On the group of real analytic diffeomorphisms of a compact real analytic manifold

Leslie, J.

doi:10.1090/S0002-9947-1982-0675073-5

On the group of real analytic diffeomorphisms of a compact real analytic manifold

Author: J. Leslie
Journal: Trans. Amer. Math. Soc. 274 (1982), 651-669
MSC: Primary 58B12; Secondary 58D05, 58F99
DOI: https://doi.org/10.1090/S0002-9947-1982-0675073-5
MathSciNet review: 675073
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we exhibit a Lie group structure on the group of real analytic diffeomorphisms of a compact real analytic manifold. Further, we show that a variant of the Kupka-Smale theorem holds for the underlying topology of the above mentioned Lie group.

[1] Ralph Abraham and Joel Robbin, Transversal mappings and flows, An appendix by Al Kelley, W. A. Benjamin, Inc., New York-Amsterdam, 1967. MR 0240836
[2] N. Bourbaki, Topologie générale, Chap. II, Hermann, Paris, 1967.
[3] J. F. Colombeau, Différentiation et bornologie, U. E. R. de Mathématiques et d’Informatique, Université de Bordeaux I, Talence, 1973 (French). Thèse de Doctorat d’État de l’auteur soutenue à l’Université de Bordeaux en Mars 1973. MR 0450969
[4] J. F. Colombeau, Sur quelques particularités du calcul différentiel dans les espaces vectoriels topologiques ou bornologiques, Rev. Roumaine Math. Pures Appl. 18 (1973), 3–17 (French). MR 0361769
[5] Jean Dieudonné, Denumerability conditions in locally convex vector spaces, Proc. Amer. Math. Soc. 8 (1957), 367–372. MR 0083687, https://doi.org/10.1090/S0002-9939-1957-0083687-X
[6] Jack Hale, Theory of functional differential equations, 2nd ed., Springer-Verlag, New York-Heidelberg, 1977. Applied Mathematical Sciences, Vol. 3. MR 0508721
[7] Philip Hartman, Ordinary differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 0171038
[8] Morris W. Hirsch and Charles C. Pugh, Stable manifolds and hyperbolic sets, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 133–163. MR 0271991
[9] H. Hogbe-Nlend, Théorie des bornologies et applications, Lecture Notes in Mathematics, Vol. 213, Springer-Verlag, Berlin-New York, 1971 (French). MR 0625157
[10] Léon Van Hove, Topologie des espaces fonctionnels analytiques et des groupes infinis de transformations, Acad. Roy. Belgique. Bull. Cl. Sci. (5) 38 (1952), 333–351 (French). MR 0050161
[11] Howard Jacobowitz, Implicit function theorems and isometric embeddings, Ann. of Math. (2) 95 (1972), 191–225. MR 0307127, https://doi.org/10.2307/1970796
[12] J. A. Leslie, On a differential structure for the group of diffeomorphisms, Topology 6 (1967), 263–271. MR 0210147, https://doi.org/10.1016/0040-9383(67)90038-9
[13] J. A. Leslie, Some Frobenius theorems in global analysis, J. Differential Geometry 2 (1968), 279–297. MR 0251750
[14] Daniel C. Lewis Jr., Invariant Manifolds Near an Invariant Point of Unstable Type, Amer. J. Math. 60 (1938), no. 3, 577–587. MR 1507339, https://doi.org/10.2307/2371599
[15] José Sebastião e Silva, Su certe classi di spazi localmente convessi importanti per le applicazioni, Rend. Mat. e Appl. (5) 14 (1955), 388–410 (Italian). MR 0070046
[16] S. Smale, Stable manifolds for differential equations and diffeomorphisms, Ann. Scuola Norm. Sup. Pisa (3) 17 (1963), 97–116. MR 0165537
[17] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 0228014, https://doi.org/10.1090/S0002-9904-1967-11798-1
[18] E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems. I, Comm. Pure Appl. Math. 28 (1975), 91–140. MR 0380867, https://doi.org/10.1002/cpa.3160280104
[19] Eduard Zehnder, Moser’s implicit function theorem in the framework of analytic smoothing, Math. Ann. 219 (1976), no. 2, 105–121. MR 0388440, https://doi.org/10.1007/BF01351894