I am currently reading Chavels "Riemannian Geometry: A Modern Introduction". On page 86 he gives the following definition of Rauch's comparison theorem:
Let $M$ be a Riemannian manifold, $\delta\in\mathbb{R}$ and $\gamma:[0,\beta]\rightarrow M$ a normed geodesic with $\gamma (0)=p$ such that $\mathcal{K}\leq\delta$ for all sections along $im(\gamma)$.
- If $Y\in \mathcal{J}^{\perp}$, then function $|Y|$ on $im(\gamma)$ satisfies $|Y|''+\delta|Y|\geq 0\quad (\star)$
- If $\Psi$ is a solution for $(\star)$ with $\Psi(0)=|Y|(0)$ and $\Psi'(0)=|Y|'(0)$, then \begin{align*} (\frac{|Y|}{\Psi})'\geq 0 \quad (1)& &|Y|\geq \Psi\quad(2) \end{align*}
- We have equality in $(2)$ on $t_0\in(0,\beta)$ iff $\mathcal{K}(Y,\gamma')=\delta$ on all $[0,t_0]$ and there exists a parallel unit vector field $X$ along $\gamma$ for which $Y(t)=\Psi(t)X(t)$ on all of $[0,t_0]$.
How does that connect to the usual definition of the theorem comparing to manifolds? I don't see the connection even if Chavel says in a remark these two formulations are related.
Edit: $\mathcal{J}^{\perp}$ are the Jacobi fields perpendicular to $\gamma'$.