nLab Cartan's closed subgroup theorem

Statement

Proposition

(Cartan's closed subgroup theorem)

If $H \subset G$ is a closed subgroup of a (finite dimensional) Lie group, then $H$ is a sub-Lie group, hence a smooth submanifold such that its group operations are smooth functions with respect to the submanifold smooth structure.

(due to Cartan 52, see Lee 12, Thm. 20.12)

Consequences

• continuous homomorphisms of Lie groups are smooth

• coset space coprojection admitting local sections

References

The statement is originally due to:

• Élie Cartan, La théorie des groupes finis et continus et l’analysis situs, Mémorial des sciences mathématiques, no. 42 (1952) , 68 p (numdam:MSM_1952__42__1_0)

Textbook accounts:

• Jeffrey M. Lee, Theorem 5.81 in: Manifolds and Differential Geometry, Graduate Studies in Mathematics 107, AMS 2009

  (doi:10.1090/gsm/107)

• John Lee, Theorem 20.12 in: Introduction to Smooth Manifolds, Second Edition, Graduate Texts in Mathematics 218 (2012), Springer (doi:10.1007/978-1-4419-9982-5, book webpage, pdf)

Lecture notes:

• Zuoqin Wang, Cartan’s closed subgroup theorem, Lecture 11 in: 李群引论, 2013 (pdf, pdf)

See also

• Wikipedia, Closed-subgroup theorem