# Masaryk University

Publication details

# On a new normalization for tractor covariant derivatives

Authors SOUČEK Vladimír SOMBERG Petr HAMMERL Matthias 2012 Article in Periodical Journal of the European Mathematical Society http://www.ems-ph.org/journals/show_abstract.php?issn=1435-9855&vol=14&iss=6&rank=5 http://dx.doi.org/10.4171/JEMS/349 General mathematics Parabolic geometry - prolongation of invariant PDE’s - BGG sequence - tractor covariant derivatives - projective geometry - conformal geometry - Grassmannian geometry 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 illustrate these results with simple examples in projective, conformal and Grassmannian geometry. Our approach is based on standard BGG techniques.