Let $M$ be a smooth manifold and $G$ a Lie group acting properly on $M$. Let $k$ be the codimension of a maximal dimensional $G$-orbit in $M$.

Is it always possible to construct $k$ functions $f_1, \dots, f_k \colon M \to \mathbb{R}$, such that $f_1, \dots , f_k$ are $G$-invariant and functionally independent on an open and dense subset of $M$?

If it is not possible, under which further assumptions could it be true?


1 Answer 1


This can be done locally. Suppose that the manifold $M$ and the Lie group $G$ are compact. Then one can say that the type of the orbits of $x$ and $y$ coincide if and only if the stabilizer of $x$ and $y$ are conjugated. There is particular type called the principal type: The type of the orbit of $x$ is principal if and only if the canonical representation of the stabilizer $G_x$ on $T_xM/V_x$) where $T_xM$ is tangent space of $M$ at $x$ and $V_x$ is the subspace of $T_xM$ tangent to the orbit) is trivial. A principal orbit has a maximal dimension and the set of principal orbit is a dense open subset.

The slice theorem of Koszul allows then to identify equivariantly a neighborhood of the orbit of $x$ to $G/G_x\times W_x$ where is an open subset of $T_xM/V_x$. So you can find functions $f_1,...,f_k$ that are independent functionally and invariant by $G$ given by the coordinates of $W_x$.

  • $\begingroup$ Thank you very much. So that way the functions $f_1, \dots f_k$ are defined locally on a slice. Is it now possible to use a partition of unity, such that these locally defined functions can be patched together to extend them to "global" functionally independent functions? (so open and dense?) $\endgroup$
    – Olorin
    Mar 16, 2016 at 17:27
  • $\begingroup$ You can extend them with partition of unity, but I am not sure they will be still invariant by $G$. $\endgroup$ Mar 16, 2016 at 17:40

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.