 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 $\mathrm{Vol}B(p,r)\le {\omega }_n{r}^n$. 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.

The Bishop–Gromov inequality is a fundamental result in Riemannian geometry that provides a comparison between the volume of geodesic balls in a Riemannian manifold and the volume of balls in a model space of constant curvature, specifically spherical or Euclidean spaces. The inequality is particularly useful in the context of manifolds with bounded sectional curvature.