Let $\mathcal M \subset \mathbb R^d$ be a smooth manifold, and for each $s \in \mathcal M$ let $T_s[\mathcal M]$ denote the tangent space of $\mathcal M$ at $s$. Also, for each $s \in \mathcal M$ let $P_s$ denote the orthogonal projection of $\mathbb R^d$ into $T_s[\mathcal M]$. What can be said about the continuity properties of $P_s$ in $s$?
In particular, I seek that $s$ has a modulus of continuity.