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Homogeneous spaces with invariant projectively flat affine connections
Author(s):
Hirohiko
Shima
Journal:
Trans. Amer. Math. Soc.
351
(1999),
4713-4726.
MSC (1991):
Primary 53C05, 53C30, 53C35, 53A15;
Secondary 17C20
Posted:
August 25, 1999
MathSciNet review:
1675234
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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:
- [A]
- Y. Agaoka, Invariant flat projective structures on homogeneous spaces, Hokkaido Math. J. 11 (1982), 125-172. MR 85g:53038
- [BK]
- H. Braun und M. Koecher, Jordan-Algebren, Grundlehren der Math. Wissenschaften, 128, Springer, Berlin, Heidelberg, New York, 1966. MR 34:4310
- [Ka]
- S. Kaneyuki, The Sylvester's Law of Inertia in Simple Graded Lie Algebras, J. Math. Soc. Japan, 50(1998), 593-614. MR 99f:17035
- [KN]
- S. Kobayashi and K. Nomizu, Foundations of differential geometry, vol.I, John Wiley & Sons, New York, 1963. MR 97c:53001b
- [K]
- M. Koecher, Jordan algebras and their applications, Lecture notes, Univ. of Minnesota, Minneapolis, 1962.
- [Ks]
- J.L. Koszul, Domaines bornés homogenes et orbites de groupes de transformations affines, Bull. Soc. Math. France, 89 (1961), 515-533. MR 26:3090
- [NP]
- K. Nomizu and U. Pinkall, On a certain class of homogeneous projectively flat manifolds, Tohoku Math. J. 39 (1987), 407-427. MR 88j:53050
- [NS]
- K. Nomizu and T. Sasaki, Affine Differential Geometry, Cambridge Univ. Press, 1994. MR 96e:53014
- [Sa]
- T. Sasaki, Hyperbolic affine hyperspheres, Nagoya Math. J. 77 (1980), 107-123. MR 81e:53037
- [Sh1]
- H. Shima, On locally symmetric homogeneous domains of completely reducible linear Lie groups, Math. Ann., 217 (1975), 93-95. MR 52:818
- [Sh2]
- H. Shima, Symmetric spaces with invariant locally Hessian structures, J. Math. Soc. Japan, 29 (1977), 581-589. MR 56:9462
- [Sh3]
- H. Shima, Homogeneous Hessian manifolds, Ann. Inst. Fourier, Grenoble, 30 (1980), 91-128. MR 82a:53054
- [V1]
- E.B. Vinberg, Homogeneous cones, Dokl. Akad. Nauk SSSR, 133 (1960), 9-12; English transl., Soviet Math. Dokl., 1 (1960), 787-790. MR 25:5077
- [V2]
- E.B. Vinberg, The theory of convex homogeneous cones, Trans. Moscow Math. Soc. 12 (1963), 340-403. MR 28:1637
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Additional Information:
Hirohiko
Shima
Affiliation:
Department of Mathematics, Yamaguchi University, Yamaguchi 753-8512, Japan
Email:
shima@po.cc.yamaguchi-u.ac.jp
DOI:
10.1090/S0002-9947-99-02523-4
PII:
S 0002-9947(99)02523-4
Keywords:
Invariant projectively flat affine connections,
centro-affine immersions,
homogeneous spaces,
symmetric spaces,
Jordan algebras
Received by editor(s):
March 15, 1996
Posted:
August 25, 1999
Copyright of article:
Copyright
1999,
American Mathematical Society
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