= Euclidean rigidity of maximal asymptotic volume ratio
{c}
{title2=$\lim_{r\to\infty}\operatorname{Vol}B(p,r)/(\omega_nr^n)=1\ \Longrightarrow\ M\cong\mathbb R^n$}
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 $dr^2+r^2g_{S^{n-1}}$, the Euclidean metric.
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