The almost Einstein operator for ( 2 , 3 , 5 ) distributions

Katja Sagerschnig; Travis Willse

Archivum Mathematicum (2017)

  • Volume: 053, Issue: 5, page 347-370
  • ISSN: 0044-8753

Abstract

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For the geometry of oriented ( 2 , 3 , 5 ) distributions ( M , ) , which correspond to regular, normal parabolic geometries of type ( G 2 , P ) for a particular parabolic subgroup P < G 2 , we develop the corresponding tractor calculus and use it to analyze the first BGG operator Θ 0 associated to the 7 -dimensional irreducible representation of 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 Θ 0 : For any ( M , ) , this kernel consists precisely of the almost Einstein scales of the Nurowski conformal structure on M that 𝐃 determines. We apply our formula for Θ 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 G 2 .

How to cite

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Sagerschnig, Katja, and Willse, Travis. "The almost Einstein operator for $(2, 3, 5)$ distributions." Archivum Mathematicum 053.5 (2017): 347-370. <http://eudml.org/doc/294548>.

@article{Sagerschnig2017,
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 $\mathbf \{D\}$ 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 $\operatorname\{G\}_2$.},
author = {Sagerschnig, Katja, Willse, Travis},
journal = {Archivum Mathematicum},
keywords = {$(2, 3, 5)$-distributions; almost Einstein; BGG operators; conformal geometry; invariant differential operators},
language = {eng},
number = {5},
pages = {347-370},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {The almost Einstein operator for $(2, 3, 5)$ distributions},
url = {http://eudml.org/doc/294548},
volume = {053},
year = {2017},
}

TY - JOUR
AU - Sagerschnig, Katja
AU - Willse, Travis
TI - The almost Einstein operator for $(2, 3, 5)$ distributions
JO - Archivum Mathematicum
PY - 2017
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 053
IS - 5
SP - 347
EP - 370
AB - 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 $\operatorname{G}_2$.
LA - eng
KW - $(2, 3, 5)$-distributions; almost Einstein; BGG operators; conformal geometry; invariant differential operators
UR - http://eudml.org/doc/294548
ER -

References

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