Riemannian Structures on Quotients in Elastic Diffeology via Q-Charts
Abstract
Following Miyamoto, we consider Q-charts, a class of subductions ϕ: X→ 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 Ψ(ϕ) 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 TnX and their fibre products TkX are the pullbacks along ϕ 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 Torα=R/(Z+αZ) completely. Let ϑ be the one-form on Torα induced by dt on R. Every Riemannian metric is a positive constant multiple of ϑ2, 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.
Related
- On Riemannian Geometry in Elastic Diffeology
- Curvature, second variation and Jacobi fields in elastic diffeology
- Learning Normal Coordinates on Riemannian Manifolds with Flow Charts
- Frayed Demazure weaves for Poisson-compatible cluster structures on Bott--Samelson charts
- Sub-Riemannian Selberg Trace Formulae for Compact Quotients of SL(2,R) and Determinants of Sub-Laplacians
BibTeX
@misc{shiobara2026riemannian,
title = {{Riemannian Structures on Quotients in Elastic Diffeology via Q-Charts}},
author = {Yusuke Shiobara},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01346},
eprint = {2610.01346},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01346},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Riemannian Structures on Quotients in Elastic Diffeology via Q-Charts
AU - Yusuke Shiobara
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01346
UR - https://arxiv.org/abs/2610.01346
AB - Following Miyamoto, we consider Q-charts, a class of subductions ϕ: X→ 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 Ψ(ϕ) 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 ϕ 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_α=R/(Z+αZ) completely. Let ϑ be the one-form on Tor_α induced by dt on R. Every Riemannian metric is a positive constant multiple of ϑ^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.
ER -