Suppose $M_{g}$ is the mapping torus $\Sigma_{g} \times [0, 1]/ (x, 0) \equiv (\tau x, 1)$, where $\Sigma_{g}$ is the hyperbolic space with genus $g,$ and $\tau : \Sigma_{g} \to \Sigma_{g}$ is an isometry map.

Now suppose $\gamma$ is a simple closed geodesic in $M_{g}$, could we find a tubular neighborhood of $\gamma$ in $M_{g}$ and with radius $\varepsilon$, $\varepsilon$ is small enough, such that the boundary torus of this tubular neighborhood has flat metric?