Narrow Results

In this paper, we present geometrical interpretations of the Lie derivative and the Lie-invariance for an affine connection, based on two geometrical constructions: the former is given on the tangent bundle and relies on the notion of linear Ehresmann connection; the latter is developed on the frame bundle and is based on the more common notion of principal connection.

We present a generalization of the construction of a principal G-bundle from a one Čech cocycle to the case of higher abelian gerbes. We prove that the sheaf of local sections of the associated bundle to a higher abelian gerbe is isomorphic to the sheaf of sections of the gerbe itself. Our main result states that equivalence classes of higher abelian gerbes are in bijection with isomorphism classes of the corresponding bundles. We also present topological characterization of those bundles. In the last section, we show that the usual notion of Ehresmann connection leads to the gerbe connection for higher ℂ*-gerbes.

We study the trivialization and the reduction of Tulczyjew’s triplet, in the presence of a symmetry and an Ehresmann connection associated to it. We thus establish a geometric pathway for the Legendre transformations on singular dynamical systems.

In 2008–2009, L. F. O. Costa and C. A. R. Herdeiro proposed a new gravito-electromagnetic analogy, based on tidal tensors. We show that connections on the tangent bundle of the space-time manifold help in finding an advantageous geometrization of their ideas. Moreover, the combination tidal tensors — connections on tangent bundle can underlie a common mathematical description of the main equations of gravity and electromagnetism.