Equivariant deformation of Morse functions.

Question

Let $f$ be a Morse function on closed smooth manifold $M$, (one can also suppose that this is a hyperbolic manifold) G is a group which acts effectively and smooth on $M$, f and G (hyperbolic metric) are equivariant in $M$. There many results about deformations of Morse functions. But is there any way to always pertrubate $f$ in a equivariant way? More precisely, is it true that for every $\epsilon$ there exists $g$ - smooth equivariant Morse function which is $|f-g|_{M} < \epsilon$, and if it is true, can one take such $g$ to be a Morse-Smale function? Any links would be much appreciated.

Answer 1

If your function is equivariant, it must take the same value at all points in an orbit. Now look at $S^2$, with $SO(3)$ acting on it: there's only one orbit, so all equivariant functions are constant. So any equivariant function is not a morse function.