[Paper Review] Riemannian Geometry of Lie Algebroids

Mohamed Boucetta|ArXiv.org|Jun 21, 2008

Homotopy and Cohomology in Algebraic Topology10 references3 citations

TL;DR

This paper introduces Riemannian Lie algebroids as a generalization of Riemannian manifolds, extending classical Riemannian geometry tools—such as the Levi-Civita connection, geodesic flow, curvature, and Jacobi fields—to the Lie algebroid setting. The key contribution is a generalization of the Hadamard-Cartan theorem: a complete Riemannian Lie algebroid with nonpositive sectional curvature is integrable and diffeomorphic to its Weinstein groupoid via the exponential map.

ABSTRACT

We introduce Riemannian Lie algebroids as a generalization of Riemannian manifolds and we show that most of the classical tools and results known in Riemannian geometry can be stated in this setting. We give also some new results on the integrability of Riemannian Lie algebroids.

Motivation & Objective

• To extend classical Riemannian geometry tools—like connections, geodesics, and curvature—to the setting of Lie algebroids.
• To define and study the Levi-Civita connection on a Riemannian Lie algebroid using Koszul-type formulas.
• To generalize the Hadamard-Cartan theorem to Riemannian Lie algebroids, establishing integrability under nonpositive curvature.
• To investigate the role of curvature and holonomy in the integrability of Riemannian Lie algebroids.
• To connect Riemannian structures on Lie algebroids to Poisson geometry via Riemann-Poisson manifolds and their integrability.

Proposed method

• Define a Riemannian Lie algebroid as a Lie algebroid equipped with a fiber-wise inner product on the bundle.
• Construct the Levi-Civita connection on a Riemannian Lie algebroid using a Koszul-type formula analogous to the classical case.
• Introduce the Sasaki metric on the total space of the algebroid to study geodesic flow and compute its divergence.
• Derive first and second variation formulas for energy and define Jacobi fields along geodesics in the algebroid setting.
• Define curvature tensor and sectional curvature for Riemannian Lie algebroids, generalizing classical Riemannian curvature concepts.
• Use Crainic-Fernandes integrability obstructions (specifically the vanishing of the characteristic class H) to prove integrability under nonpositive curvature.

Experimental results

Research questions

• RQ1 Can the Levi-Civita connection and associated geometric tools be generalized from Riemannian manifolds to Lie algebroids?
• RQ2 What is the behavior of geodesic flow and divergence in the context of Riemannian Lie algebroids, and how does it differ from the classical Liouville theorem?
• RQ3 How can curvature concepts such as sectional curvature and Jacobi fields be defined and used in the Lie algebroid setting?
• RQ4 Under what conditions is a Riemannian Lie algebroid integrable, and how does curvature influence this?
• RQ5 What is the relationship between Riemann-Poisson manifolds and the integrability of their associated cotangent Lie algebroids?

Key findings

• The Levi-Civita connection on a Riemannian Lie algebroid satisfies O’Neill-type formulas analogous to those in Riemannian submersions.
• The divergence of the geodesic flow with respect to the Sasaki metric does not vanish in general, contrasting with the classical Liouville theorem.
• The first and second variation formulas for energy and the concept of Jacobi fields are successfully generalized to the Lie algebroid setting.
• A Riemannian Lie algebroid with vanishing characteristic class H is integrable, as shown via the Crainic-Fernandes integrability criterion.
• A complete Riemannian Lie algebroid with nonpositive sectional curvature is integrable and the exponential map from the algebroid to the Weinstein groupoid is a diffeomorphism.
• Riemann-Poisson manifolds—Poisson structures compatible with a Riemannian metric via the Levi-Civita connection—yield integrable cotangent Lie algebroids.