# Principal orbit type theorem

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 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

• tom Dieck, Tammo (1987), Transformation groups, de Gruyter Studies in Mathematics, 8, Berlin: Walter de Gruyter & Co., pp. 42–43, ISBN 3-11-009745-1