let $X$ and $Y$ be two $(n-1)$ dimensional manifolds in $\mathbb{R}^n$ of class $C^1$ and $C^\infty$ respectively, and denotes by $\mu^{(1)}$ and $\mu^{(2)}$ the two outer unit normal vectors to $X$ and $Y$ respectively.
Question: is true that $$\mathcal{H}^{n-1} \left\{ z \in X \cap Y: \mu^{(1)}(z)=\mu^{(2)}(z)\right\} =0 $$
Thanks.