Flatness, Menger curvature, and parametrization

We show that on linearly locally contractible (LLC) manifolds, the beta numbers (which describe unilateral flatness) are comparable to the theta numbers (which describe bilateral flatness), quantitatively. As an application, we show that if $M\subset {\mathbb{R}}^n$ is a compact LLC $m$-manifold with finite Menger $p$-energy for some $p>m(m+2)$, then $M$ is in fact a $C^{1,\alpha }$ manifold. We also show that the bound $m(m+2)$ is critical by constructing, for each $n\ge 3$, an LLC $n$-sphere in ${\mathbb{R}}^{n+1}$ that has finite Menger $p$-energy for every $p<m(m+2)$ but is not even quasisymmetrically equivalent to the standard $n$-sphere.