nLab Bochner linearization theorem

This entry is about Bochner’s theorem on transformation groups. For the statement in harmonic analysis see at Bochner's theorem.

Idea

The Bochner linearization theorem [Bochner 1945] states that for a differentiable group action of a compact Lie group on a differentiable manifold there exists around each fixed point a (similarly differentiable) local coordinate chart on which the action is linear.

(The analogous statement for analytic manifolds and analytic group actions is attributed by Bochner 1945 to Henri Cartan, referencing Martin 1944.)

This linearization theorem is the special case (for fixed loci of dimension$=0$) of the existence of equivariant tubular neighbourhoods, see there for more.

Related concepts

• transformation group

• equivariant tubular neighbourhood

• orbifold

References

The original article:

• Salomon Bochner: Compact Groups of Differentiable Transformation, Annals of Mathematics 46 3 (1945) 372-381 [doi:10.2307/1969157, jstor:1969157]

following

• W. T. Martin: Mappings by means of systems of analytic functions of several complex variables, Bulletin of the American Math. Society 50 1 (1944) 5-19.

See also:

• Nicolas Bourbaki: Prop. 5 §9.3 Chapter IX, of: Lie Groups and Algebras, Masson (1982), Springer (2007) [scan]