AHS-structures and affine holonomies

Čap, Andreas

doi:10.1090/S0002-9939-08-09722-0

AHS-structures and affine holonomies
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by Andreas Čap

Proc. Amer. Math. Soc. 137 (2009), 1073-1080

DOI: https://doi.org/10.1090/S0002-9939-08-09722-0

Published electronically: October 22, 2008

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Abstract:

We show that a large class of non-metric, non-symplectic affine holonomies can be realized, uniformly and without case by case considerations, by Weyl connections associated to the natural AHS–structures on certain generalized flag manifolds.

References

Marcel Berger, Sur les groupes d’holonomie homogène des variétés à connexion affine et des variétés riemanniennes, Bull. Soc. Math. France 83 (1955), 279–330 (French). MR 79806
M. Cahen, L.J. Schwachhöfer, special symplectic connections, preprint, math.DG/0402221
Andreas Čap and Hermann Schichl, Parabolic geometries and canonical Cartan connections, Hokkaido Math. J. 29 (2000), no. 3, 453–505. MR 1795487, DOI 10.14492/hokmj/1350912986
Andreas Čap and Jan Slovák, Weyl structures for parabolic geometries, Math. Scand. 93 (2003), no. 1, 53–90. MR 1997873, DOI 10.7146/math.scand.a-14413
A. Čap, J. Slovák, and V. Souček, Invariant operators on manifolds with almost Hermitian symmetric structures. II. Normal Cartan connections, Acta Math. Univ. Comenian. (N.S.) 66 (1997), no. 2, 203–220. MR 1620484
Shoshichi Kobayashi and Tadashi Nagano, On filtered Lie algebras and geometric structures. I, J. Math. Mech. 13 (1964), 875–907. MR 0168704
Sergei Merkulov and Lorenz Schwachhöfer, Classification of irreducible holonomies of torsion-free affine connections, Ann. of Math. (2) 150 (1999), no. 1, 77–149. MR 1715321, DOI 10.2307/121098
Keizo Yamaguchi, Differential systems associated with simple graded Lie algebras, Progress in differential geometry, Adv. Stud. Pure Math., vol. 22, Math. Soc. Japan, Tokyo, 1993, pp. 413–494. MR 1274961, DOI 10.2969/aspm/02210413