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Principal orbit type theorem

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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 isotopy group is the subgroup of G fixing some point of M.
  • An isotopy type is a conjugacy class of isotopy groups.
  • The principal orbit type theorem states that there is a unique isotopy type such that the set of points of M fixed by a subgroup H of the isotopy type is open and dense.
  • The principal orbit type is the space G/H, where H is a subgroup in the isotpy type above.

    • References

    Principal orbit type theorem Wikipedia
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