[Paper Review] An introduction to Cartan geometries

Benjamin McKay|arXiv (Cornell University)|Feb 28, 2023

Homotopy and Cohomology in Algebraic Topology4 citations

TL;DR

This paper provides a comprehensive, graduate-level introduction to Cartan geometries, presenting them as geometric structures infinitesimally modeled on homogeneous spaces G/P, generalizing Riemannian geometry. It develops the theory using Lie groups, principal bundles, Maurer–Cartan forms, connections, and curvature, establishing foundational tools and proving key results on developability, completeness, and holonomy, with the central contribution being a systematic framework for understanding Cartan geometries through their model symmetries and curvature invariants.

ABSTRACT

We explain what Cartan geometries are, aiming at an audience of graduate students familiar with manifolds, Lie groups and differential forms.

Motivation & Objective

• To provide a self-contained, accessible introduction to Cartan geometries for graduate students with background in manifolds, Lie groups, and differential forms.
• To unify and clarify the foundational concepts of Cartan geometries—especially curvature, development, and holonomy—by building from Lie group theory and homogeneous spaces.
• To establish global properties of Cartan geometries by analogy with their homogeneous models, particularly in terms of completeness and automorphism groups.
• To bridge classical differential geometry with modern geometric structures through the moving frame method and Cartan's equivalence method.
• To provide a rigorous foundation for further study of parabolic geometries, G-structures, and geometric structures with large symmetry groups.

Proposed method

• Uses the Maurer–Cartan form on Lie groups to define the infinitesimal structure of Cartan geometries via the soldering form and connection 1-form.
• Applies the moving frame method to analyze geometric invariants and curvature tensors, particularly through the curvature map associated to the Cartan connection.
• Employs principal bundle theory and connections (including Cartan connections) to formalize the geometric structure of Cartan geometries.
• Introduces the concept of development of curves to study completeness and flatness, linking local geometry to global topology.
• Utilizes Lie algebra representations and invariant theory to analyze symmetries and automorphism groups of Cartan geometries.
• Applies results from ordinary differential equations and Lie equations to study orbits of vector fields and the integrability of geometric structures.

Experimental results

Research questions

• RQ1 How can Cartan geometries be systematically constructed as infinitesimal models of homogeneous spaces G/P?
• RQ2 What conditions ensure that a Cartan geometry is developable or complete, and how do these relate to curvature and holonomy?
• RQ3 How do automorphism groups of Cartan geometries relate to the symmetries of their model homogeneous spaces?
• RQ4 In what sense do Cartan geometries generalize Riemannian geometry, and how do curvature and holonomy invariants extend classical notions?
• RQ5 What role does the soldering form play in relating Cartan geometries to G-structures and classical geometric structures?

Key findings

• Cartan geometries are characterized by a Cartan connection—a principal connection with values in the Lie algebra of the model group G, satisfying specific normalization and curvature conditions.
• The curvature of a Cartan geometry is a tensorial invariant that measures the failure of the geometry to be locally isomorphic to the model homogeneous space G/P.
• A Cartan geometry is flat if and only if its curvature vanishes identically, and such geometries are locally isomorphic to the model space via the development map.
• Completeness of a Cartan geometry is equivalent to the completeness of the development map, and this property is preserved under certain geometric constraints.
• The holonomy group of a Cartan geometry is isomorphic to the holonomy group of the associated Cartan connection, and it acts on the model space via the linear holonomy representation.
• The automorphism group of a Cartan geometry acts transitively on the underlying manifold if and only if the geometry is locally homogeneous, and such automorphisms preserve the Cartan connection and curvature.