On the Local Theory of Continuous Infinite Pseudo Groups I*

Masatake Kuranishi

doi:10.1017/S0027763000006747

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The local theory of continuous (infinite) pseudo-groups of transformations was originated by S. Lie, and developed by himself, F. Engel, E. Vessiot, E. Cartan, etc. In the beginning, the definition was not clear and we can find several different definitions in the papers of pioneers. In 1902, E. Cartan introduced a definition using his theory of exterior differential systems and made an extensive study in his series of papers [1], [2], and [31 The writer will adopt his definition in this series of papers. A continuous pseudo-group of transformations is, roughly speaking, a collection of real (or complex) analytic homeo-morphisms of domains in a real (or complex) euclidean space, which is closed under the operations of composition and inverse, and which forms the general solutions of a system of partial differential equations.

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A part of this series of papers was done at the Institute for Advanced Study, Princeton, in 1955-56, and at the University of Chicago in 1956-57. At the University of Chicago, the writer was partially supported by the United States Air Force under Contract No. AF 18 (600)-1383 monitered by the Office of Scientific Research.

References

[1] Cartan, E., Sur la structure des groupes infinis de transformations, Ann. Ec. Norm. Sup. 21 (1904), pp. 153–206 and 22 (1905), pp. 219–308.Google Scholar

[2] Cartan, E., Les sous-groupes des groupes continus de transformations, ibid. 25 (1908), pp. 57–194.Google Scholar

[3] Cartan, E., Les groupes de transformations continus, infinis, simples, ibid. 26 (1909), pp. 93–161.Google Scholar

[4] Cartan, E., Seminaire Ec. Norm. Sup. 1950–51 (multilithed).Google Scholar

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On the Local Theory of Continuous Infinite Pseudo Groups I*

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