Suppose $(M, g)$ is a Riemannian manifold, $\gamma:[0, b] \rightarrow M$ is a unit-speed geodesic segment, and $J$ is any normal Jacobi field along $\gamma$ such that $J(0)=0$.
- If all sectional curvatures of $M$ are bounded above by a constant $c$, then
$$|J(t)| \geq s_c(t)\left|D_t J(0)\right|$$
for all $t \in\left[0, b_1\right]$, where $b_1=b$ if $c \leq 0$, and $b_1=\min (b, \pi R)$ if $c=1 / R^2>0$.
- If all sectional curvatures of $M$ are bounded below by a constant $c$, then
$$|J(t)| \leq s_c(t)\left|D_t J(0)\right|$$
Recall:
Jacobi equations $$ D_{t}^{2}J+R\big(J,\gamma^{\prime}\big)\gamma^{\prime}=0. $$
Comparison theorems are all on upper or lower bounds on sectional curvatures.
Why we need "$\gamma$ is a unit-speed geodesic segment"?
BTW, this theorem is very useful, I would like to ask if there are more specific results in ordinary differential equations textbook, for example, $R\big(J,\gamma^{\prime}\big)$ is monotonic.