When "$(\varepsilon,\delta)$-geodesic" cannot be a loop?

Question

EDIT: Let $M$ be a smooth compact Riemannian manifold. Let $\varepsilon,\delta>0$. I call a smooth curve $\gamma\colon [a,b]\to M$ an $(\varepsilon,\delta)$-geodesic if for any $t_1<t_2<t_1+\delta$ one has $$dist(\gamma(t_1),\gamma(t_2))\leq length(\gamma[t_1,t_2])\leq (1+\varepsilon)dist(\gamma(t_1),\gamma(t_2)),$$ where $dist$ denotes the Riemannian distance between the two points, and $length(\gamma[t_1,t_2])$ is the length of the curve $\gamma$ between $t_1$ and $t_2$.

Question. Given $M$, do there exist $\varepsilon,\delta, l>0$ such that no $(\varepsilon,\delta)$-geodesic of length less than $l$ is a loop, i.e. $\gamma(a)\ne \gamma(b)$?

Remark. Clearly $0$-geodesics are usual geodesics. If the length of a geodesic is less than the injectivity radius then it cannot be a loop.

This question is a more precise version of "Almost geodesics" in Riemannian manifolds which cannot be loops

Answer 1

Assembling my comments into a partial answer:

You may be interested in the theorem that "in the presence of negative curvature, local quasi-geodesics are in fact global quasi-geodesics". Here is a MathStackExchange answer that gives a proof. The comment on the answer gives a reference.