弹性微分学中商对象的黎曼结构：基于Q-图
Riemannian Structures on Quotients in Elastic Diffeology via Q-Charts
- Shinshu University（信州大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明弹性微分学的黎曼结构沿Q-图下降，并应用于无理环面，完全确定其黎曼几何，包括度量、联络和测地流性质。
AI中文摘要：
沿用宫本的方法，我们考虑Q-图，这是一类微分学空间的细分映射$\phi\colon X\to Y$，Blohmann的弹性质沿此类映射下降。我们证明弹性微分学的黎曼装置——垂直、水平和全联络，黎曼度量，Levi-Civita联络以及测地线——也沿Q-图下降。更精确地说，$Y$上的结构与$X$上在纤维保持变换的伪群$\Psi(\phi)$下不变的结构一一对应，并且无挠性、有效性、平坦性、完全性、Levi-Civita性质和测地线方程在两个方向上都保持。关键输入是迭代切丛$T^nX$及其纤维积$T_kX$是沿$\phi$从$Y$上相应丛的拉回。对于微分学离散群的主作用商，我们全局提升测地线，证明$\R$作为曲线对象的弹性空间类对此类商封闭，并证明完备测地流下降。作为应用，我们完全确定了无理环面$\Tor_\alpha=\R/(\Z+\alpha\Z)$的黎曼几何。设$\vartheta$为$\Tor_\alpha$上由$\R$上的$dt$诱导的一次形式。每个黎曼度量都是$\vartheta^2$的正常数倍，垂直联络构成一个单参数族的平坦有效联络族，其中唯一一个成员是每个度量的唯一Levi-Civita联络。该联络具有完备测地流，但诱导的伪距离恒为零。
英文摘要：
Following Miyamoto, we consider Q-charts, a class of subductions $ϕ\colon X\to Y$ 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 $Ψ(ϕ)$ 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 $\Tor_α=\R/(\Z+α\Z)$ completely. Let $\vartheta$ be the one-form on $\Tor_α$ 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.
