Previous |  Up |  Next


jet prolongation; principal prolongation; Cartan connection
We discuss frame bundles and canonical forms for geometries modeled on homogeneous spaces. Our aim is to introduce a geometric picture based on the non-holonomic jet bundles and principal prolongations as introduced in [Kolář, 71]. The paper has a partly expository character and we focus on very general aspects only. In the final section, various links to known results on the parabolic geometries are given briefly and some directions for further investigations are roughly indicated.
[1] Alekseevsky, D.V.; Michor, P.W.: Differential geometry of Cartan connections. ESI Preprint 39, Publ. Math. Debrecen 47 (1995), 349–375. MR 1362298
[2] Baston, R. J.: Almost Hermitian symmetric manifolds, I: Local twistor theory. Duke Math. J. 63 (1991), 81–111. MR 1106939 | Zbl 0724.53019
[3] Čap, A.; Schichl, H.: paper in preparation.
[4] Čap, A.; Slovák, J.: On local flatness of AHS–manifolds. to appear in Rendiconti Circ. Mat. Palermo, Proceedings of the Winter School Geometry and Physics, Srní1995.
[5] Čap, A.; Slovák, J.; Souček, V.: Invariant Operators on Manifolds with Almost Hermitian Symmetric Structures, I. Invariant Differentiation. electronically available at, Preprint ESI 186 (1994), 31 pp. MR 1474550
[6] Čap, A.; Slovák, J.; Souček, V.: Invariant Operators on Manifolds with Almost Hermitian Symmetric Structures, II. Normal Cartan connections. electronically available at, Preprint ESI 194 (1995), 16 pp. MR 1620484
[7] Kobayashi, S.: Transformation groups in differential geometry. Springer-Verlag, Berlin, Heidelberg, New York, 1972. MR 0355886 | Zbl 0246.53031
[8] Kolář, I.: Canonical forms on the prolongations of principal fiber bundles. Rev. Roumaine Math. Pures Appl. 16 (1971), 1091–1106. MR 0301668
[9] Kolář, I.: Higher order torsions of spaces with Cartan Connection. Cahiers Topologie Géom. Différentielle 12 (1971), 137–146. MR 0315619
[10] Kolář, I.: Generalized $G$-structures and $G$-structures of higher order. Bollettino U. M. Ital. 12, Suppl. 3 (1975), 245–256. MR 0445424
[11] Kolář, I.: A generalization of the torsion form. Časopis pro pěstování matematiky 100 (1975), 284–290. MR 0383287
[12] Kolář, I.; Michor, P. W.; Slovák, J.: Natural operations in differential geometry. Springer-Verlag, Berlin Heidelberg New York, 1993. MR 1202431
[13] Kriegl, A.; Michor, P. W.: The Convenient Setting for Global Analysis. to appear, Surveys and Monographs, AMS, Providence, 1997. MR 1471480
[14] Libermann, P.: Sur les prolongements des fibrés principaux et des groupoides différentiables banachiques. Analyse global, Sém. Mat. Supérieures, No. 42 (Été, 1969) (1971), 7–108. MR 0356117 | Zbl 0248.53031
[15] Morimoto, T.: Geometric structures on filtered manifolds. Hokkaido Math. J. 22 (1993), 263–347. MR 1245130 | Zbl 0801.53019
[16] Slovák, J.: The principal prolongation of first order $G$-structures. Proceedings of the Winter School Geometry and Physics, Srní 1994, Supplemento ai Rendiconti Circ. Mat. Palermo 39 (1996), 123–131. MR 1396607
[17] Tanaka, N.: On differential systems, graded Lie algebras and pseudo-groups. J. Math. Kyoto Univ. 10 (1970), 1–82. MR 0266258 | Zbl 0206.50503
[18] Tanaka, N.: On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections. Japanese J. Math. 2 (1976), 131–190. MR 0589931 | Zbl 0346.32010
[19] Tanaka, N.: On the equivalence problems associated with simple graded Lie algebras. Hokkaido Math. J. 8 (1979), 23–84. MR 0533089 | Zbl 0409.17013
Partner of
EuDML logo