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

TL;DR

This paper investigates the geometric properties, specifically curvatures, of measure-preserving diffeomorphism groups on non-orientable surfaces like the Klein bottle and projective plane, with applications to weather modeling.

Contribution

It extends Arnold's approach to compute and estimate curvatures and Ricci curvatures for these groups, providing new insights into their geometric structure and applications.

Findings

Computed curvatures for Klein bottle and projective plane

Provided asymptotic behavior and normalized Ricci curvatures

Estimated weather unpredictability in natural models

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.