Let $(M,g)$ be a connected Riemannian manifold of dimension $n>1$. Then the Hopf-Rinow theorem states that $(M,g)$ is geodesically complete if and only if $(M,d\_g)$ is complete as a metric space ($d\_g$ is the induced intrinsic metric).


I need to know if a similar result is true under weaker hypothesis:


1) Suppose $(M,d)$ is a connected metric space, then is it true that if $(X,d)$ is geodetically complete then it is also complete?


2) If the answer to the previous question is no, would it change something if we consider $(M,d\_g)$, where $d\_g$ is the intrinsic induced metric of the Riemannian metric $g=\sum\_{i,j=1}^ng\_{ij}dx\_idx\_j$ where the $g\_{ij}$ are just continuous, not $C^\infty$ (in this case I already know from this other question [Geodesics for non differentiable riemannian metric](https://mathoverflow.net/questions/251012/geodesics-for-non-differentiable-riemannian-metric) that $(M,d\_g)$ is locally geodesic).


Thank you!



Generalizations of Hopf-Rinow theorem
It seems that the answer to 1 is no: [The Hopf-Rinow Theorem is false in infinite Dimensions](http://blms.oxfordjournals.org/content/7/3/261.full.pdf). If you add the assumption that $M$ is a locally compact length space then the answer is "yes" by Theorem 2.5.28 in [this book](http://www.math.psu.edu/petrunin/papers/alexandrov/bbi.pdf). I think this is strong enough to answer "yes" to 2.


