SIGMA 5 (2009), 094, 21 pages      arXiv:0906.0560      https://doi.org/10.3842/SIGMA.2009.094
Contribution to the Special Issue “Élie Cartan and Differential Geometry”

On Tanaka's Prolongation Procedure for Filtered Structures of Constant Type

Igor Zelenko
Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA

Received June 02, 2009, in final form September 29, 2009; Published online October 06, 2009

Abstract
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.

Key words: G-structures; filtered structures; generalized Spencer operator; prolongations.

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References

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, Paris, Gautier-Villars, 1953, 927-1010).
2. Cartan É., Sur les variétés à connexion projective, Bull. Soc. Math. France 54 (1924), 205-241.
3. Kuzmich O., Graded nilpotent Lie algebras in low dimensions, Lobachevskii J. Math. 3 (1999), 147-184.
4. Morimoto T., Geometric structures on filtered manifolds, Hokkaido Math. J. 22 (1993), 263-347.
5. Morimoto T., Lie algebras, geometric structures, and differential equations on filtered manifolds, in Lie Groups, Geometric Structures and Differential Equations - One Hundred Years after Sophus Lie (Kyoto/Nara, 1999), Adv. Stud. Pure Math., Vol. 37, Math. Soc. Japan, Tokyo, 2002, 205-252.
6. Morimoto T., Cartan connection associated with a subriemannian structure, Differential Geom. Appl. 26 (2008), 75-78.
7. Singer I.M., Sternberg S., The infinite groups of Lie and Cartan. I. The transitive groups, J. Analyse Math. 15 (1965), 1-114.
8. Spencer D.C., Overdetermined systems of linear partial differential equations, Bull. Amer. Math. Soc. 75 (1969), 179-239.
9. Sternberg S., Lectures on differential geometry, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964.
10. Tanaka N., On differential systems, graded Lie algebras and pseudogroups, J. Math. Kyoto. Univ. 10 (1970), 1-82.
11. Tanaka N., On the equivalence problems associated with simple graded Lie algebras, Hokkaido Math. J. 8 (1979), 23-84.
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.
14. Yamaguchi K., Differential systems associated with simple graded Lie algebras, in Progress in Differential Geometry, Adv. Stud. Pure Math., Vol. 22, Math. Soc. Japan, Tokyo, 1993, 413-494.