For any $p\in P$, there is some sufficiently small neighborhood $U\subset P$ of $p$ and a tubular neighborhood $\mathcal{U}\subset Q$ with $U=\mathcal{U}\cap P$ such that the function $r$ defined by (3.12) satisfies the following properties:

1. (1)

   $\nabla r=\partial_{r}$ holds in $\mathcal{U}\setminus U$. In particular, $r$ is the normal distance function in $\mathcal{U}$, i.e., $r(q)=d(q,P)$ for all $q\in\mathcal{U}$.

2. (2)

   $\partial_{r}$ is orthogonal to $\partial_{x_{i}}$’s in $\mathcal{U}\subset Q$, and hence

   (3.13)

   $$\langle\partial_{x_{i}},r\partial_{r}\rangle=\langle\partial_{x_{i}},\sum_{\alpha=1}^{k_{0}}y_{\alpha}\partial_{y_{\alpha}}\rangle=\sum_{\alpha=1}^{k_{0}}y_{\alpha}g_{i\alpha}=0,\ 1\leq i\leq m-k_{0}.$$