Let $(M,g)$ be a complete Riemannian manifold, and let $k \in \mathbb{R}$. Assume that every sectional curvature of $M$ is bounded above by $k$, meaning that for every point $p \in M$ and every two-dimensional subspace $\sigma \subset T_pM$,
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\begin{align*}
K_M(\sigma) \leq k.
\end{align*}
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Define $\operatorname{conj}(M)$ to be the infimum of all positive numbers $T > 0$ for which there exists a unit-speed geodesic $\gamma: [0,T] \to M$ such that $\gamma(T)$ is conjugate to $\gamma(0)$ along $\gamma$, with the convention $\operatorname{conj}(M)=\infty$ if no such $T$ exists.