[Paper Review] Connections and jet fields

Arturo Echeverrı́a-Enrı́quez, Miguel C. Muñoz‐Lecanda+1 more|arXiv (Cornell University)|Mar 28, 2018

Nonlinear Waves and Solitons1 references3 citations

TL;DR

This paper provides a comprehensive review of connections in fiber bundles and jet bundles, establishing their equivalence to first- and second-order systems of partial differential equations (PDEs) via multivector fields and jet fields. It demonstrates how connections—particularly linear and metric connections—can be characterized through horizontal and vertical splittings of tangent bundles, with key results linking covariant derivatives to horizontal lifts and the complete lift of vector fields in the tangent bundle of a manifold.

ABSTRACT

In this review paper we discuss the different interpretations of the concept of connection in a fiber bundle and in a jet bundle, and relate it with first and second-order systems of partial differential equations (PDE's) and multivector fields. As particular cases we analyze the concepts of linear connections and connections in a manifold.

Motivation & Objective

• To unify and clarify multiple equivalent interpretations of connections in fiber bundles, emphasizing their geometric and algebraic structures.
• To establish the correspondence between connections in jet bundles and second-order partial differential equations (SOPDEs), particularly through holonomic jet fields.
• To analyze linear connections and connections on manifolds using the framework of jet bundles and multivector fields.
• To demonstrate the role of horizontal and vertical splittings in defining covariant derivatives and parallel transport.
• To provide a geometric foundation for understanding connections in theoretical physics and differential geometry via jet-theoretic formalism.

Proposed method

• Utilizes first-order jet bundles $J^1 ilde{ au}$ to represent jet prolongations of sections, with local coordinates $(x^ u, y^i, y^i_ u)$ for $J^1 ilde{ au}$.
• Defines connections on fiber bundles via horizontal subbundles $\mathrm{H}(\nabla) \subset \mathrm{T}(J^1\tilde{\tau})$, with local frame $\left\{ \frac{\partial}{\partial x^\nu} - \Gamma^\rho_{\nu\mu} v^\mu \frac{\partial}{\partial v^\rho} \right\}$.
• Introduces the horizontal and vertical projections $\mathfrak{h}, \mathfrak{v}$ on $\mathrm{T}(\mathrm{T}M)$, enabling decomposition of vector fields and defining the covariant derivative.
• Applies the complete lift $Y^C$ of a vector field $Y$ to $\mathrm{T}M$, decomposed into horizontal and vertical parts to express $\nabla_X Y$.
• Characterizes connections via the splitting $\mathrm{T}(\mathrm{T}M) = \mathrm{V}(\tau) \oplus \mathrm{H}(\nabla)$, with $\mathrm{H}(\nabla)$ determined by Christoffel symbols $\Gamma^\rho_{\nu\mu}$.
• Uses the Liouville vector field and its properties to characterize linear connections, particularly in the context of tangent bundles.

Experimental results

Research questions

• RQ1 How are connections in fiber bundles equivalent to first-order systems of PDEs through jet fields and multivector fields?
• RQ2 What is the precise geometric relationship between connections in $J^1\pi$ and second-order PDEs (SOPDEs) via holonomic jet fields?
• RQ3 How can linear connections on a manifold be characterized using the horizontal and vertical splitting of $\mathrm{T}(\mathrm{T}M)$?
• RQ4 In what way do the complete lift of vector fields and horizontal projections yield the covariant derivative?
• RQ5 How does the use of jet bundles and multivector fields unify the treatment of connections and PDEs in geometric mechanics?

Key findings

• Connections in fiber bundles are equivalent to first-order systems of PDEs through the integrability of jet fields and the existence of holonomic sections in $J^1\pi$.
• A connection on a manifold induces a horizontal subbundle $\mathrm{H}(\nabla) \subset \mathrm{T}(\mathrm{T}M)$, with local frame $\left\{ \frac{\partial}{\partial x^\nu} - \Gamma^\rho_{\nu\mu} v^\mu \frac{\partial}{\partial v^\rho} \right\}$, defining the covariant derivative.
• The covariant derivative $\nabla_X Y$ is geometrically realized as $[\mathrm{T}_p\tau(\mathfrak{h} \circ Y^C \circ X)](p)$, where $Y^C$ is the complete lift of $Y$.
• The horizontal and vertical projections $\mathfrak{h}, \mathfrak{v}$ on $\mathrm{T}(\mathrm{T}M)$ decompose any vector field, and their local expressions are explicitly given in terms of Christoffel symbols.
• Linear connections on a vector bundle are characterized by the splitting $\mathrm{T}(\mathrm{T}M) = \mathrm{V}(\tau) \oplus \mathrm{H}(\nabla)$, with $\mathrm{H}(\nabla)$ determined by the connection coefficients $\Gamma^\rho_{\nu\mu}$.
• The paper establishes that the complete lift $Y^C$ of a vector field $Y$ splits into horizontal and vertical components, and the vertical part yields the covariant derivative $\nabla_X Y$ via $\mathfrak{V}(Y^C)$.