Geometry associated with semisimple flat homogeneous spaces

Ochiai, Takushiro

doi:10.1090/S0002-9947-1970-0284936-6

Geometry associated with semisimple flat homogeneous spaces
HTML articles powered by AMS MathViewer

by Takushiro Ochiai PDF

Trans. Amer. Math. Soc. 152 (1970), 159-193 Request permission

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.

References

Armand Borel, On the curvature tensor of the Hermitian symmetric manifolds, Ann. of Math. (2) 71 (1960), 508–521. MR 111059, DOI 10.2307/1969940
A. Borel and F. Hirzebruch, Characteristic classes and homogeneous spaces. I, Amer. J. Math. 80 (1958), 458–538. MR 102800, DOI 10.2307/2372795

Oevres complètes

17

Oevres complètes

17

Shiing-Shen Chern, Pseudo-groupes continus infinis, Géométrie différentielle. Colloques Internationaux du Centre National de la Recherche Scientifique, Strasbourg, 1953, Centre National de la Recherche Scientifique, Paris, 1953, pp. 119–136 (French). MR 0063377
Charles Ehresmann, Les connexions infinitésimales dans un espace fibré différentiable, Colloque de topologie (espaces fibrés), Bruxelles, 1950, Georges Thone, Liège; Masson & Cie, Paris, 1951, pp. 29–55 (French). MR 0042768
Victor Guillemin, The integrability problem for $G$-structures, Trans. Amer. Math. Soc. 116 (1965), 544–560. MR 203626, DOI 10.1090/S0002-9947-1965-0203626-7
Victor Guillemin and Shlomo Sternberg, Deformation theory of pseudogroup structures, Mem. Amer. Math. Soc. 64 (1966), 80. MR 211421
Sigurđur Helgason, Differential geometry and symmetric spaces, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. MR 0145455
Shôshichi Kobayashi, On connections of Cartan, Canadian J. Math. 8 (1956), 145–156. MR 77978, DOI 10.4153/CJM-1956-018-8
Shoshichi Kobayashi, Theory of connections, Ann. Mat. Pura Appl. (4) 43 (1957), 119–194. MR 96276, DOI 10.1007/BF02411907
Shoshichi Kobayashi, Canonical forms on frame bundles of higher order contact, Proc. Sympos. Pure Math., Vol. III, American Mathematical Society, Providence, R.I., 1961, pp. 186–193. MR 0126810
Shoshichi Kobayashi and Tadashi Nagano, On projective connections, J. Math. Mech. 13 (1964), 215–235. MR 0159284
Shoshichi Kobayashi and Tadashi Nagano, On filtered Lie algebras and geometric structures. I, J. Math. Mech. 13 (1964), 875–907. MR 0168704
Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol I, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1963. MR 0152974
Bertram Kostant, Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math. (2) 74 (1961), 329–387. MR 142696, DOI 10.2307/1970237
Yozô Matsushima and Shingo Murakami, On vector bundle valued harmonic forms and automorphic forms on symmetric riemannian manifolds, Ann. of Math. (2) 78 (1963), 365–416. MR 153028, DOI 10.2307/1970348
Yozô Matsushima and Shingo Murakami, On certain cohomology groups attached to Hermitian symmetric spaces, Osaka Math. J. 2 (1965), 1–35. MR 184255

Cohomology groups of vector-valued forms on symmetric spaces

Tadashi Nagano, The projective transformation on a space with parallel Ricci tensor, K\B{o}dai Math. Sem. Rep. 11 (1959), 131–138. MR 109330
Tadashi Nagano, The conformal transformation on a space with parallel Ricci tensor, J. Math. Soc. Japan 11 (1959), 10–14. MR 124010, DOI 10.2969/jmsj/01110010
Tadashi Nagano, Transformation groups on compact symmetric spaces, Trans. Amer. Math. Soc. 118 (1965), 428–453. MR 182937, DOI 10.1090/S0002-9947-1965-0182937-8
Takushiro Ochiai, Classification of the finite nonlinear primitive Lie algebras, Trans. Amer. Math. Soc. 124 (1966), 313–322. MR 204480, DOI 10.1090/S0002-9947-1966-0204480-0
Takushiro Ochiai, On the automorphism group of a $G$-structure, J. Math. Soc. Japan 18 (1966), 189–193. MR 195019, DOI 10.2969/jmsj/01820189

"Sophus Lie” de l’École Normale Supérieure

17

I. M. Singer and Shlomo Sternberg, The infinite groups of Lie and Cartan. I. The transitive groups, J. Analyse Math. 15 (1965), 1–114. MR 217822, DOI 10.1007/BF02787690
Noboru Tanaka, Projective connections and projective transformations, Nagoya Math. J. 12 (1957), 1–24. MR 105154
Noboru Tanaka, Conformal connections and conformal transformations, Trans. Amer. Math. Soc. 92 (1959), 168–190. MR 124011, DOI 10.1090/S0002-9947-1959-0124011-X
Noboru Tanaka, On the equivalence problems associated with a certain class of homogeneous spaces, J. Math. Soc. Japan 17 (1965), 103–139. MR 188930, DOI 10.2969/jmsj/01720103

Zur Infinitesimalgeometrie Einordnung der projektiven und der konformen Auffassung

1921