Is a certain inverse image under an equivariant map a submanifold?

Question

Suppose $G$ is a Lie group and $K$ a maximal compact subgroup of $G$, and suppose $G$ acts smoothly and properly on a manifold $X$.

Question 1: Suppose we are given a $G$-equivariant smooth map $f:X\rightarrow G/K$. Is $S:=f^{-1}(eK)$ necessarily a submanifold of $X$?

Question 2: If $X$ is now a manifold with boundary, then is $S:=f^{-1}(eK)$ a submanifold with boundary?

Thanks.

Answer 1

Suppose $f:X\to G/K$ is smooth and $G$-equivariant, with $G$ a Lie group and $K$ a closed subgroup. Then, in particular, $f$ must be a submersion: if $p\in X$ and $g\in G$, then $f(gp)=gf(p)$, so $f(\exp(t\xi)p)=\exp(t\xi)f(p)$.

Thus, $f^{-1}(gK)$ is a submanifold for every $gK\in G/K$.