= Spherical rigidity of maximal total volume
{title2=$\operatorname{Vol}M=\operatorname{Vol}S_k^n\ \Longrightarrow\ M\cong S_k^n$}
Under the hypotheses of <Bishop volume comparison with a positive-curvature model>, the <Bonnet-Myers theorem> gives diameter at most $D$. If total <Riemannian volume> equals $v_k(D)$, every comparison ball ratio is one. In a small normal ball the pointwise radial density bound therefore becomes equality. Writing $h=\partial_r\log J$, the traced <radial Riccati equation for distance spheres> and <Cauchy-Schwarz inequality> force the radial operator to equal $(s_k'/s_k)I$. The angular metric is consequently $s_k(r)^2g_{S^{n-1}}$, so the metric is locally round. The <classification of complete positive constant-curvature manifolds> gives a quotient $S_k^n/\Gamma$, whose volume is $v_k(D)/|\Gamma|$. Equality forces $\Gamma$ trivial.
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