2010/03/31 by Matthias Hammerl, Hammerl, Matthias, Petr Somberg +5 · 2 citations
Mathematics · Physics and Astronomy · #53A20 #53A30 #53A55 #58A32 #58J70 #Advanced Differential Equations and Dynamical Systems #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.1003.6090
openalex publication_date 2010/03/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \na^\om on the corresponding tractor bundle V, where \om is the normal Cartan connection. The first operator D0 in the sequence is overdetermined and it is well known that \na^\om 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 \na 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\na^\om by differential terms. Finally we demonstrate these results on simple examples in projective and Grassmannian geometry. Our approach is based on standard techniques of the BGG machinery.