On some aspects of the geometry of non integrable distributions and applications

We consider a regular distribution D in a Riemannian manifold (M, g). The LeviCivita connection on (M, g) together with the orthogonal projection allow to endow the space of sections of D with a natural covariant derivative, the intrinsic connection. Hence we have two different covariant derivatives for sections of D, one directly with the connection in (M, g) and the other one with this intrinsic connection. Their difference is the second fundamental form of D and we prove it is a significant tool to characterize the involutive and the totally geodesic distributions and to give a natural formulation of the equation of motion for mechanical systems with constraints. The two connections also give two different notions of curvature, curvature tensors and sectional curvatures, which are compared in this paper with the use of the second fundamental form.Peer ReviewedPostprint (author's final draft

On some aspects of the geometry of non integrable distributions and applications

We consider a regular distribution $\mathcal{D}$ in a Riemannian manifold $(M,g)$. The Levi-Civita connection on $(M,g)$ together with the orthogonal projection allow to endow the space of sections of $\mathcal{D}$ with a natural covariant derivative, the intrinsic connection. Hence we have two different covariant derivatives for sections of $\mathcal{D}$, one directly with the connection in $(M,g)$ and the other one with this intrinsic connection. Their difference is the second fundamental form of $\mathcal{D}$ and we prove it is a significant tool to characterize the involutive and the totally geodesic distributions and to give a natural formulation of the equation of motion for mechanical systems with constraints. The two connections also give two different notions of curvature, curvature tensors and sectional curvatures, which are compared in this paper with the use of the second fundamental form.Comment: 23 page

A geometrical analysis of the field equations in field theory

In this review paper we give a geometrical formulation of the field equations in the Lagrangian and Hamiltonian formalisms of classical field theories (of first order) in terms of multivector fields. This formulation enables us to discuss the existence and non-uniqueness of solutions, as well as their integrability.Comment: 14 pages. LaTeX file. This is a review paper based on previous works by the same author

Remarks on multisymplectic reduction

The problem of reduction of multisymplectic manifolds by the action of Lie groups is stated and discussed, as a previous step to give a fully covariant scheme of reduction for classical field theories with symmetries.Comment: 9 pages. Some comments added in the section "Discussion and outlook" and in the Acknowledgments. New references are added. Minor mistakes are correcte