# Principal orbit type theorem

From HandWiki

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

Original source: https://en.wikipedia.org/wiki/Principal orbit type theorem.
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