Background:
As shown in [1] and [2], for a closed smooth submanifold $M$ of $\mathbb R^d$, the domain $D_M$ of the projection map $P_M:D_M\rightarrow M$ has a dense interior $\Omega_M$ over which $P_M|_{\Omega_M}$ is smooth.
Now, let $G<O(d)$ be a compact group whose standard action on $\mathbb R^d$ is free. For $z\in\mathbb R^d$, let $M_z = G\cdot z$. Then, $M_z$ is a closed smooth manifold and we have $x\in D_{M_z} \iff z\in D_{M_x}$ due to freeness. I am wondering whether manifolds arising as orbits have a more refined projection geometry:
Question 1: For free group actions and perhaps any orthogonal representation, is it the case that $D_{M_x} = \Omega_{M_x}$?
Question 2: For free group actions, is it the case that $x\in \Omega_{M_z}\iff z\in \Omega_{M_x}$? If not, does the statement hold when $x$ is arbitrary fixed and $z$ is generic?
[1] Dudek, Ewa, and Konstanty Holly. "Nonlinear orthogonal projection." Annales Polonici Mathematici. Vol. 59. No. 1. Polska Akademia Nauk. Instytut Matematyczny PAN, 1994.
[2] Leobacher, Gunther, and Alexander Steinicke. "Existence, uniqueness and regularity of the projection onto differentiable manifolds." Annals of global analysis and geometry 60.3 (2021): 559-587.
