We can equip the (co)tangent bundle of a Riemannian manifold (B,g) with a Saski metric $\hat{g}$ (see, for example, "On the geometry of tangent bundles" by Gudmunssun & Kappos) that looks like \begin{equation*} \hat{g} = \pi^*g\oplus g_V \end{equation*} where $\pi: T^{(*)}B \to B$ is the projection and $g_V$ is the metric on the vertical directions of the fibration. In certain cases (i.e. when $B$ is affine), we can compactify by taking the quotient of $T^{(*)}B$ by a (dual) lattice $\Gamma^{(*)}$ to obtain the non-singular torus fibrations $T^{(*)}B/\Gamma^{(*)} \to B$.
In physics, there is a (I don't know how well-defined) notion of "time-compactification" by passing from a Riemannian manifold $(M,\hat{g}_+)$ to a pseudo-Riemannian manifold $(M,\hat{g}_-)$ via a "Wick rotation" (https://en.wikipedia.org/wiki/Wick_rotation). In flat coordinates, if \begin{equation*} \hat{g}_+ = \sum_i dx_i^2 + \sum_j dy_j^2 \ , \end{equation*} then a Wick rotation is equivalent to the procedure of substituting $y_j \to \sqrt{-1} y_j$ since \begin{equation*} \hat{g}_- = \sum_i dx_i^2 - \sum_j dy_j^2 \ . \end{equation*}
I would like to know whether this can be related to the procedure of taking the alternative metric \begin{equation} \hat{g}' = \pi^*g\oplus (-1)g_V \end{equation} on $T^{(*)}B$ and whether this can somehow be seen as equivalent to compactification by taking the quoitient with respect to $\Gamma^{(*)}$.
