Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On isomorphic classical diffeomorphism groups. I


Author: Augustin Banyaga
Journal: Proc. Amer. Math. Soc. 98 (1986), 113-118
MSC: Primary 58D05; Secondary 22E65, 57R50
DOI: https://doi.org/10.1090/S0002-9939-1986-0848887-5
MathSciNet review: 848887
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ ({M_i},{\alpha _i})$, $ i = 1,2$, be two smooth manifolds equipped with symplectic, contact or volume forms $ {\alpha _i}$. We show that if a group isomorphism between the automorphism groups of $ {\alpha _i}$ is induced by a bijective map between $ {M_i}$, then this map must be a $ {C^\infty }$ diffeomorphism which exchanges the structures $ {\alpha _i}$. This generalizes a theorem of Takens.


References [Enhancements On Off] (What's this?)

  • [1] A. Banyaga, Sur la structure du groupe des diffeomorphismes qui preservent une forme symplectique, Comment. Math. Helv. 53 (1978), 174-227. MR 490874 (80c:58005)
  • [2] -, On isomorphic classical diffeomorphism groups. II, preprint, 1985.
  • [3] W. M. Boothby, Transitivity of automorphisms of certain geometric structures, Trans. Amer. Math. Soc. 137 (1969), 93-100. MR 0236961 (38:5254)
  • [4] Th. Brocker and K. Janich, Introduction to differential topology, Cambridge Univ. Press, New York, 1982. MR 674117 (83i:58001)
  • [5] R. P. Filipkiewicz, Isomorphisms between diffeomorphism groups, Ergodic Theory Dynamical Systems 2 (1982), 159-171. MR 693972 (84j:58039)
  • [6] F. Klein, Erlanger programm, Math. Ann. 43 (1893), 63.
  • [7] V. V. Lychagin, Local classification of non linear first order partial differential equations, Russian Math. Surveys 30 (1975), 105-175.
  • [8] D. Montgomery and L. Zippin, Topological transformation groups, Wiley, Chichester, 1955. MR 0073104 (17:383b)
  • [9] H. Omori, Infinite dimensional Lie transformation groups, Lecture Notes in Math, vol. 427, Springer-Verlag, Berlin and New York, 1974. MR 0431262 (55:4263)
  • [10] L. E. Pursell and M. E. Shanks, The Lie algebra of smooth manifolds, Proc. Amer. Math. Soc. (1954), 468-472. MR 0064764 (16:331a)
  • [11] F. Takens, Characterization of a differentiable structure by its group of diffeomorphisms, Bol. Soc. Brasil. Mat. 10 (1979), 17-25. MR 82e-58027. MR 552032 (82e:58027)
  • [12] W. Thurston, On the structure of volume preserving diffeomorphisms, unpublished.
  • [13] Mr. Wechsler, Homeomorphism groups of certain topological spaces, Ann. of Math. 2 (1955), 360-373. MR 0072453 (17:287d)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 58D05, 22E65, 57R50

Retrieve articles in all journals with MSC: 58D05, 22E65, 57R50


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1986-0848887-5
Keywords: Automorphism of a geometric structure, Erlanger Program, symplectic forms, volume forms, contact structures, $ \omega $-transitivity
Article copyright: © Copyright 1986 American Mathematical Society

American Mathematical Society