Let $(M^n,g)$ be a complete Riemannian manifold with $\operatorname{Ric}_g\ge (n-1)k\,g$, fix $p\in M$, and let $0<r<R<\rho_k$, where

\begin{align*} \rho_k:= \begin{cases} \pi/\sqrt{k},&k>0,\\ \infty,&k\leq 0. \end{cases} \end{align*}

Let $\operatorname{sn}_k:(0,\rho_k)\to(0,\infty)$ be the model sine function solving $u''+ku=0$ with $u(0)=0$ and $u'(0)=1$, and let $\operatorname{ct}_k:(0,\rho_k)\to\mathbb{R}$ be defined by $\operatorname{ct}_k(t)=\operatorname{sn}_k'(t)/\operatorname{sn}_k(t)$. Let $V_k:(0,\rho_k)\to(0,\infty)$ be the model ball-volume function

\begin{align*} V_k(\rho)=\mathcal{H}^{n-1}(S^{n-1})\int_0^\rho \operatorname{sn}_k(t)^{n-1}\,d\mathcal{L}^1(t). \end{align*}

Assume that

\begin{align*} \frac{\operatorname{Vol}_g(B(p,R))}{V_k(R)}=\frac{\operatorname{Vol}_g(B(p,r))}{V_k(r)}. \end{align*}

For $\xi\in S_pM$, let $\operatorname{cut}_p(\xi)\in(0,\infty]$ be the cut time of the unit-speed geodesic $t\mapsto\exp_p(t\xi)$, and define

\begin{align*} \mathcal A_{\mathrm{reg}}=\{(t,\xi):\xi\in S_pM,\ r<t<R,\ t<\operatorname{cut}_p(\xi)\}. \end{align*}

Let $J:\mathcal A_{\mathrm{reg}}\to(0,\infty)$ denote the smooth radial Jacobian density for the Riemannian measure in geodesic polar coordinates, so that the measure is $J(t,\xi)\,d\mathcal{L}^1(t)\,d\sigma_p(\xi)$ on $\mathcal A_{\mathrm{reg}}$, where $d\sigma_p$ is the Riemannian measure on $S_pM$. Then for $d\sigma_p$-almost every $\xi$ and every $t$ in each connected component of $\{s\in(r,R):(s,\xi)\in\mathcal A_{\mathrm{reg}}\}$, the quotient $J(t,\xi)/\operatorname{sn}_k(t)^{n-1}$ is constant. On this regular set the shape operator of the geodesic spheres has trace $(n-1)\operatorname{ct}_k(t)$, the radial Ricci curvature is $(n-1)k$, and the equality case in the radial Riccati inequality holds along those radial geodesics. No assertion is made across the cut locus or outside the annulus.