A constructive generalised Goursat normal form

Details A constructive generalised Goursat normal form A constructive generalised Goursat normal form

2004-04-21

arxiv.org/abs/math/0404377v1

Mathematics

Differential Geometry

33 pages

Scientific paper

We provide necessary and sufficient conditions on the derived type of a vector field distribution $\Cal V$ in order that it be locally equivalent to a partial prolongation of the contact distribution $\Cal C^{(1)}_q$, on the first order jet bundle of maps from $\Bbb R$ to $\Bbb R^q$, $q\geq 1$. This result fully generalises the classical Goursat normal form. Our proof is constructive: it is proven that if $\Cal V$ is locally equivalent to a partial prolongation of $\Cal C^{(1)}_q$ then the explicit construction of contact coordinates algorithmically depends upon the integration of a sequence of geometrically defined and algorithmically determined integrable Pfaffian systems on the ambient manifold.

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