I know from Hilbert projection theorem that for any non-empty, convex and closed set $C$ of a Hilbert space $X$ that there exists a projection map that maps every element in $X$ to a unique element in $C$.
My question is that, is there any extensions of this theorem (in any direction) that concerns compact submanifolds? For example, the unit circle is a compact manifold which is not convex. I am wondering if there is a projection tool on this type of manifolds.