Let $(M,g)$ be a connected $d$-dimensional Riemannian manifold equipped; let $(X_{M,g},d_g)$ denote its associated metric (length) space.  Under what conditions is $(X_{M,g},d_g)$ [Ahlfors $d$-regular][1]; i.e. each $\mathcal{H}^d(B_{d_g}(x,r))\in \Theta( r^d)$?

Here $\mathcal{H}^d$ is the $d$-dimensiona Hausdorff outer measure and $B_{d_g}(x,r)$ is a metric ball about some $x \in M$ of radius $r>0$.

I guess, I'm looking for some type of Ricci-curvature constraint...but maybe my intuition is off...


  [1]: https://mathoverflow.net/questions/319955/origin-of-term-ahlfors-david-regular