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Journal of the European Mathematical Society

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Volume 14, Issue 6, 2012, pp. 1859–1883
DOI: 10.4171/JEMS/349

Published online: 2012-10-10

On a new normalization for tractor covariant derivatives

Matthias Hammerl[1], Petr Somberg[2], Vladimír Souček[3] and Josef Šilhan[4]

(1) Universität Wien, Austria
(2) Charles University, Prague, Czech Republic
(3) Charles University, Prague, Czech Republic
(4) Masaryk University, Brno, Czech Republic

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 $\nabla^\omega$ on the corresponding tractor bundle $V,$ where $\omega$ is the normal Cartan connection. The first operator $D_0$ in the sequence is overdetermined and it is well known that $\nabla^\omega$ 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 $d^{\nabla^\omega}$ 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.

Keywords: Parabolic geometry, prolongation of invariant overdetermined PDE's, BGG sequence, tractor covariant derivatives

Hammerl Matthias, Somberg Petr, Souček Vladimír, Šilhan Josef: On a new normalization for tractor covariant derivatives. J. Eur. Math. Soc. 14 (2012), 1859-1883. doi: 10.4171/JEMS/349