On a new normalization for tractor covariant derivatives

• Autores: Matthias Hammerl, Petr Somberg, Vladimir Soucek, Josef Silhan
• Localización: Journal of the European Mathematical Society, ISSN 1435-9855, Vol. 14, Nº 6, 2012, págs. 1859-1883
• Idioma: inglés
• DOI: 10.4171/jems/349
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• Resumen

  • A regular normal parabolic geometry of type G/P on a manifold M gives rise to sequences D i of invariant differential operators, known as the curved version of the BGG resolution. These sequences are constructed from the normal covariant derivative ? ? on the corresponding tractor bundle V, where ? is the normal Cartan connection. The first operator D 0 in the sequence is overdetermined and it is well known that ? ? yields the prolongation of this operator in the homogeneous case M=G/P . Our first main result is the curved version of such a prolongation. This requires a new normalization of the tractor covariant derivative on V . Moreover, we obtain an analogue for higher operators D i . In that case one needs to modify the exterior covariant derivative d ? ? by differential terms. Finally we demonstrate these results on simple examples in projective, conformal and Grassmannian geometry. Our approach is based on standard techniques of the BGG machinery.