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 $$ L r^n\leq \mu(B(x,r))\leq U r^n \qquad(\boldsymbol{*}) $$ 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/reference to this fact?
2. Can the constants $L,U$ for $(\boldsymbol{*})$ be stated using the curvature and dimension of $(M,g)$?

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 $(M,g)$?

Let $(M,g)$ be a compact connected $d$-dimensional Riemannian manifold equipped; let $(X_{M,g},d_g)$ denote its associated metric (length) space. In the comments (to the original formulation of this post), it was mentioned that $(X_{M,g},d_g)$ is Ahlfors $d$-regular; meaning that: there exist constants $c_L,c_U>0$ for which $$ c_L r^d\leq \mathcal{H}^d(B_{d_g}(x,r))\leq c_U r^d \qquad(\boldsymbol{1}) $$ where $\mathcal{H}^d$ is the $d$-dimensional Hausdorff measure thereon and $B(x,r)$ denotes the ball in $(X_{M,g},d_g)$ about $x\in X$ of radius $r>0$. (I.e. each $\mathcal{H}^d(B_{d_g}(x,r))\in \Theta( r^d)$).*

1. Where can I find a proof/reference to this fact?
2. Can the constants $c_L,c_U>0$ for $(\boldsymbol{1})$ be estated using the curvature and dimension of $(M,g)$?

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 $$ L r^n\leq \mu(B(x,r))\leq U r^n \qquad(\boldsymbol{*}) $$ 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/reference to this fact?
2. Can the constants $L,U$ for $(\boldsymbol{*})$ be stated using the curvature and dimension of $(M,g)$?