True or false: "Any edge-weighted undirected graph can be isometrically embedded into some Riemannian manifold".
"isometric embedding" here means that for any pair of nodes, their shortest path distance in the graph equals their geodesic distance in the Riemannian manifold.The question seems equivalent to asking whether any finite discrete metric space can be isometrically embedded into a Riemannian manifold.
This is already false for the graph which consists of a root and three nodes connected to it by edges. The proof follows from uniqueness of continuation of geodesics in Riemannian manifolds.