 Bishop volume comparison with a positive-curvature model

For a complete connected $n$-dimensional Riemannian manifold with $\mathrm{Ric}\ge (n-1)kg$, $k>0$, put $D=\pi /\sqrt{k}$ andHere ${\omega }_{n-1}$ is the area of the unit $(n-1)$-sphere, and $v_k(r)$ is the Riemannian volume of a radius-$r$ ball in the round curvature-$k$ sphere. The Bishop-Gromov inequality asserts that $V_p(r)/v_k(r)$ is nonincreasing for $0<r<D$, tends to one at zero, and is at most one. In geodesic polar coordinates, the radial volume density before the cut time is at most $s_k(t{)}^{n-1}$. The bound uses trace Ricci curvature; with normalized Ricci curvature it reads $\overline{\mathrm{Ric}}\ge kg$.