Let $g_{ij}(x) = \delta_{ij} + b_{ij}(x)$ where $b_{ij}$ is a smooth periodic function on $\mathbb{R}^3$ and $|b_{ij}(x)| \leq 0.01$ (an alternative description is that I am smoothly perturbing the torus $\mathbb{R}^3/\mathbb{Z}^3$ by a small amount).
I am curious about the shapes of geodesics $\gamma(s)$ (considered as functions $\gamma:[0,\infty)\to \mathbb{R}^3$) with respect to this metric. In particular, suppose that $\gamma$ is a geodesic with initial velocity $\dot{\gamma}(0) = e$.
Is it the case that $\langle \dot{\gamma}(s), e\rangle\geq 0$ (using the unperturbed Euclidean inner product) for all times $s$? I am not confident either way if this should be true or false.
Can there be ``trapped'' geodesics that never leave some ball in $\mathbb{R}^3$? This seems unlikely to me but I can't prove that the geodesics escape.