Publication details

On a new normalization for tractor covariant derivatives



Year of publication 2012
Type Article in Periodical
Magazine / Source Journal of the European Mathematical Society
MU Faculty or unit

Faculty of Science

Field General mathematics
Keywords Parabolic geometry - prolongation of invariant PDE’s - BGG sequence - tractor covariant derivatives - projective geometry - conformal geometry - Grassmannian geometry
Description 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.
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