Geodesic
The almost Einstein operator for $(2, 3, 5)$ distributions
Archivum mathematicum, Tome 53 (2017) no. 5, pp. 347-370 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 $\mathbf{D}$ 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 $\operatorname{G}_2$.
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 $\mathbf{D}$ 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 $\operatorname{G}_2$.
DOI : 10.5817/AM2017-5-347
Classification : 53A30, 53A40, 53C25, 58A30, 58J60
Keywords: $(2, 3, 5)$-distributions; almost Einstein; BGG operators; conformal geometry; invariant differential operators 
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Sagerschnig, Katja; Willse, Travis. The almost Einstein operator for $(2, 3, 5)$ distributions. Archivum mathematicum, Tome 53 (2017) no. 5, pp. 347-370. doi: 10.5817/AM2017-5-347

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