Providing an appropriate definition of a horizontal subbundle of a Lie algebroid will lead to construction of a better framework on Lie algebriods. In this paper, we give a new and natural definition of a horizontal subbundle using the prolongation of a Lie algebroid and then we show that any linear connection on a Lie algebroid generates a horizontal subbundle and vice versa. The same correspo...

Given a Dirac subbundle and an isotropic subbundle, we provide a canonical method to obtain a new Dirac subbundle. When the original Dirac subbundle is Courant involutive this construction has interesting applications, unifying and generalizing some results on the reduction of Dirac structures previously found in the literature.

A subbundle of a Hermitian vector bundle (E, h) can be metrically and differentiably defined by the orthogonal projection onto this subbundle. A weakly holomorphic subbundle of a Hermitian holomorphic bundle is, by definition, an orthogonal projection π lying in the Sobolev space L1 of L 2 sections with L2 first order derivatives in the sense of distributions, which satisfies furthermore (Id − ...

In a geometric point of view, a nonlinear control system, affine in the controls, is thought of as an affine subbundle of the tangent bundle of the state space. In deriving conditions for local controllability from this point of view, one should describe those properties of the affine subbundle that either ensure or prohibit local controllability. In this paper, second-order conditions of this ...

These lecture notes were written to accompany a minicourse given at the School and Workshop on Dynamical Systems at IMPA (Rio de Janeiro), January 2006. Warning: These notes are unedited lecture notes and do contain errors! Proceed with caution... Let f : M → M be a (pointwise) partially hyperbolic diffeomor-phism. Recall what partially hyperbolic means. At every point in M , there exist tangen...

We establish a version of the complex Frobenius theorem in the context of a complex subbundle S of the complexified tangent bundle of a manifold, having minimal regularity. If the subbundle S defines the structure of a Levi-flat CR-manifold, it suffices that S be Lipschitz for our results to apply. A principal tool in the analysis is a precise version of the Newlander-Nirenberg theorem with par...

The aim of the paper is to prove that every smooth generalized vector subbundle D of a vector bundle E is the image of a smooth endomorphism Φ on the fibers of E that induces authomorphisms of fibers of D and is called here a natural anchor. Two new constructions of finite sets of smooth generators of the fibers of D are obtained, using any finite set of generators of the module Γ(E). Natural a...

We establish a version of the complex Frobenius theorem in the context of a complex subbundle S of the complexified tangent bundle of a manifold, having minimal regularity. If the subbundle S defines the structure of a Levi-flat CR-manifold, it suffices that S be Lipschitz for our results to apply. A principal tool in the analysis is a precise version of the Newlander-Nirenberg theorem with par...