In mathematics, the principal orbit type theorem states that compact

Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addit ...

acting smoothly on a connected differentiable

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...

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 (mathematics)