Let $(M,g)$ be a compact connected $n$-dimensional Riemannian manifold; let $(X,d)$ denote its associated metric (length) space. A comment on the original formulation of this post mentioned that $(X,d)$ is Ahlfors $n$-regular, meaning that: there exist constants $L,U>0$ for which $$ L r^n\leq \mu(B(x,r))\leq U r^n$$ where $\mu$ is the $n$-dimensional Hausdorff measure and $B(x,r)$ denotes the ball in $(X,d)$ about $x\in X$ of radius $r>0$ (i.e. each $\mu(B(x,r))\in \Theta( r^n)$).

1. Where can I find a proof or reference for this fact?
2. Can the constants $L,U$ be stated using the curvature and dimension of $(M,g)$?