Homogeneous spaces with invariant projectively flat affine connections

Shima, Hirohiko

doi:10.1090/S0002-9947-99-02523-4

Homogeneous spaces with invariant projectively flat affine connections
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Trans. Amer. Math. Soc. 351 (1999), 4713-4726 Request permission

Abstract:

We characterize invariant projectively flat affine connections in terms of affine representations of Lie algebras, and show that a homogeneous space admits an invariant projectively flat affine connection if and only if it has an equivariant centro-affine immersion. We give a correspondence between semi-simple symmetric spaces with invariant projectively flat affine connections and central-simple Jordan algebras.

References

Yoshio Agaoka, Geometric invariants associated with flat projective structures, J. Math. Kyoto Univ. 22 (1982/83), no. 4, 701–718. MR 685526, DOI 10.1215/kjm/1250521676
Hel Braun and Max Koecher, Jordan-Algebren, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band 128, Springer-Verlag, Berlin-New York, 1966 (German). MR 0204470
Soji Kaneyuki, The Sylvester’s law of inertia in simple graded Lie algebras, J. Math. Soc. Japan 50 (1998), no. 3, 593–614. MR 1626338, DOI 10.2969/jmsj/05030593
Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol. I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1996. Reprint of the 1963 original; A Wiley-Interscience Publication. MR 1393940
M. Koecher, Jordan algebras and their applications, Lecture notes, Univ. of Minnesota, Minneapolis, 1962.
Jean-Louis Koszul, Domaines bornés homogènes et orbites de groupes de transformations affines, Bull. Soc. Math. France 89 (1961), 515–533 (French). MR 145559
Katsumi Nomizu and Ulrich Pinkall, On a certain class of homogeneous projectively flat manifolds, Tohoku Math. J. (2) 39 (1987), no. 3, 407–427. MR 902579, DOI 10.2748/tmj/1178228287
Katsumi Nomizu and Takeshi Sasaki, Affine differential geometry, Cambridge Tracts in Mathematics, vol. 111, Cambridge University Press, Cambridge, 1994. Geometry of affine immersions. MR 1311248
Takeshi Sasaki, Hyperbolic affine hyperspheres, Nagoya Math. J. 77 (1980), 107–123. MR 556312
Hirohiko Shima, On locally symmetric homogeneous domains of completely reducible linear Lie groups, Math. Ann. 217 (1975), no. 1, 93–95. MR 379914, DOI 10.1007/BF01363245
Hirohiko Shima, Symmetric spaces with invariant locally Hessian structures, J. Math. Soc. Japan 29 (1977), no. 3, 581–589. MR 451175, DOI 10.2969/jmsj/02930581
Hirohiko Shima, Homogeneous Hessian manifolds, Ann. Inst. Fourier (Grenoble) 30 (1980), no. 3, 91–128. MR 597019
È. B. Vinberg, Homogeneous cones, Soviet Math. Dokl. 1 (1960), 787–790. MR 0141680
È. B. Vinberg, The theory of homogeneous convex cones, Trudy Moskov. Mat. Obšč. 12 (1963), 303–358 (Russian). MR 0158414