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)



A locally closed set with a smooth group structure is a Lie group

Author: Armando Machado
Journal: Proc. Amer. Math. Soc. 84 (1982), 303-307
MSC: Primary 22E15; Secondary 58A05
MathSciNet review: 637188
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove the following result. Let $ V$ be a smooth manifold and let $ G \subset V$ be a locally closed set with a group structure such that both multiplication and inversion are smooth maps; then $ G$ is an imbedded smooth submanifold of $ V$. This result is a generalization of the well-known fact that a closed subgroup of a Lie group is itself a Lie group, because we are not assuming any group structure in the manifold $ V$.

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

  • [1] N. Aronszajn, Subcartesian and subriemannian spaces, Notices Amer. Math. Soc. 14 (1967), 111.
  • [2] G. Bouligand, Introduction à la géométrie infinitésimale directe, Vuibert, Paris, 1932.
  • [3] Haïm Brezis, On a characterization of flow-invariant sets, Comm. Pure Appl. Math. 23 (1970), 261–263. MR 0257511
  • [4] Armando Machado, Sur les ensembles invariants par la coulée d’un champ de vecteurs, Proceedings of the sixth conference of Portuguese and Spanish mathematicians, Part I (Santander, 1979), 1979, pp. 155–159 (French, with English summary). MR 754578
  • [5] Charles D. Marshall, Calculus on subcartesian spaces, J. Differential Geometry 10 (1975), no. 4, 551–573. MR 0394742
  • [6] P. Stefan, Accessible sets, orbits, and foliations with singularities, Proc. London Math. Soc. (3) 29 (1974), 699–713. MR 0362395

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 22E15, 58A05

Retrieve articles in all journals with MSC: 22E15, 58A05

Additional Information

Keywords: Closed subgroup of a Lie group, strictly tangent vector, flow-invariant set
Article copyright: © Copyright 1982 American Mathematical Society