Geodesic
On a new normalization for tractor covariant derivatives
Journal of the European Mathematical Society, Tome 14 (2012) no. 6, pp. 1859-1883

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A regular normal parabolic geometry of type G/P on a manifold M gives rise to sequences Di​ of invariant differential operators, known as the curved version of the BGG resolution. These sequences are constructed from the normal covariant derivative ∇ω on the corresponding tractor bundle V, where ω is the normal Cartan connection. The first operator D0​ in the sequence is overdetermined and it is well known that ∇ω 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 Di​. In that case one needs to modify the exterior covariant derivative d∇ω 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.
DOI : 10.4171/jems/349
Classification : 58-XX, 53-XX, 00-XX
Keywords: Parabolic geometry, prolongation of invariant overdetermined PDE's, BGG sequence, tractor covariant derivatives 
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Matthias Hammerl; Petr Somberg; Vladimír Souček; Josef Šilhan. On a new normalization for tractor covariant derivatives. Journal of the European Mathematical Society, Tome 14 (2012) no. 6, pp. 1859-1883. doi: 10.4171/jems/349

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