In mathematics, the principal orbit type theorem states that compact

Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...

acting smoothly on a connected

differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ...

has a principal orbit type.

Definitions

Suppose ''G'' is a compact Lie group acting smoothly on a connected differentiable manifold ''M''. *An isotropy group is the subgroup of ''G'' fixing some point of ''M''. *An isotropy type is a conjugacy class of isotropy groups. *The principal orbit type theorem states that there is a unique isotropy type such that the set of points of ''M'' with isotropy groups in this isotropy type is open and dense. *The principal orbit type is the space ''G''/''H'', where ''H'' is a subgroup in the isotropy type above.

References

*{{citation, mr=0889050 , last=tom Dieck, first= Tammo , title=Transformation groups , series=de Gruyter Studies in Mathematics, volume= 8, publisher= Walter de Gruyter & Co., place= Berlin, year= 1987, isbn= 3-11-009745-1 , pages=42–43 Lie groups Group actions Theorems in differential geometry