Let $M$ be a complete Riemannian manifold with bounded sectional curvature and $G$ a compact connected Lie group acts smoothly on $M$. Consider the fixed point set $F$, it is of course a submanifold of $M$ by the slice theorem. Let $\{F_i\}$ be the connected components of $F$. Then for each $i$, is there a sequence of Riemannian manifolds $\{M_j\},j\in\mathbb{N}$ with $M_0=M$ such that $\{M_j\}$ collapses to $F_i$ while keeping their sectional curvatures uniformly bounded?

If in general such a sequence does not exist, how about the case $G=T$? Here $T$ is a finite-dimensional torus.

`collapsing'? Is it true that each Euclidean space`

collapses' to a point? [If not, take T = S^1 acting in standard linear fashion on the plane to get a counterexample. $\endgroup$ – Richard Montgomery Sep 19 '12 at 17:58