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Geometry associated with semisimple flat homogeneous spaces


Author: Takushiro Ochiai
Journal: Trans. Amer. Math. Soc. 152 (1970), 159-193
MSC: Primary 53.52
DOI: https://doi.org/10.1090/S0002-9947-1970-0284936-6
MathSciNet review: 0284936
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Abstract: Our object is Cartan connections with semisimple flat homogeneous spaces as standard spaces. We study these from the viewpoint of $ G$-structures of second order. This allows us especially to treat classical projective and conformal connections in the unifying manner. We also consider its application to the problem of certain geometric transformations.


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  • [1] A. Borel, On the curvature tensor of the Hermitian symmetric manifolds, Ann. of Math. (2) 71 (1960), 508-521. MR 22 #1923. MR 0111059 (22:1923)
  • [2] A. Borel and F. Hirzebruch, Characteristic classes and homogeneous spaces. I, Amer. J. Math. 80 (1958), 458-538. MR 21 #1586. MR 0102800 (21:1586)
  • [3] E. Cartan, ``Les espaces a connexion conforme,'' en Oevres complètes. Part III, Vol. I: Divérs, géométrie différentielle, Gauthier-Villars, Paris, 1955, pp. 749-797. MR 17, 697.
  • [4] -, ``Sur les variétés a connexion projective,'' en Oevres complètes. Part III, Vol. I: Divérs, géométrie différentielle, Gauthier-Villars, Paris, 1955, pp. 825-861. MR 17, 697.
  • [5] S. S. Chern, Pseudo-groupes continus infinis, géométrie différentielle, Colloq. Internat. Centre National Recherche Scientifique (Strasbourg, 1953) Centre National de la Recherche Scientifique, Paris, 1953, pp. 119-136. MR 16, 112. MR 0063377 (16:112a)
  • [6] C. Ehresmann, Les connexions infinitésimales dans un espace fibré différentiable, Colloque de Topologie (espaces fibrés) (Bruxelles, 1950) Georges Thone, Liège et Masson, Paris, 1951, pp. 29-55. MR 13, 159. MR 0042768 (13:159e)
  • [7] V. W. Guillemin, The integrability problem for $ G$-structures, Trans. Amer. Math. Soc. 116 (1965), 544-560. MR 34 #3475. MR 0203626 (34:3475)
  • [8] V. W. Guillemin and S. Sternberg, Deformation theory of pseudogroup structures, Mem. Amer. Math. Soc. No. 64 (1966). MR 35 #2302. MR 0211421 (35:2302)
  • [9] S. Helgason, Differential geometry and symmetric spaces, Pure and Appl. Math., vol. 12, Academic Press, New York, 1962. MR 26 #2986. MR 0145455 (26:2986)
  • [10] S. Kobayashi, On connections of Cartan, Canad. J. Math. 8 (1956), 145-156. MR 17, 1126. MR 0077978 (17:1126f)
  • [11] -, Theory of connections, Ann. Math. Pura Appl. (4) 43 (1957), 119-194. MR 20 #2760. MR 0096276 (20:2760)
  • [12] -, Canonical forms on frame bundles of higher order contact, Proc. Sympos. Pure Math., vol. 3, Amer. Math. Soc., Providence, R. I., 1961, pp. 186-193. MR 23 #A4104. MR 0126810 (23:A4104)
  • [13] S. Kobayashi and T. Nagano, On projective connections, J. Math. Mech. 13 (1964), 215-235. MR 28 #2501. MR 0159284 (28:2501)
  • [14] -, On filtered Lie algebras and geometric structures. I, II, J. Math. Mech. 13 (1964), 875-907; ibid., 14 (1965), 513-521. MR 29 #5961; MR 32 #2512. MR 0168704 (29:5961)
  • [15] S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol. I, Interscience, New York, 1963. MR 27 #2945. MR 0152974 (27:2945)
  • [16] B. Kostant, Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math. (2) 74 (1961), 329-387. MR 26 #265. MR 0142696 (26:265)
  • [17] Y. Matsushima and S. Murakami, On vector bundle valued harmonic forms and automorphic forms on symmetric riemannian manifolds, Ann. of Math. (2) 78 (1963), 365-416. MR 27 #2997. MR 0153028 (27:2997)
  • [18] -, On certain cohomology groups attached to Hermitian symmetric spaces, Osaka J. Math. 2 (1965), 1-35. MR 32 #1728. MR 0184255 (32:1728)
  • [19] S. Murakami, Cohomology groups of vector-valued forms on symmetric spaces, Lecture Note, Univ. of Chicago, Chicago, Ill., 1966.
  • [20] T. Nagano, The projective transformation on a space with parallel Ricci tensor, Kōdai Math. Sem. Rep. 11 (1959), 131-138. MR 22 #216. MR 0109330 (22:216)
  • [21] T. Nagano, The conformal transformation on a space with parallel Ricci tensor, J. Math. Soc. Japan 11 (1959), 10-14. MR 23 #A1330. MR 0124010 (23:A1330)
  • [22] -, Transformation groups on compact symmetric spaces, Trans. Amer. Math. Soc. 118 (1965), 428-353. MR 32 #419. MR 0182937 (32:419)
  • [23] T. Ochiai, Classification of the finite nonlinear primitive Lie algebras, Trans. Amer. Math. Soc. 124 (1966), 313-322. MR 34 #4320. MR 0204480 (34:4320)
  • [24] -, On the automorphism group of a $ G$-structure, J. Math. Soc. Japan 18 (1966), 189-193. MR 33 #3224. MR 0195019 (33:3224)
  • [25] "Sophus Lie'' de l'École Normale Supérieure 1954/55, Séminaire théorie des algèbres de Lie. Topologie des groupes de Lie, Secrétariat mathématique, Paris, 1955. MR 17, 384.
  • [26] I. M. Singer and S. Sternberg, The infinite group of Lie and Cartan. I. The transitive groups, J. Analyse Math. 15 (1965), 1-114. MR 36 #911. MR 0217822 (36:911)
  • [27] N. Tanaka, Projective connections and projective transformations, Nagoya Math. J. 12 (1957), 1-24. MR 21 #3899. MR 0105154 (21:3899)
  • [28] -, Conformal connections and conformal transformations, Trans. Amer. Math. Soc. 92 (1959), 168-190. MR 23 #A1331. MR 0124011 (23:A1331)
  • [29] -, On the equivalence problems associated with a certain class of homogeneous spaces, J. Math. Soc. Japan 17 (1965), 103-139. MR 32 #6358. MR 0188930 (32:6358)
  • [30] H. Weyl, Zur Infinitesimalgeometrie Einordnung der projektiven und der konformen Auffassung, Göttingen. Nachr. 1921, 99-112.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1970-0284936-6
Keywords: Graded Lie algebra, Spencer cohomology, semisimple Lie algebra, harmonic class, semisimple flat homogeneous space, bundle of higher order contact, linear connection, Cartan connection, normal Cartan connection, Weyl tensor
Article copyright: © Copyright 1970 American Mathematical Society

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