Jet groupoids, natural bundles and the Vessiot equivalence method

• Jetgruppoide, Natürliche Bündel und die Vessiotsche Äquivalenzmethode

Lorenz, Arne; Plesken, Wilhelm (Thesis advisor)

Aachen : Publikationsserver der RWTH Aachen University (2009)
Dissertation / PhD Thesis

Aachen, Techn. Hochsch., Diss., 2009

Abstract

The present thesis deals with the equivalence of differential geometric objects. Based on a work of Vessiot published in 1903, an equivalence method is developed. It is intended to be an alternative to Cartan's well-known approach. In addition to theoretical aspects, an implementation of the methods is presented. The Vessiot equivalence method is formulated in the language of differential geometry. To describe geometric objects, the central concept of natural bundles is introduced. They are used to test equivalence of different geometric objects. For this, the symmetries of the objects and invariants are necessary. Lie and Vessiot showed that coordinate transfomations of natural bundles can be used to describe symmetries of geometric objects in terms of partial differential equations. In general, these equations are not integrable, i.e. by formal differentiation and elimination of highest order derivatives additional equations of lower order can be obtained. In the present thesis, the known methods are extended to check integrability. Furthermore it is shown how to complete the equations to an integrable system efficiently. In all steps, natural bundles are used. For the description of partial differential equations, a geometric approach of Spencer is used. It relies on the jet formalism and a differential equation is considered as a manifold. In the case of equations for symmetries, this manifold has a groupoid structure, which is important to decide integrability with the help of natural bundles. During the completion of the symmetry equations to integrability, invariants occur. The Vessiot equivalence method computes a generating set of invariants. With the help of symmetries and invariants it is possible to check whether two geometric objects are equivalent. Vessiot's equivalence method is compared to Cartan's approach. It is possible to give an interpretation of central constructions of Cartan in Vessiot's context. Furthermore, the Vessiot equivalence method is applied to the example of linear partial differential operators in order to calculate generating sets of invariants under gauge transformations.