On natural derivatives and the curvature formula in fibre bundles

TL;DR

This paper provides a new proof of the curvature formula in fibre bundles using natural derivatives and tensoriality, extending known results from linear to nonlinear connections.

Contribution

It introduces a direct proof of the curvature formula in fibre bundles based on natural derivatives, generalizing from linear to nonlinear connections.

Findings

Derived a curvature formula using natural derivatives

Extended linear connection results to nonlinear connections

Provided a tensoriality-based proof approach

Abstract

In a fibre bundle, natural derivatives of a section are defined as tangent vector fields on the image of a section of the fibre bundle. A local extension to vector fields in the tangent bundle leads to a direct proof of the formula expressing the curvature of a connection in terms of covariant derivatives. The result is based on a tensoriality argument and extends to nonlinear connections on fibre bundles a well-known formula for linear connections on vector bundles.