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Keywords:
spinorial geometry; metrisability problem; equivalence problem; first BGG operator
spinorial geometry; metrisability problem; equivalence problem; first BGG operator
Summary:
Almost c-spinorial geometry arises as an interesting example of the metrisability problem for parabolic geometries. It is a complex analogue of real spinorial geometry. In this paper, we first define the type of parabolic geometry in question, then we discuss its underlying geometry and its homogeneous model. We compute irreducible components of the harmonic curvature and discuss the conditions for regularity. In the second part of the paper, we describe the linearisation of the metrisability problem for Hermitian and skew-Hermitian metrics, state the corresponding first BGG equations and present explicit formulae for their solutions on the homogeneous model.
References:
[1] Calderbank, D.M.J., Diemer, T.: Differential invariants and curved Bernstein-Gelfand-Gelfand sequences. J. Reine Angew. Math. 537 (2001), 67–103. MR 1856258 | Zbl 0985.58002
[2] Calderbank, D.M.J., Diemer, T., Souček, V.: Ricci-corrected derivatives and invariant differential operators. Differential Geom. Appl. 23 (2) (2005), 149–175. DOI 10.1016/j.difgeo.2004.07.009 | MR 2158042 | Zbl 1082.58037
[3] Calderbank, D.M.J., Eastwood, M.G., Matveev, V.S., Neusser, K.: C-projective geometry. 2015, arXiv:1512.04516v1 [math.DG].
[4] Calderbank, D.M.J., Slovák, J., Souček, V.: Subriemannian metrics and parabolic geometries. private communication.
[5] Čap, A., Gover, A.R., Hammerl, M.: Normal BGG solutions and polynomials. Internat. J. Math. 23 (2012), 29pp. MR 3005570 | Zbl 1263.53016
[6] Čap, A., Slovák, J.: Parabolic geometries. I. Mathematical Surveys and Monographs, vol. 154, American Mathematical Society, Providence, RI, 2009, Background and general theory. DOI 10.1090/surv/154/03 | MR 2532439 | Zbl 1183.53002
[8] Eastwood, M.: Notes on projective differential geometry. Symmetries and overdetermined systems of partial differential equations, IMA Vol. Math. Appl., vol. 144, Springer, New York, 2008, pp. 41–60. MR 2384705 | Zbl 1186.53020
[9] Eastwood, M., Matveev, V.: Metric connections in projective differential geometry. Symmetries and overdetermined systems of partial differential equations, IMA Vol. Math. Appl., vol. 144, Springer, New York, 2008, pp. 339–350. MR 2384718 | Zbl 1144.53027
[10] Hammerl, M., Somberg, P., Souček, V., Šilhan, J.: On a new normalization for tractor covariant derivatives. J. Eur. Math. Soc. (JEMS) 14 (6) (2012), 1859–1883. DOI 10.4171/JEMS/349 | MR 2984590 | Zbl 1264.58029
[11] Onishchik, A.L.: Lectures on real semisimple Lie algebras and their representations. ESI Lectures in Mathematics and Physics, European Mathematical Society (EMS), Zürich, 2004. MR 2041548 | Zbl 1080.17001
[12] Púček, R.: Applications of invariant operators in real parabolic geometries. Master's thesis, Charles University, 2016, https://is.cuni.cz/webapps/zzp/detail/166442/
[13] Šilhan, J.: Cohomology of Lie algebras - online service. http://web.math.muni.cz/~silhan/lie/
[14] Slovák, J., Souček, V.: Invariant operators of the first order on manifolds with a given parabolic structure. Global analysis and harmonic analysis (Marseille-Luminy, (1999), Sémin. Congr., vol. 4, Soc. Math. France, Paris, 2000, pp. 251–276. MR 1822364 | Zbl 0998.53021
[15] Yamaguchi, K.: Geometry of linear differential systems towards contact geometry of second order. Symmetries and overdetermined systems of partial differential equations, IMA Vol. Math. Appl., vol. 144, 2008, pp. 151–203. MR 2384710 | Zbl 1146.58003
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