Euclidean distance matrices $\mathbb{EDM}$ enjoy a lot of nice properties such as being symmetric positive semidefinite and fixed-rank. For a point set $Y\in\mathbb{R}^{d\times n}$, a Euclidean distance matrix $D\in\mathbb{EDM}$ admits a low rank decomposition:$$ \begin{equation} D = 1 \text{diag}(G)^{\top} + \text{diag}(G)1^{\top} - 2G \end{equation} $$where $G=YY^{\top}$ is the Gram matrix and $\mathrm{rank}(D)<=d+2$. If points are in general position, we can also write: $\text{rank}(D)=d+2$. These properties are quite useful in many applications.
I am wondering if the more general geodesic distance matrices would also share some of these nice properties or other ones. In particular the question relates to the case where $d=3$ (for example, I would compute the approximate geodesics on meshes).
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