Are there any smooth manifolds $M$ with the following property: There exist a realizing metric $d$ (i.e $d$ induces the topology on $M$), and $d$ is smooth on all of $M \times M$? If not, is it possibe to guarantee smoothness of the function $x \mapsto d(x,y)$ (for a fixed $y$ ), or smoothness of $d^2$ even on a compact manifold? (I am trying to see if we can achieve "improved smoothness" if we do not force the metric to be Riemannian) Of course, such a metric cannot be induced by a Riemannian metric. (see [here][1] and [here][2]). [1]:http://mathoverflow.net/questions/21295/smoothness-of-distance-function-in-riemannian-manifolds [2]: http://math.stackexchange.com/questions/967849/differentiability-of-the-distance-function