Curvature of Measure-Preserving Diffeomorphism Groups of Non-Orientable Surfaces

TL;DR

The paper develops a curvature-based analysis of the measure-preserving diffeomorphism groups for non-orientable surfaces, focusing on the Klein bottle and the real projective plane by exploiting their orientation double covers. Using explicit bases for the corresponding SVect algebras and Arnold-type curvature formulas, it derives sectional and normalized Ricci curvatures in strategically chosen directions, uncovering predominantly negative curvatures with regions of positivity in RP^2. It further analyzes asymptotics of sectional curvatures as mode indices grow and computes concrete Ricci-curvature values, establishing a rich geometric picture for SDiff on these non-orientable surfaces. Finally, the authors translate these geometric findings into estimates of weather-forecast unreliability for toy-model Earth shapes, highlighting distinct digits-of-accuracy requirements between Klein bottle and RP^2 scenarios and contrasting them with the sphere and torus cases.

Abstract

We study curvatures of the groups of measure-preserving diffeomorphisms of non-orientable compact surfaces. For the cases of the Klein bottle and the real projective plane we compute curvatures, their asymptotics and the normalized Ricci curvatures in many directions. Extending the approach of V. Arnold, and A. Lukatskii we provide estimates of weather unpredictability for natural models of trade wind currents on the Klein bottle and the projective plane.

Figures (9)

• Figure 1: Lattice $\Gamma$ in $\mathbb{R}^2$. The green and blue arrows are the identifications defining the torus $\mathbb T$. Identifying the red arrow with the green arrows gives the Klein bottle $\mathbb K$, whose double cover is the torus $\mathbb T$.
• Figure 2: A vector field $v = \sin(x_2)\partial_{x_1} \leftrightarrow \xi_{(0,1)} =-\cos(x_2)$ on the torus in the subalgebra $\mathop{\mathrm{S_0Vect}}\nolimits_I(\mathbb T)$, i.e. where $I_*v = v\circ I$. Thus this gives a vector field on the Klein bottle
• Figure 3: The value of $C(\xi_{(20,10)}, \eta_l)$ along the $x_3$-axis for $\eta_l$ with various $l = (l_1, l_2)\in J^{\Im}$ satisfying $\left\lVert l\right\rVert\leq 50$. On the "cross" passing through $l_1 = 20$, $l_2 = 10$, the hypothesis of Corollary \ref{['cor:neg.C.basis']} is not fulfilled. The sequence $\eta_{(m, 10)}$ from Table \ref{['tab:pos.seq']} is going along $l_2 = 10$. (The colours are to distinguish individual points.)
• Figure 4: The vectors $k, \ l, \ -l, \ \overline{l}, \ -\overline{l}$ and corresponding angles $\kappa, \ \lambda, \ {\pi + \lambda} , \ -\lambda, \ \pi -\lambda$ between them and the positive $x$-axis.
• Figure 5: Here $R=25$ and $k = (20, 10)$ as in Figure \ref{['fig:blanket']}. The green points correspond to the indices $l$ such that $\mathcal{E}_l^{\Re} \in A_R$, the blue to the indices $l$ such that $\mathcal{E}_l^{\Re} \in \mathcal{B}_R \setminus A_R$ and the red to the $l$ such that $\mathcal{E}_l^{\Re} \notin \mathcal{B}$.

Theorems & Definitions (61)

• Definition 2.1
• Example 2.2
• Example 2.3
• Definition 2.4
• Lemma 2.5
• proof
• Corollary 2.6
• Definition 3.1
• Proposition 3.2
• proof