Curvature Decomposition for Riemannian Lie Algebroids

Abstract

Classical curvature formulas for Lie groups and Riemannian submersions are recast in terms of Lie groupoids with right-invariant source-fibre metrics and Riemannian Lie algebroids. We derive a sectional-curvature formula for such groupoids extending the Arnold-Milnor ``1-2-3-4'' formula. For transitive Riemannian Lie algebroids, we factor the base and vertical Levi-Civita connections through Lie algebroids of metric derivations and obtain Lie-algebroid analogues of O'Neill's curvature formulas. Applied to the action of the diffeomorphism group of the torus on densities, these formulas recover Arnold's curvature of the measure-preserving diffeomorphism group and give explicit Wasserstein sectional curvatures at the uniform density. Finite-dimensional examples include the rotational action algebroid of the round sphere and the WGS84 ellipsoid, and a smooth family of negative curvatures on non-abelian isotropy groups over the real projective line.