Let $(M,g)$ be a Riemannian manifold, let $\gamma:[0,a]\to M$ be a unit-speed geodesic, and let $J$ be a transverse Jacobi field with $J(0)=0$ and $|D_tJ(0)|=1$. Let $I_k\subset(0,a]$ be a comparison interval on which $\operatorname{sn}_k(t)>0$, with $I_k\subset(0,\pi/\sqrt{k})$ when $k>0$ and $I_k\subset(0,\infty)$ when $k\le0$.

If $K_M(\dot\gamma(t),J(t))\le k$ wherever $J(t)\neq0$, then

\begin{align*} |J(t)| \ge \operatorname{sn}_k(t) \end{align*}

for every $t\in I_k$. If $K_M(\dot\gamma(t),J(t))\ge k$ and there are no conjugate points to $\gamma(0)$ on the comparison interval, then

\begin{align*} |J(t)| \le \operatorname{sn}_k(t) \end{align*}

for every $t\in I_k$.