On Parallel Sections of a Vector Bundle

TL;DR

This paper investigates conditions under which a smooth vector bundle with a connection has non-trivial local parallel sections, using an algebraic approach based on a derived flag and the Frobenius Theorem.

Contribution

It introduces a new algebraic method employing a derived flag to determine the existence of local parallel sections in vector bundles.

Findings

Provides criteria for the existence of local parallel sections

Uses a derived flag construction to analyze vector bundles

Applies Frobenius Theorem in a novel algebraic context

Abstract

We consider when a smooth vector bundle endowed with a connection possesses non-trivial, local parallel sections. This is accomplished by means of a derived flag of subsets of the bundle. The procedure is algebraic and rests upon the Frobenius Theorem.