JournalsjemsVol. 14, No. 6pp. 1859–1883

On a new normalization for tractor covariant derivatives

  • Matthias Hammerl

    Universität Wien, Austria
  • Petr Somberg

    Charles University, Prague, Czech Republic
  • Vladimír Souček

    Charles University, Prague, Czech Republic
  • Josef Šilhan

    Masaryk University, Brno, Czech Republic
On a new normalization for tractor covariant derivatives cover
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Abstract

A regular normal parabolic geometry of type G/PG/P on a manifold MM gives rise to sequences DiD_i of invariant differential operators, known as the curved version of the BGG resolution. These sequences are constructed from the normal covariant derivative ω\nabla^\omega on the corresponding tractor bundle V,V, where ω\omega is the normal Cartan connection. The first operator D0D_0 in the sequence is overdetermined and it is well known that ω\nabla^\omega yields the prolongation of this operator in the homogeneous case M=G/PM = 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 VV. Moreover, we obtain an analogue for higher operators DiD_i. In that case one needs to modify the exterior covariant derivative dωd^{\nabla^\omega} 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.

Cite this article

Matthias Hammerl, Petr Somberg, Vladimír Souček, Josef Šilhan, On a new normalization for tractor covariant derivatives. J. Eur. Math. Soc. 14 (2012), no. 6, pp. 1859–1883

DOI 10.4171/JEMS/349