Let $(X,d)$ be a connected geodesic metric space. When does there there exists a covering map $\pi:H\rightarrow X$ which is a *local-isometry* where $H$ is either a Hilbert space or a Euclidean space?


* **Strengthened:** If $(X,d)$ is simply connected and is the length space induced by a Riemannian geometry and if we only consider finite-dimensional $H$ then by [the classical Cartan-Hadamard Theorem](https://en.wikipedia.org/wiki/Cartan%E2%80%93Hadamard_theorem) we know that $(X,d)$ must be non-positive curvature.
* **Relaxed:** If instead we weaken the problem, so that H is only required to be Hadamard then $(X,d)$ must also be Hadamard (by the metric version of the Cartan-Hadamard Theorem).


However, what is know of $(X,d)$ if we specifically want $H$ to be Hilbert or Euclidean?



Spaces satisfying a strong Cartan-Hadamard theorem
Note that Hilbert spaces (of all dimensions finite or infinite) are the only geodesic spaces with extendable geodesics which are flat in the sense of Alexandrov.
Therefore $X$ has to have extendable geodesics + it has to be locally flat (in the sense of Alexandrov).
By Cartan--Hadamard theorem, these two conditions are also sufficient.


