I would like to pose a question regarding the distance between a geodesic $\gamma(t)$ and a perturbed geodesic $\gamma\_{\epsilon}(t)$ on a Riemannian manifold. Specifically, is the distance controlled by a bound on the associated Jacobi field?


Following @Deane and @Otis (With heartfelt thanks), I'd like to attempt this question from a different perspective.. It may not be as strict, but I will do my best to gradually refine it. Starting from the definition (**Actually, I want to point out that when $t$ is sufficiently large, the distance between the geodesic still has an upper bound**) :


Let $ M $ be a Riemannian manifold and there exists a smooth map
$\Gamma: [0,\epsilon] \times [0,t] \rightarrow M$,
such that
\begin{align\*}
 \Gamma(0,t) = \gamma(t), \quad
 \Gamma(\epsilon,t) = \gamma\_\epsilon(t),
\end{align\*}


Let $\sigma(s)$ denote a unit speed minimizing geodesic from $\gamma(0)$ to $\gamma\_\epsilon(0)$, and $\epsilon$ be the length of $\sigma$.


Suppose we have a bound on the Jacobi field $\partial\_s\Gamma(s,t)$ for each $s \in (0,\epsilon)$ of the form
\begin{equation}
 |\partial\_s\Gamma| \leq C.
\end{equation}
Note: This is essential; see @Deane's answer for details.


Try to demonstrate that (It also holds when $t$ is sufficiently large.)


\begin{equation}
\frac{1}{2} d^2 (\gamma\_\epsilon(t),\gamma(t))\leq C \epsilon.
\end{equation}


**Attempts (Update)**


Let


\begin{equation\*}
 D(\epsilon) = \frac{1}{2}\int\_0^\epsilon d^{{2}}(\gamma\_s(t),\gamma(t)) ds.
\end{equation\*}


For each $ s \in (0, \epsilon) $, the Jacobi field $ \partial\_s \Gamma(s, t) $ satisfies the bound
\begin{equation\*}
 \left|\partial\_s \Gamma\right| \leqslant C,
\end{equation\*}
where $C>0$ is a constant.


\begin{align\*}
 (D'(\epsilon)) & = \frac{1}{2} d^2 (\gamma(t),\gamma\_\epsilon(t))= \frac{1}{2}
 \int\_0^\epsilon \langle \nabla d^{2}(\gamma(t),\gamma\_s(t)), \partial\_s\Gamma(s,t) \rangle ds\\
 & \leq \int\_0^\epsilon \langle d(\gamma(t),\gamma\_s(t)) , \vert \partial\_s\Gamma(s,t) \vert \rangle ds \\
 & \leq C \int\_0^\epsilon d(\gamma(t),\gamma\_s(t)) ds.
\end{align\*}


According to the inequality $\frac{1}{2} d^2 (\gamma(t),\gamma\_\epsilon(t)) \leq C \int\_0^\epsilon d(\gamma(t),\gamma\_s(t)) ds$, we can obtain the following bound \begin{equation\*}d(\gamma(t),\gamma\_\epsilon(t)) \leq C\epsilon. \end{equation\*}



Compute distance between geodesics and perturbed geodesics on a Riemannian manifold via Jacobi field $\vert J \vert$
You should formulate your question in more detail and rigor. Let me try to do this here and elaborate on Otis's comments.


Let $M$ be a Riemannian manifold and $\gamma: [0,1] \rightarrow M$ be a constant speed geodesic. You say that $\gamma\_\epsilon: [0,1] \rightarrow M$ is a perturbed geodesic but do not say what that means. I'll assume that this means there is a smooth map
$$ \Gamma: [0,\epsilon] \times [0,1] \rightarrow M $$
such that for any $t\in [0,1]$,
\begin{align\*}
\Gamma(0,t) &= \gamma(t)\\
\Gamma(\epsilon,t) &= \gamma\_\epsilon(t)
\end{align\*}
and for each $s \in [0,\epsilon]$, the curve
$$t \mapsto \Gamma(s,t)$$ is a constant speed geodesic.


Your goal is to estimate the distance between $\gamma(t)$ and $\gamma\_\epsilon$ and propose to use the observation that
$$
d(\gamma(t),\gamma\_\epsilon(t)) \le \ell(\Gamma(\cdot,t)) = \int\_{0}^{\epsilon} |\partial\_s\Gamma(s,t)|\,ds.
$$
By the Cauchy-Schwarz inequality, the following bound also holds:
$$
(d(\gamma(t),\gamma\_\epsilon(t)))^2 \le \int\_{0}^{\epsilon} |\partial\_s\Gamma(s,t)|^2\,ds.
$$


Along each geodesic $\Gamma(s,\cdot)$, the vector field $\partial\_s\Gamma(s,\cdot)$ is a Jacobi field. You denote
$$ J = \partial\_s\Gamma(0,\cdot). $$
It is clear that a bound on $|J(t)|=|\partial\_s\Gamma(0,t)|$ does not imply any bound on $|\partial\_s\Gamma(s,t)|$ for $0 < s \le \epsilon$. As Otis observes, this implies that your assumptions are not enough to obtain a bound on $d(\gamma(t),\gamma\_\epsilon(t))$.


If, on the other hand, if you have a bound on the Jacobi field $\partial\_s\Gamma(s,\cdot)$ for each $s \in (0,\epsilon)$ of the form
$$ |\partial\_s\Gamma| \le C, $$
then you obtain trivially that
$$ d(\gamma(t),\gamma\_\epsilon(t)) \le C\epsilon. $$


If you assume a sectional curvature bound, then, as Otis observes, you still need to assume more. One possible assumption consists of (sufficiently small) bounds on $d(\gamma(0),\gamma\_\epsilon(0))$ and $d(\gamma(1),\gamma\_\epsilon(1))$. Standard comparison theorems then imply the bounds you want. There are other assumptions you could make that imply similar bounds.


Last, I want to say again that if you want to ask a question on math.stackexchange.com (where I think this question really belongs) or MathOverflow, you should write it as carefully as I have done above. Often, by doing only that, you can see what's going on more clearly.


