Let $d\in\mathbb N$, $k\in\{1,\ldots,d\}$ and $f:\mathbb R^d\to\mathbb R^k$ be differentiable. Assume $0$ is a regular value of $f$.
We know that $M:=\{f=0\}$ is a $(d-k)$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$ and $T_x\:M={\operatorname{span}\left\{\nabla f_1(x),\ldots,\nabla f_k(x)\right\}}^\perp$ for all $x\in M$.
Let $x\in M$, $v\in T_x\:M$, $t>0$ and $y:=x+tv$. Note that $y\in T_x\:M$ and this point can be interpreted as being obtained by moving in direction $v$ with step size $t$.
As we moved along the tangential space of $M$, we moved away from $M$. Now I would like to project $y$ back to $M$ obtaining a point $z\in M$ in this way. Is there a nice form for this projection or a nice characterization for a system of equations induced which we would need to solve to obtain $z$?