Weyl geometry

Weyl geometry
Name	Weyl geometry
Field	Differential geometry, Mathematical physics
Introduced	1918
Founder	Hermann Weyl

Weyl geometry is a type of geometric framework extending Riemannian geometry by allowing a pointwise scale or length connection that varies under parallel transport. It was introduced to unify ideas in mathematical physics and differential geometry and has influenced developments in general relativity, gauge theory, and modern formulations in conformal geometry and fiber bundle theory.

Definition and basic concepts

Weyl geometry defines a manifold equipped with an equivalence class of metrics related by local rescalings together with a one-form connection encoding how lengths change under parallel transport; this contrasts with Riemannian geometry where lengths remain invariant. The structure consists of a smooth manifold, a conformal class of metric tensors, and a Weyl connection determined by a one-form often denoted by ω or φ; under a local change by a scalar function the metric and the one-form transform in tandem, preserving the Weyl class. Fundamental notions include Weyl covariant derivative, length curvature, and compatibility conditions between the connection and conformal structure, which are formalized using methods from tensor calculus and principal bundle theory.

Historical development

The geometry was proposed in 1918 by Hermann Weyl as part of an attempt to unify Albert Einstein's general relativity with James Clerk Maxwell's electromagnetic theory; Weyl introduced a scale gauge invariance inspired by ideas in philosophy of mathematics and mathematical physics. The initial proposal drew critique from Albert Einstein and led to refinements; later work by Élie Cartan and others recast Weyl's ideas within the broader context of connection theory and Lie group methods. In the mid-20th century, Weyl geometry informed the development of modern gauge theory by figures such as Chen Ning Yang, Robert Mills, and influenced mathematical expositions by Shing-Tung Yau and Michael Atiyah, while later applications and reinterpretations appeared in research by Paul Dirac, Roger Penrose, and Erik Verlinde.

Mathematical structure

Formally, a Weyl manifold is a pair (M, [g], ∇) where M is a smooth manifold, [g] is a conformal class of metrics, and ∇ is a torsion-free affine connection satisfying ∇g = −2ω ⊗ g for a one-form ω. The transformation law under conformal rescaling g → e^{2σ}g sends ω → ω + dσ, reflecting a local gauge transformation property analogous to those in electromagnetism and Yang–Mills theory. The space of Weyl connections on a given conformal manifold is an affine space modeled on the one-forms on M, while special choices, such as closed Weyl forms or exact Weyl forms, correspond to integrable or trivializable scale connections studied by authors including A. Lichnerowicz and Nikolai Vilenkin. Techniques from fiber bundle theory, connection forms, and Cartan geometry are used to analyze global and local properties, holonomy, and moduli of Weyl structures.

Connections and curvature

The Weyl connection defines curvature tensors generalizing the Riemann curvature tensor and introducing the length curvature two-form F = dω, which measures nonintegrability of local length standards; F plays a role analogous to the electromagnetic field strength in Maxwell equations. Decomposition of curvature into conformal (Weyl) curvature, Ricci-type parts, and scalar curvature yield invariants relevant to classification theorems in the style of Schur's lemma and the Ambrose–Singer theorem. Scalar invariants and Bianchi identities adapt to the Weyl setting, and energy-momentum-like quantities can be defined using adapted notions of divergence that involve the Weyl one-form; these constructions are discussed in works by Claude Itzykson, Robert Geroch, and André Lichnerowicz.

Physical applications

Weyl geometry has been explored as a framework for unifying interactions and for reformulating scale-invariant and conformal models in cosmology, quantum field theory, and alternative gravity theories. Early attempts by Hermann Weyl and Paul Dirac proposed connections with electromagnetism and varying fundamental constants; later approaches placed Weyl structures at the heart of conformal gravity models studied by Philip Mannheim and in scalar–tensor theories related to Brans–Dicke theory. In modern particle physics and string theory contexts, Weyl invariance appears in worldsheet formulations, while in cosmology Weyl geometry has been invoked in scenarios addressing dark energy and inflation by authors including Sean Carroll and Claudio Bunster. Experimental constraints from precision tests in astrophysics and laboratory measurements constrain many Weyl-inspired modifications, and ongoing work explores effective field theory realizations and phenomenological signatures.

Generalizations and related geometries

Weyl geometry sits among a family of generalized geometries including Cartan geometry, conformal geometry, Finsler geometry, and metric-affine geometries studied by A. N. Tod and Eugenio Calabi. Extensions include Weyl–Cartan geometries with torsion, integrable Weyl structures with closed scale forms, and gauge-theoretic formulations on principal GL(n, R)-bundles. Relations to Kähler geometry and complex manifolds arise in locally conformally Kähler spaces, while links to noncommutative geometry and generalized connections have been developed by scholars such as Alain Connes and Nigel Hitchin. Contemporary research investigates moduli spaces of Weyl structures, stability conditions in geometric analysis, and interactions with the AdS/CFT correspondence explored by Juan Maldacena.

Category:Differential geometry