For $x\in X$, we define the harmonic radius $r_{h}(x)$ so that $r_{h}(x)=0$ if no neighborhood of $x$ is a Riemannian manifold. Otherwise we define $r_{h}(x)$ to be the largest $r>0$ such that there exists a mapping $\Phi:B_{r}(0^{n})\to X$ such that:

1. (1)

   $\Phi(0)=x$ with $\Phi$ is a diffeomorphism onto its image.

2. (2)

   $\Delta_{g}x^{\ell}=0$, where $x^{\ell}$ are the coordinate functions and $\Delta_{g}$ is the Laplace Beltrami operator.

3. (3)

   If $g_{ij}=\Phi^{*}g$ is the pullback metric, then

   $\displaystyle||g_{ij}-\delta_{ij}||_{C^{0}(B_{r}(0^{n}))}+r||\partial_{k}g_{ij}||_{C^{0}(B_{r}(0^{n}))}\leq 10^{-3}\,.$

   (2.8)