Address:
Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
E-mail:
katja.sagerschnig@univie.ac.at
travis.willse@univie.ac.at
Abstract: For the geometry of oriented $(2, 3, 5)$ distributions $(M, )$, which correspond to regular, normal parabolic geometries of type $(\operatorname{G}_2, P)$ for a particular parabolic subgroup $P < \operatorname{G}_2$, we develop the corresponding tractor calculus and use it to analyze the first BGG operator $\Theta _0$ associated to the $7$-dimensional irreducible representation of $\operatorname{G}_2$. We give an explicit formula for the normal connection on the corresponding tractor bundle and use it to derive explicit expressions for this operator. We also show that solutions of this operator are automatically normal, yielding a geometric interpretation of $\ker \Theta _0$: For any $(M, )$, this kernel consists precisely of the almost Einstein scales of the Nurowski conformal structure on $M$ that $$ determines. We apply our formula for $\Theta _0$ (1) to recover efficiently some known solutions, (2) to construct a distribution with root type $[3, 1]$ with a nonzero solution, and (3) to show efficiently that the conformal holonomy of a particular $(2, 3, 5)$ conformal structure is equal to $\operatorname{G}_2$.
AMSclassification: primary 53A30; secondary 53A40, 53C25, 58A30, 58J60.
Keywords: (2, 3, 5)-distributions, almost Einstein, BGG operators, conformal geometry, invariant differential operators.
DOI: 10.5817/AM2017-5-347