Let $M$ be a differentiable manifold, and $G$ a Lie group acting smoothly on $M$. Under which condition - if any - is the set of fixed points of the action a submanifold of $M$?
My thoughts so far: if we could determine this set as the preimage of some non-critical value of a diffeomorphism, we would be done. In the easy case $M=\mathbb R^n$, one can consider the function $g\cdot x - x$, with $g\in G$ fixed and $x\in M$ varying, and then pick the preimage of zero. In any case, this gives a larger set than those of all the fixed points: an intersection would be needed, and then who knows what may be happening?
I'm particularly interested in compact Lie groups, and I tried looking for counterexamples, but I can't find any. I have the feeling it shouldn't be too hard to find a set of fixed points self-intersecting in some point (I'm thinking of sth like the axes in $\mathbb R^2$), but I probably shouldn't be trusting feelings.
References, hints, generic intuitions are very welcome!