On a new normalization for tractor covariant derivatives

Abstract

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 $\na^\om$ on the corresponding tractor bundle $V,$ where $\om$ is the normal Cartan connection. The first operator $D_0$ 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 $\tilde{\na}$ 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 $d^{\na^\om}$ 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.

22 pages

References in corpus (2)

• Yang-Mills detour complexes and conformal geometry
• Invariant Prolongation of BGG-Operators in Conformal Geometry