Rauch comparison theorem with other initial conditions? The statement of the Rauch comparison theorem (see e.g. https://en.wikipedia.org/wiki/Rauch_comparison_theorem) involves two normal Jacobi fields $J$ and $\tilde{J}$ along two unit speed geodesics $\gamma$ and $\tilde{\gamma}$ in their respective manifolds $M$ and $\tilde{M}$. As a hypothesis, $J(0)=\tilde{J}(0)=0$ and $\|D_tJ(0)\|=\|\tilde{D}_t\tilde{J}(0)\|$. Also, conjugate points must be avoided in $\tilde{\gamma}$.
I've been looking for an identical statement but taking a non-zero initial condition (EDIT): $\|J(0)\|=\|\tilde{J}(0)\|=a\neq0$. Nevertheless, I didn't find anything and I don't even know if it's possible.
The only similar thing I found is on the page 149 of the Sakai's book Riemannian geometry (1996). Here, an extended result to Rauch is given but it involves an hypersurface $N$, $N$-Jacobi fields and focal points. I also found somewhere Rauch as a corollary of the Sturm's theorem on differential equations, but the version I'm looking for was not deduced from there (I don't know if it's possible just by taking another initial condition in the differential equation problem).
To sum up, I just want the same statement as above but with a non-zero initial condition (maybe with some change on the condition of avoiding conjugate points also, I don't know). Any reference is welcome!
Thank you in advance!
 A: The initial condition you suggested, $J(0)=\tilde J(0)$, doesn't make sense. Those two vectors are tangent to different manifolds, so they can't be equal.
However, there is a version of the Rauch comparison theorem that allows for nonzero initial conditions. One version of the theorem is as follows:
Theorem: Suppose $J$ and $\tilde{J}$ are normal Jacobi fields along unit-speed geodesics $\gamma$ and $\tilde{\gamma}$ in their respective Riemannian manifolds $M$ and $\widetilde{M}$. Suppose also that the sectional curvatures satisfy $K(\tilde \sigma)\ge K(\sigma)$ for all tangent $2$-planes $\sigma$ containing $\gamma'(t)$ and $\tilde\sigma$ containing $\tilde\gamma'(t)$, and $\tilde\gamma(t)$ is not a focal point of the geodesic submanifold perpendicular to $\tilde\gamma'(0)$ for any $t$. If $\|J(0)\|= \|\tilde J(0)\|$, $J’(0)=0$, and $\tilde J’(0)=0$, then $\|J(t)\|\ge \|\tilde J(t)\|$ for all $t$.
The statement and proof of this theorem (actually, a slightly more general version of it) can be found in Cheeger & Ebin's Comparison Theorems in Riemannian Geometry, Theorem 1.34.
