# On a new normalization for tractor covariant derivatives

Matthias Hammerl; Petr Somberg; Vladimír Souček; Josef Šilhan

Journal of the European Mathematical Society (2012)

- Volume: 014, Issue: 6, page 1859-1883
- ISSN: 1435-9855

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topHammerl, 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 -

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