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parabolic geometries; relative BGG conctruction; relative tractor calculus; Legendrean contact structures; Lagrangean contact structures; invariant differential operators; partial connections
For a manifold $M$ endowed with a Legendrean (or Lagrangean) contact structure $E\oplus F \subset TM$, we give an elementary construction of an invariant partial connection on the quotient bundle $TM/F$. This permits us to develop a naïve version of relative tractor calculus and to construct a second order invariant differential operator, which turns out to be the first relative BGG operator induced by the partial connection.
[1] Čap, A., Slovák, J.: Parabolic geometries. I. Mathematical Surveys and Monographs, vol. 154, AMS, Providence, RI, 2009, Background and general theory. DOI 10.1090/surv/154/03 | MR 2532439 | Zbl 1183.53002
[2] Čap, A., Souček, V.: Relative BGG sequences: I. Algebra. J. Algebra 463 (2016), 188–210. DOI 10.1016/j.jalgebra.2016.06.007 | MR 3527545
[3] Čap, A., Souček, V.: Relative BGG sequences; II. BGG machinery and invariant operators. Adv. Math. 320 (2017), 1009–1062. DOI 10.1016/j.aim.2017.09.016 | MR 3709128
[4] Takeuchi, M.: Legendrean contact structures on projective cotangent bundles. Osaka J. Math. 31 (4) (1994), 837–860. MR 1315010
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