Let $(M,g)$ be a Riemannian manifold, $p$ a point on the manifold and $v \in T_p M$. Let $\gamma$ be the geodesic starting at $p$ in the direction $v$. There exists a time $t_f$ such that there exists a Fermi coordinate system adapted to $\gamma$ up to time $t_f$.

**My question:** Does there exist a lower bound for $t_f$ in terms of the 2-jet of $g$ at $p$?

That is, I have solid estimates on $g$ up to its second derivatives at $p$: $$\|g\| + \|\nabla g\| + \|\nabla^2 g\| \le h \qquad (1)$$ for some $h$. I would like to show that there exists $f(h)$ such that $$t_f \ge f(h)$$ for all Riemannian metrics satisfying (1) at $p$.

**Edit** (Mar 19): Taking the helpful advice of Anton, Deane, TK and Willie into account, I've reworded the question:

Let $U = B(0,r)$ be the closed Euclidean ball of radius $r$ in $\mathbb R^n$. Write $$\lambda = \inf_{x \in U} \inf_{\|v\|=1} \langle v,v \rangle_{g(x)}$$ as the minimum eigenvalue of the metric in $U$, and suppose that $$\frac{1}{\lambda} + \|g\|_{C^2(U)} \le h.$$ This is a more refined version of (1) above. Since $$\ddot \gamma^k = -\Gamma^k_{ij} \dot \gamma^i \dot \gamma^j,$$ our estimate gives a control on the acceleration of a geodesic $\gamma$ in $U$, so there exists a minimum self-intersection time $t_i$ depending on $h$ and $r$ (i.e., if $t, t' \le t_i$ then $\gamma(t) \ne \gamma(t')$).

Does this imply the existence of a uniform lower bound on $t_f$ (depending only on $r$ and $h$)? If so, can we relax the control on the second derivative of $g$?

More succinctly: Are existence and non-self-intersection of a geodesic the only obstructions to the existence of Fermi coordinates?