= Bishop volume comparison with a positive-curvature model
{c}
{title2=$V_p(r)/v_k(r)\text{ is nonincreasing}$}
For a complete connected $n$-dimensional <Riemannian manifold> with $\operatorname{Ric}\geq(n-1)kg$, $k>0$, put $D=\pi/\sqrt k$ and
$$
s_k(t)=\frac{\sin(\sqrt k\,t)}{\sqrt k},\qquad v_k(r)=\omega_{n-1}\int_0^r s_k(t)^{n-1}\,dt.
$$
Here $\omega_{n-1}$ 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 $\overline{\operatorname{Ric}}\geq kg$.
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