In mathematics, the principal orbit type theorem states that compact Lie group acting smoothly on a connected differentiable manifold has a principal orbit type.

Definitions

Suppose ''G'' is a compact Lie group acting smoothly on a connected differentiable manifold ''M''. *An

isotropy group In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...

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)