Geodesic
On Tanaka's Prolongation Procedure for Filtered Structures of Constant Type
Symmetry, integrability and geometry: methods and applications, Tome 5 (2009) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We present Tanaka's prolongation procedure for filtered structures on manifolds discovered in [Tanaka N., J. Math. Kyoto. Univ. 10 (1970), 1–82] in a spirit of Singer–Sternberg's description of the prolongation of usual $G$-structures [Singer I. M., Sternberg S., J. Analyse Math. 15 (1965), 1–114; Sternberg S., Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964]. This approach gives a transparent point of view on the Tanaka constructions avoiding many technicalities of the original Tanaka paper.
Keywords: $G$-structures; filtered structures; generalized Spencer operator; prolongations. 
@article{SIGMA_2009_5_a93,
     author = {Igor Zelenko},
     title = {On {Tanaka's} {Prolongation} {Procedure} for {Filtered} {Structures} of {Constant} {Type}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2009},
     volume = {5},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2009_5_a93/}
}
TY  - JOUR
AU  - Igor Zelenko
TI  - On Tanaka's Prolongation Procedure for Filtered Structures of Constant Type
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2009
VL  - 5
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2009_5_a93/
LA  - en
ID  - SIGMA_2009_5_a93
ER  - 
%0 Journal Article
%A Igor Zelenko
%T On Tanaka's Prolongation Procedure for Filtered Structures of Constant Type
%J Symmetry, integrability and geometry: methods and applications
%D 2009
%V 5
%U http://geodesic.mathdoc.fr/item/SIGMA_2009_5_a93/
%G en
%F SIGMA_2009_5_a93
Igor Zelenko. On Tanaka's Prolongation Procedure for Filtered Structures of Constant Type. Symmetry, integrability and geometry: methods and applications, Tome 5 (2009). http://geodesic.mathdoc.fr/item/SIGMA_2009_5_a93/

[1] Cartan É., “Les systémes de Pfaff á cinq variables et les équations aux dérivées partielles du second ordre”, Ann. Sci. École Norm. Sup. (3), 27 (1910), 109–192 ; reprinted in his Oeuvres completes, Partie II, Vol. 2, Gautier-Villars, Paris, 1953, 927–1010 | MR | Zbl

[2] Cartan É., “Sur les variétés à connexion projective”, Bull. Soc. Math. France, 54 (1924), 205–241 | MR

[3] Kuzmich O., “Graded nilpotent Lie algebras in low dimensions”, Lobachevskii J. Math., 3 (1999), 147–184 | MR | Zbl

[4] Morimoto T., “Geometric structures on filtered manifolds”, Hokkaido Math. J., 22 (1993), 263–347 | MR | Zbl

[5] Morimoto T., “Lie algebras, geometric structures, and differential equations on filtered manifolds”, Lie Groups, Geometric Structures and Differential Equations – One Hundred Years after Sophus Lie (Kyoto/Nara, 1999), Adv. Stud. Pure Math., 37, Math. Soc. Japan, Tokyo, 2002, 205–252 | MR | Zbl

[6] Morimoto T., “Cartan connection associated with a subriemannian structure”, Differential Geom. Appl., 26 (2008), 75–78 | DOI | MR | Zbl

[7] Singer I. M., Sternberg S., “The infinite groups of Lie and Cartan. I. The transitive groups”, J. Analyse Math., 15 (1965), 1–114 | DOI | MR | Zbl

[8] Spencer D. C., “Overdetermined systems of linear partial differential equations”, Bull. Amer. Math. Soc., 75 (1969), 179–239 | DOI | MR | Zbl

[9] Sternberg S., Lectures on differential geometry, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964 | MR | Zbl

[10] Tanaka N., “On differential systems, graded Lie algebras and pseudogroups”, J. Math. Kyoto. Univ., 10 (1970), 1–82 | MR | Zbl

[11] Tanaka N., “On the equivalence problems associated with simple graded Lie algebras”, Hokkaido Math. J., 8 (1979), 23–84 | MR | Zbl

[12] Tresse A., Détermination des invariants ponctuels de l'équation différentielle ordinaire du second ordre $y''=w(x,y,y')$, S. Hirkei, Leipzig, 1896

[13] Tresse A., “Sur les invariants différentielles des groupes continus de transformations”, Acta Math., 18 (1894), 1–88 | DOI | MR

[14] Yamaguchi K., “Differential systems associated with simple graded Lie algebras”, Progress in Differential Geometry, Adv. Stud. Pure Math., 22, Math. Soc. Japan, Tokyo, 1993, 413–494 | MR | Zbl