Cartan connection

Cartan connection
Name	Cartan connection
Field	Differential geometry
Introduced	1920s
Introduced by	Élie Cartan
Related	Klein geometry, principal bundle, Ehresmann connection, Riemannian connection

Cartan connection A Cartan connection is a geometrical tool that encodes how a curved space models infinitesimally on a homogeneous model space. Developed in the early 20th century, it generalizes notions from Élie Cartan, Felix Klein's Erlangen Program, and Hermann Weyl's work on connections, unifying ideas from Riemannian geometry, projective geometry, and conformal geometry.

Introduction

Cartan connections arose from Élie Cartan's efforts to extend Felix Klein's classification of homogeneous spaces to manifolds that are only locally modeled on a homogeneous model such as Euclidean space, hyperbolic space, or projective space. The theory connects to principal bundles studied by Charles Ehresmann and to classical constructions by Ludwig Schläfli and Bernhard Riemann, and it influenced later developments by Hermann Weyl, André Lichnerowicz, and Shiing-Shen Chern.

Definition and formulation

A Cartan connection is typically defined on a principal bundle with structure group a Lie group G over a manifold M, modeled on a homogeneous space G/H where H is a closed subgroup studied by Wilhelm Killing and Élie Cartan (senior). Formally one gives a g-valued 1-form on the bundle satisfying properties that generalize the Maurer–Cartan form from Maurice Fréchet and Élie Cartan's original works. The formulation uses concepts from Lie group theory, Lie algebra cohomology, and the theory of principal bundles as developed by Shiing-Shen Chern and Charles Ehresmann. Equivalent descriptions employ associated bundles, soldering forms, and reductions akin to constructions of Elie Cartan used in the study of G-structures by Charles Ehresmann and A. E. Fischer.

Examples and special cases

Important examples include the Levi-Civita connection on a Riemannian manifold viewed relative to the orthonormal frame bundle associated to Bernhard Riemann's foundations, the Cartan formulation of conformal geometry linked to work of Hermann Weyl and Élie Cartan, and projective Cartan connections related to Shiing-Shen Chern and Wilhelm Blaschke's projective differential geometry. Other special cases include the flat Maurer–Cartan form on Lie groups such as SO(n), the Cartan–Bott connections appearing in studies influenced by Raoul Bott, and parabolic Cartan connections studied in relation to Borel subgroup theory in Élie Cartan's classification of symmetric spaces.

Curvature, torsion, and invariants

Curvature of a Cartan connection generalizes the curvature tensors introduced by Bernhard Riemann and the torsion concepts studied by Élie Cartan; it satisfies a structure equation analogous to the Maurer–Cartan equation familiar from Sophus Lie's theory. Invariants arising from curvature and torsion connect to characteristic classes studied by Shiing-Shen Chern and Atiyah–Singer index theorem-related work of Michael Atiyah and Isadore Singer. The curvature encodes obstruction to local equivalence with the homogeneous model used by Felix Klein and its cohomological aspects relate to results by Jean-Louis Koszul and Bertram Kostant.

Holonomy and geometric structures

Holonomy groups of Cartan connections generalize the classical holonomy groups classified by Marcel Berger for Riemannian connections and relate to special geometric structures such as G2 structures and Spin(7) structures studied in the context of S. S. Chern and Edward Witten's interests. The study of holonomy intertwines with representation theory of Lie groups and results of Bertram Kostant and Robert Bryant, and has implications for locally homogeneous geometries considered in works by William Thurston and Dennis Sullivan.

Construction and equivalence methods

Constructing Cartan connections often involves prolongation and reduction techniques pioneered by Élie Cartan and later formalized by Nicolas Bourbaki-influenced geometers and by Charles Ehresmann and Shiing-Shen Chern. Equivalence methods use moving frames developed by Élie Cartan and later algorithmized by Peter Olver and Marius Crainic, while normalization conditions and regularity use representation-theoretic inputs from Bertram Kostant and parabolic geometry frameworks by Thomas H. Parker and Andreas Čap. The existence and uniqueness results are linked to integrability theorems akin to those of Newlander–Nirenberg and to prolongation theorems appearing in works by Nijenhuis.

Applications and significance in geometry and physics

Cartan connections provide a unifying language for geometries appearing in general relativity as in Élie Cartan's reformulations of Einstein field equations and in gauge-theoretic approaches influenced by Yang–Mills theory of Chen Ning Yang and Robert Mills. They underpin modern treatments of conformal and projective invariants used in analyses by André Lichnerowicz and Conrad Vogan and appear in the mathematical foundations of string theory investigated by Edward Witten and Michael Green. In geometric representation theory and index theory, Cartan connections inform constructions studied by Michael Atiyah, Isadore Singer, and Bertram Kostant, and they remain central in research programs of contemporary geometers such as Andreas Čap and Jan Slovák.

Category:Differential geometry