Let $(M,g)$ be a complete Riemannian manifold, let $p \in M$, and let $R > 0$. Assume that every sectional curvature $K_M(\sigma)$ of every two-dimensional tangent plane $\sigma \subset T_qM$ with $q \in B(p,R)$ satisfies
For $k \in \mathbb{R}$, define the model sine function $\operatorname{sn}_k: I_k \to \mathbb{R}$ by
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\begin{align*}
\operatorname{sn}_k(t)
=
\begin{cases}
\frac{1}{\sqrt{k}}\sin(\sqrt{k}t), & k > 0,\\
t, & k = 0,\\
\frac{1}{\sqrt{-k}}\sinh(\sqrt{-k}t), & k < 0,
\end{cases}
\end{align*}
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where $I_k = (0,\pi/\sqrt{k})$ if $k>0$ and $I_k=(0,\infty)$ if $k \le 0$.
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Let $v \in T_pM$ satisfy $0 < |v|_g = r < R$, assume $r \in I_{k_2}$, and assume that $\exp_p$ is nonsingular at $v$. If $w \in T_pM$ satisfies $(v,w)_g = 0$, then
Equivalently, if $e \in T_pM$ is a unit vector, $u \in T_pM$ satisfies $(e,u)_g=0$, $0<r<R$, $r \in I_{k_2}$, and $\exp_p$ is nonsingular at $re$, then