On a new normalization for tractor covariant derivatives

Hammerl, Matthias, et al. "On a new normalization for tractor covariant derivatives." Journal of the European Mathematical Society 014.6 (2012): 1859-1883. <http://eudml.org/doc/277291>.

@article{Hammerl2012,
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 $\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 demonstrate these results on simple examples in projective, conformal and Grassmannian geometry. Our approach is based on standard techniques of the BGG machinery.},
author = {Hammerl, Matthias, Somberg, Petr, Souček, Vladimír, Šilhan, Josef},
journal = {Journal of the European Mathematical Society},
keywords = {parabolic geometry; prolongation of invariant overdetermined PDE's; BGG sequence; tractor covariant derivatives; parabolic geometry; prolongation of overdetermined PDE's; BGG sequence; tractor covariant derivatives},
language = {eng},
number = {6},
pages = {1859-1883},
publisher = {European Mathematical Society Publishing House},
title = {On a new normalization for tractor covariant derivatives},
url = {http://eudml.org/doc/277291},
volume = {014},
year = {2012},
}

TY - JOUR
AU - Hammerl, Matthias
AU - Somberg, Petr
AU - Souček, Vladimír
AU - Šilhan, Josef
TI - On a new normalization for tractor covariant derivatives
JO - Journal of the European Mathematical Society
PY - 2012
PB - European Mathematical Society Publishing House
VL - 014
IS - 6
SP - 1859
EP - 1883
AB - 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 demonstrate these results on simple examples in projective, conformal and Grassmannian geometry. Our approach is based on standard techniques of the BGG machinery.
LA - eng
KW - parabolic geometry; prolongation of invariant overdetermined PDE's; BGG sequence; tractor covariant derivatives; parabolic geometry; prolongation of overdetermined PDE's; BGG sequence; tractor covariant derivatives
UR - http://eudml.org/doc/277291
ER -