Riemannian Structures on Quotients in Elastic Diffeology via Q-Charts

Following Miyamoto, we consider Q-charts, a class of subductions $\phi \text{\hspace{0.17em}⁣}:X\to Y$ 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 $\mathrm{\Psi }(\phi )$ 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 $T^{n}X$ and their fibre products $T_{k}X$ 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 $\mathbb{R}$ 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 ${\text{Tor}}_{\alpha }=\mathbb{R}\mathrm{/}(\mathbb{Z}+\alpha \mathbb{Z})$ completely. Let $\vartheta$ be the one-form on ${\text{Tor}}_{\alpha }$ induced by $dt$ on $\mathbb{R}$. 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.