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*}

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