= Bishop-Gromov inequality
{c}
{title2=$r\longmapsto\operatorname{Vol}B(p,r)/(\omega_nr^n)\text{ is nonincreasing}$}
{wiki=Bishop–Gromov_inequality}
For a complete connected $n$-dimensional <Riemannian manifold> with nonnegative <Ricci curvature>, the displayed ratio is nonincreasing, tends to one at zero, and is at most one. In particular $\operatorname{Vol}B(p,r)\leq\omega_nr^n$. In <geodesic polar coordinates>, the radial <Jacobi field> determinant is at most the Euclidean density $r^{n-1}$ until the cut time. The trace of the <radial Riccati equation for distance spheres>, with <Cauchy-Schwarz inequality>, proves this comparison. Equality for every radius forces the Euclidean polar metric and absence of a <cut locus>.
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