Compute distance between geodesics and perturbed geodesics on a Riemannian manifold via Jacobi field $\vert J \vert$

Question

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*}

Answer 1

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.