Suppose $\mathcal M_{1},\ \mathcal{M}_{2}$ are $(n-1)$-manifold embedded in $\mathbb R^n$. $d_i$ is the geodesic distance defined on $\mathcal M_i$.
What is the name of the distance and metric defined by: $d(p_1,p_2)=\inf \{d_1(p_1,p)+d_2(p_2,p)|p\in\mathcal M_1\cap\mathcal M_2\}$?
What are some of the interesting properties of this metric? I learnt the metric of the product manifold $\mathcal M_1\times\mathcal M_2$ but they are not the same thing. Are there any connection of them?