Let $M$ be a connected manifold equipped with a connection $\nabla$. By Hopf-Rinow theorem, we know that if $M$ is complete then for any $x,y$ there exist a curve $\gamma:[0,1] \to M$ such that $\gamma(0) = x, \gamma(1) = y$ and $\nabla_{\gamma'(t)} \gamma'(t)=0$ for all $t$. This is a way to say that $\gamma$ is a geodesic.

Suppose now that $M$ is possibly non complete. Given a threeshold $\varepsilon$, is it always possible to find a $\gamma$ between two fixed points $x,y$ such that $$ \frac{\lVert \nabla_{\gamma'(t)} \gamma'(t) \rVert }{\lVert \gamma' \rVert^2 } < \varepsilon $$

I kind of solved the case in which $M$ is of the form $\mathbb{R}^n \setminus \cup_{i=1}^k N_i $, where $N_i$ are submanifolds of codimension at least 2. In this case you can take a segment from $x,y$ and perturb it to be transverse to each $N_i$ in the $C_2$ topology (see Hirsch, differential topology, transversality chapter), so that $\gamma''$ will be almost zero and $\gamma'$ almost costantly $(y-x)$. Since $\dim \gamma + \dim N_i < \dim M = n$, transversality means $\textrm{Im} \gamma \subset M$. In my case this is enough to conclude, so this is just a curiosity :) Maybe something in the spirit of calculus of variations?