Riemannian Structures on Quotients in Elastic Diffeology via Q-Charts

Abstract

Following Miyamoto, we consider Q-charts, a class of subductions $\phi\colon X\to Y$ of diffeological spaces along which Blohmann's elasticity descends. We show that the Riemannian apparatus of elastic diffeology---vertical, horizontal and full connections, Riemannian metrics, Levi-Civita connections and geodesics---also descends along Q-charts. More precisely, structures on $Y$ correspond bijectively to structures on $X$ that are invariant under the pseudogroup $\Psi(\phi)$ of fibre-preserving transitions, and torsion-freeness, effectivity, flatness, fullness, the Levi-Civita property and the geodesic equation are preserved in both directions. The key input is that the iterated tangent bundles $T^nX$ and their fibre products $T_kX$ are the pullbacks along $\phi$ of the corresponding bundles over $Y$. For quotients by principal actions of diffeologically discrete groups, we lift geodesics globally, prove that the class of elastic spaces on which $\R$ is a curve object is closed under such quotients, and show that complete geodesic flows descend. As an application, we determine the Riemannian geometry of the irrational tori $\Tor_\alpha=\R/(\Z+\alpha\Z)$ completely. Let $\vartheta$ be the one-form on $\Tor_\alpha$ induced by $dt$ on $\R$. Every Riemannian metric is a positive constant multiple of $\vartheta^2$, and the vertical connections form a one-parameter family of flat effective connections, a single member of which is the unique Levi-Civita connection of every metric. This connection has a complete geodesic flow, yet the induced pseudo-distance vanishes identically.