Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 092, 41 pages      arXiv:0909.4728      https://doi.org/10.3842/SIGMA.2009.092
Contribution to the Special Issue “Élie Cartan and Differential Geometry”

Existence and Construction of Vessiot Connections

Dirk Fesser a and Werner M. Seiler b
a) IWR, Universität Heidelberg, INF 368, 69120 Heidelberg, Germany
b) AG ''Computational Mathematics'', Universität Kassel, 34132 Kassel, Germany

Received May 05, 2009, in final form September 14, 2009; Published online September 25, 2009

Abstract
A rigorous formulation of Vessiot's vector field approach to the analysis of general systems of partial differential equations is provided. It is shown that this approach is equivalent to the formal theory of differential equations and that it can be carried through if, and only if, the given system is involutive. As a by-product, we provide a novel characterisation of transversal integral elements via the contact map.

Key words: formal integrability; integral element; involution; partial differential equation; Vessiot connection; Vessiot distribution.

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