Curvature of Lie Groupoids with Source-Fibre Metric and Riemannian Lie Algebroids

Abstract

Classical curvature formulas for Lie groups, principal bundles, and Riemannian submersions are usually treated as separate theories. This paper shows that they can be understood within a common framework using Lie groupoids with source-fibre metrics and their infinitesimal counterparts, Riemannian Lie algebroids. We derive a sectional curvature formula for Lie groupoids with right-invariant source metrics, extending the Arnold-Milnor ``1-2-3-4'' formula for Lie groups, and we establish a Lie algebroid version of O'Neill's curvature formulas for Riemannian submersion Lie algebroids. The examples include the action of diffeomorphisms on densities on the torus, linking the Euler equation to the Wasserstein space of densities and optimal transport, as well as a geometric model for the rotational configuration space of the Earth.