forms and multivector fields is introduced. All of these constructions will be shown to be compatible with reduction. Finally, we apply this to obtain a reduction procedure for Lie (bi-)algebroids and Dirac manifolds.