Diffeomorphisms of Curved Manifolds

TL;DR

This paper derives a formula for the curvature of the Lie group SDiff(M) of volume-preserving diffeomorphisms on a manifold, extending known results and discussing implications for specific cases like the 2-sphere.

Contribution

It provides a new expression for the curvature of SDiff(M) and extends Lukatskii's formula to locally Euclidean manifolds, connecting previous findings with new theoretical insights.

Findings

Derived curvature expression for SDiff(M)

Extended Lukatskii's formula to Euclidean cases

Discussed implications for SDiff(S^2)

Abstract

We obtain an expression for the curvature of the Lie group SDiff$\cal M$ and use it to derive Lukatskii's formula for the case where $\cal M$ is locally Euclidean. We discuss qualitatively some previous findings for SDiff$S^{2}$ in conjunction with our result.