Suppose a compact, connected Lie group $G$ acts isometrically on a complete Riemannian manifold $M$. Equip $G$ with a bi-invariant metric. Then for every $x\in M$ we have a linear map $\phi_x:\mathfrak{g}\rightarrow T_x M$ obtained by differentiating the action of $G$ on $M$.
Consider the map $||\phi||:M\rightarrow\mathbb{R}$ defined by $$x\mapsto ||\phi_x||,$$ where $||\,\cdot||$ denotes the linear operator norm.
Question 1: Suppose $\{x_i\}_{i\in\mathbb{N}}$ is a sequence of points in $M$ such that $||\phi||(x_i)\rightarrow 0$. Then:
a) Does $\{x_i\}$ converge to a point $x_0\in M$ (ie. is it a Cauchy sequence in the norm induced by the Riemannian metric on $M$)?
b) If so, is $x_0$ necessarily a fixed point of the $G$-action?
Thoughts: It seems to me that the answer to b) is yes, ie. $||\phi_{x_0}||=0$ if and only if $x_0$ is a fixed point. But I'm not sure about a), which seems to be equivalent to asking whether the map $||\phi||$ is proper.
Question 2: Suppose the fixed point set $M^G\subseteq M$ is compact. Let $U$ be a neighbourhood of $M^G$. Does there exist $c>0$ such that $||\phi_x||\geq c$ for all $x\in M\backslash U$?