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 $\Gamma^{(*)}$.

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    $\begingroup$ A couple of notes. Wick rotation does not a priori have anything to do with compactification, though sometimes the two come together. You can find a mathematically precise notion of Wick rotation in this answer. I'm not sure that what you are doing qualifies, as you are not doing an analytic continuation anywhere. Though, perhaps there is a way to reinterpret your construction in terms of complexification and analytic continuation. $\endgroup$ – Igor Khavkine Aug 3 '15 at 20:55
  • $\begingroup$ @Igor, thanks for the link. Perhaps you could explain how your view of the Wick rotation presented there is ("sometimes") related to compactification. How does a noncompact submanifold get smoothly deformed to a compact submanifold of the complexification? $\endgroup$ – harry Aug 3 '15 at 21:39
  • $\begingroup$ Simple. The complexification $M_\mathbb{C}$ is not unique. For example, the flat metric on $\mathbb{R}$ embeds into both $\mathbb{C}$ (with a non-compact Wick rotation) and $\mathbb{C}/i\mathbb{Z}$ (with a compact Wick rotation). $\endgroup$ – Igor Khavkine Aug 3 '15 at 23:53

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