On the metric Koll\'ar-Pardon problem

Let $(M, g)$ be a compact real analytic Riemannian manifold and $\pi \colon \widetilde{M} \to M$ its universal cover. Assume that $\widetilde{M}$ can be realised as a manifold definable in an o-minimal structure $\Sigma$ expanding $\mathbb{R}_{\mathrm{an}}$ in such a way that the pullback metric $\widetilde{g}:=\pi^*g$ is $\Sigma$-definable. For instance, this is the case when $\widetilde{M}$ can be realised as a semi-algebraic submanifold in $\mathbb{R}^n$ in such a way that the coefficients of the metric $\widetilde{g}$ are semi-algebraic. We show that there exists a definable smooth map $\widetilde{M} \to \widetilde{K}$ to a compact simply connected $\Sigma$-definable space $\widetilde{K}$ such that its regular fibres are Riemann locally homogeneous with respect to the metric $\widetilde{g}$. We deduce that under these assumptions $\pi_1(M)$ is quasi-isometric to a locally homogeneous space. In the case when $M$ is aspherical we show that $(\widetilde{M}, \widetilde{g})$ is a homogeneous Riemannian manifold. A similar result in the setting of complex algebraic geometry was earlier conjectured by Koll\'ar and Pardon (\cite{KP}). Using our results, we prove the conjecture of Koll\'ar-Pardon in the special case of smooth aspherical varieties admitting a bi-definable K\"ahler metric and discuss the analogues of this conjecture in other branches of geometry.