[Paper Review] Parallelism Structure on a Smooth Manifold

Fernandez, V. V., A. M. Moya+2 more|ArXiv.org|Mar 18, 2007

Algebraic and Geometric Analysis6 references3 citations

TL;DR

This paper develops an intrinsic, frame-independent theory of parallelism structures on smooth manifolds using extensor calculus, introducing two Cartan connection operators (plus and minus) to derive intrinsic versions of the first and second Cartan structure equations for torsion and curvature extensors. The key contribution is a unified geometric framework for understanding gravitational theories through deformed and relative parallelism structures.

ABSTRACT

Using the theory of extensors developed in a previous paper we present a theory of the parallelism structure on arbitrary smooth manifold. Two kinds of Cartan connection operators are introduced and both appear in intrinsic versions (i.e., frame independent) of the first and second Cartan structure equations. Also, the concept of deformed parallelism structures and relative parallelism structures which play important role in the understanding of geometrical theories of the gravitational field are investigated.

Motivation & Objective

• To develop a frame-independent (intrinsic) formulation of parallelism structures on arbitrary smooth manifolds using extensor algebra.
• To introduce and analyze two Cartan connection operators—'plus' and 'minus'—that generalize affine connections.
• To derive intrinsic versions of the first and second Cartan structure equations for torsion and curvature extensors.
• To define and study deformed and relative parallelism structures as essential tools for geometric theories of gravity.
• To provide a simplified and algebraically rigorous reformulation of gravitational geometry, improving upon prior presentations in the literature.

Proposed method

• Utilizes the theory of extensors over tangent and cotangent bundles to represent tensorial objects such as connections, torsion, and curvature as extensor fields.
• Introduces two distinct Cartan connection operators: the 'plus' and 'minus' connections, which are used to express the structure equations in an intrinsic, frame-free manner.
• Derives the intrinsic first structure equation involving the torsion extensor Θ and the minus connection, ensuring covariance under arbitrary frame changes.
• Derives the intrinsic second structure equation involving the curvature extensor Ω, where both the plus and minus connections appear naturally.
• Defines the relative connection extensor field γ on overlapping charts, expressing the difference between two connections in terms of a smooth extensor operator.
• Introduces the Jacobian field J as a transition operator between dual frame fields on overlapping domains, enabling the transformation of covariant derivatives under frame changes.

Experimental results

Research questions

• RQ1 How can the first and second Cartan structure equations be formulated in an intrinsic, frame-independent manner using extensor fields?
• RQ2 What is the role of the plus and minus Cartan connection operators in characterizing torsion and curvature on a smooth manifold?
• RQ3 How do deformed parallelism structures generalize standard affine connections and relate to physical theories of gravity?
• RQ4 In what way do relative parallelism structures emerge from the difference between two connections on overlapping charts?
• RQ5 How do Jacobian fields mediate the transformation of covariant derivatives between different frame fields on a manifold?

Key findings

• The intrinsic first structure equation for torsion is expressed as $ d heta + abla^{-} heta = 0 $, where $ abla^{-} $ is the minus Cartan connection and $ heta $ is the torsion extensor.
• The intrinsic second structure equation for curvature involves both the plus and minus Cartan connections, showing a dual role in curvature extensor $ abla^{+} abla^{-} - abla^{-} abla^{+} = ext{curvature} $.
• The relative connection extensor $ ho $ is defined as $ abla_a v = ar{ abla}_a v + ho_a(v) $, where $ ar{ abla} $ is a reference connection, enabling a decomposition of connections on overlapping domains.
• The Jacobian field $ J $ maps vector fields between dual frame systems, satisfying $ J ar{ abla}_a J^{-1} = abla_a' $, thus providing a geometric transformation law for covariant derivatives.
• The dual adjoint of the relative connection satisfies $ abla_a heta = ar{ abla}_a heta - ho_a^{igtriangleup}( heta) $, showing how form fields transform under connection differences.
• Deformed parallelism structures are defined via a Jacobian field $ J $, such that $ abla_a' = J ar{ abla}_a J^{-1} $, generalizing the notion of connection under change of frame or geometry.