arXiv subjects

J. B. Sancho

Publications and source records attributed to J. B. Sancho.

5 recordsLinked to original sources

arXiv2016

Peetre-Slovák's theorem revisited

In 1960, J. Peetre proved the finiteness of the order of linear local operators. Later on, J. Slovák vastly generalized this theorem, proving the finiteness of the order of a broad class of (non-linear) local operators. In this paper, we use the language of sheaves and ringed spaces to prove a simpler version of Slovák's result. The statement we prove, adapting Slovák's original ideas, deals with local operators defined between the sheaves of smooth sections of fibre bundles, and thus covers many of the applications of Slovák's theorem.

math.DG↗

arXiv2014

Natural operations on differential forms

We prove that the only natural operations between differential forms are those obtained using linear combinations, the exterior product and the exterior differential. Our result generalises work by Palais and Freed-Hopkins. As an application, we also deduce a theorem, originally due to Kolar, that determines those natural differential forms that can be associated to a connection on a principal bundle.

math.DG↗

arXiv2012

Energy and electromagnetism of a differential form

Let X be a smooth manifold of dimension 1+n endowed with a lorentzian metric g, and let T be the electromagnetic energy tensor associated to a 2-form F. In this paper we characterize this tensor T as the only 2-covariant natural tensor associated to a lorentzian metric and a 2-form that is independent of the unit of scale and satisfies certain condition on its divergence. This characterization is motivated on physical grounds, and can be used to justify the Einstein-Maxwell field equations. More generally, we characterize in a similar manner the energy tensor associated to a differential form of arbitrary order k. Finally, we develop a generalized theory of electromagnetism where charged particles are not punctual, but of an arbitrary fixed dimension p. In this theory, the electromagnetic field F is a differential form of order 2+p and its electromagnetic energy tensor is precisely the energy tensor associated to F.

gr-qc↗

arXiv2011

A characterization of the electromagnetic stress-energy tensor

In a previous paper, we pointed out how a dimensional analysis of the stress-energy tensor of the gravitational field allows to derive the field equation of General Relativity. In this note, we comment an analogous reasoning in presence of a 2-form, that allows to characterize the so called electromagnetic stress-energy tensor.

gr-qc↗

arXiv2009

Moduli spaces for finite-order jets of Riemannian metrics

We construct the moduli space of r-jets at a point of Riemannian metrics on a smooth manifold. The construction is closely related to the problem of classification of jet metrics via differential invariants. The moduli space is proved to be a differentiable space which admits a finite canonical stratification into smooth manifolds. A complete study on the stratification of moduli spaces is carried out for metrics in dimension n=2.

math.DG↗