Let $(M^{n},g,p)$ be a Riemannian manifold and $p\in M^{n}$. Given $r>0$, $\epsilon>0$, $k\in\mathbb{N}$, $\alpha\in(0,1)$, we say $(M^{n},g,p)$ is $(r,k+\alpha,\epsilon)$-regular at $p$ if the metric $g$ is at least $C^{k+\alpha}$ in $B_{2r}(p)$ and satisfies the following property: let $(\widetilde{B_{2r}(p)},\tilde{p})$ be the Riemannian universal cover of $B_{2r}(p)$, then $B_{r}(\tilde{p})$ is diffeomorphic to a disc $\mathbb{D}^{n}\subset\mathbb{R}^{n}$ such that $g$ in coordinates satisfies