Let $(M,g)$ be a complete Riemannian manifold, let $k>0$, and let
paragraph
admin
\begin{align*}
\gamma:[0,b) \to M
\end{align*}
latex_env
admin
be a unit-speed geodesic. Assume that for every $s \in [0,b)$ and every $2$-plane $\sigma \subset T_{\gamma(s)}M$ with $\dot{\gamma}(s) \in \sigma$, the sectional curvature satisfies
paragraph
admin
\begin{align*}
K_g(\sigma) \geq k.
\end{align*}
latex_env
admin
Let
paragraph
admin
\begin{align*}
J:[0,b) \to TM
\end{align*}
latex_env
admin
be a Jacobi field along $\gamma$ such that $J(s) \perp \dot{\gamma}(s)$ for all $s \in [0,b)$ and $J(0)=0$. Define
paragraph
admin
\begin{align*}
\operatorname{sn}_k:(0,\pi/\sqrt{k}) &\to \mathbb{R} \\
t &\mapsto \frac{\sin(\sqrt{k}\,t)}{\sqrt{k}}.
\end{align*}
latex_env
admin
If $t \in (0,\min\{b,\pi/\sqrt{k}\})$ and $\gamma(0)$ has no conjugate point to $\gamma(s)$ along $\gamma$ for any $s \in (0,t)$, then