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.
Related
- Local Topology of Riemannian Manifolds with Lower Intermediate Ricci Curvature Bounds
- Hopf type theorem for surfaces of constant weighted mean curvature in Riemannian manifolds
- Biharmonic Wintgen ideal submanifolds in Riemannian manifolds of constant sectional curvature
- Interior gradient estimates for mean curvature equations on Riemannian manifolds
- Pólya--Szegö Inequality on Submanifolds of Riemannian Manifolds with Nonnegative Curvature and Applications
BibTeX
@misc{langoen2026curvature,
title = {{Curvature Decomposition for Riemannian Lie Algebroids}},
author = {René Langøen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.04568},
eprint = {2606.04568},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.04568},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Curvature Decomposition for Riemannian Lie Algebroids
AU - René Langøen
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2606.04568
UR - https://arxiv.org/abs/2606.04568
AB - 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.
ER -