I read a paper who states:
Consider an embedded orientable surface $\Sigma \subset \mathbf{R}^3$ and $\Sigma \in C^2$ with exterior unit normal $n$. Let $\rho$ be the signed distance function such that $\nabla \rho = n$ on $\Sigma$ and let $p(x) = x - \rho(x) n(p(x))$ be the closest point mapping. ...The tangential gradient $\nabla_\Sigma$ is defined by $\nabla_\Sigma=P_\Sigma \nabla$, where $\nabla$ is the $\mathbf{R}^3$ gradient and $P_\Sigma(x) = I - n(x) \otimes n(x)$ is the projection onto the tangent plane $T_\Sigma(x)$ of $\Sigma$ at $x$.
Later it gives an identity without proof:
...Then we use the identity $Dp = P_\Sigma - \rho \mathcal{H}$, where $\mathcal{H}$ is the Hessian of the distance function $\mathcal{H} = \nabla \otimes \nabla \rho$.
Why does this identity hold true? I cannot derive it from the previous settings.
And later without proof it says:
...Then we use the fact that $P_\Sigma - \rho \mathcal{H} = (P_\Sigma-\rho \mathcal{H})P_\Sigma$.
But I cannot understand the fact as well.
As for the paper, you may find it via the following two links: