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@PHDTHESIS{Lorenz:50704,
      author       = {Lorenz, Arne},
      othercontributors = {Plesken, Wilhelm},
      title        = {{J}et groupoids, natural bundles and the {V}essiot
                      equivalence method},
      address      = {Aachen},
      publisher    = {Publikationsserver der RWTH Aachen University},
      reportid     = {RWTH-CONV-113237},
      pages        = {270 S.},
      year         = {2009},
      note         = {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.},
      keywords     = {Lie-Gruppoid (SWD) / Natürliches Bündel (SWD) /
                      G-Struktur (SWD) / Äquivalenz (SWD) / Formale Äquivalenz
                      (SWD)},
      cin          = {114410 / 110000},
      ddc          = {510},
      cid          = {$I:(DE-82)114410_20140620$ / $I:(DE-82)110000_20140620$},
      shelfmark    = {22A22 * 53C10 * 58H05 * 53C15},
      typ          = {PUB:(DE-HGF)11},
      urn          = {urn:nbn:de:hbz:82-opus-27635},
      url          = {https://publications.rwth-aachen.de/record/50704},
}