 Euclidean rigidity of maximal asymptotic volume ratio

The Bishop-Gromov inequality forces every volume ratio to equal one if the asymptotic ratio is one. Equality in the radial Jacobian comparison forces the radial operator $A(X)={\nabla }_X{\partial }_r$ of each distance sphere to be $r^{-1}I$ (it is the negative of the outward shape operator in the convention $S=-\nabla \nu$), by the traced radial Riccati equation for distance spheres and equality in Cauchy-Schwarz inequality. Any finite cut time would remove a positive cone of radial volume at a later radius, contradicting equality. Thus the Riemannian exponential map is global and one-to-one, and the polar metric is $d{r}^2+r^2{g}_{S^{n-1}}$, the Euclidean metric.