Lipschitz-Free Spaces over Manifolds and the Metric Approximation Property

Let $\mathrm{\parallel }\cdot \mathrm{\parallel }$ be a norm on ${\mathbb{R}}^N$ and let $M$ be a closed $C^1$-submanifold of ${\mathbb{R}}^N$. Consider the pointed metric space $(M,d)$, where $d$ is the metric given by $d(x,y)=\mathrm{\parallel }x-y\mathrm{\parallel }$, $x,y\in M$. Then the Lipschitz-free space $\mathcal{F}(M)$ has the Metric Approximation Property.