On the flows of linear fields

TL;DR

This paper proves that linear vector fields on vector bundles generate flows by bundle isomorphisms, ensuring global solutions to linear ODEs, and provides applications including triviality of vector bundles and Lie algebroid fiber isomorphisms.

Contribution

It offers detailed proofs of flow properties of linear fields, applications to vector bundle triviality, and Lie algebroid fiber isomorphisms, enhancing theoretical understanding.

Findings

Linear vector fields generate flows by bundle isomorphisms.

Global solutions exist for linear non-autonomous ODEs.

Vector bundles over contractible bases are smoothly trivial.

Abstract

This note provides a detailed proof of the fact that a linear vector field on a vector bundle has a flow by vector bundle isomorphisms. It implies then easily the existence of global solutions to linear non-autonomous ODE's, with a standard time-dependent flow construction. As a further application, a simple proof of the smooth triviality of vector bundles over contractible bases is given. Finally, a detailed elementary proof of the isomorphy (as Lie algebras) of all fibers of the kernel of a transitive Lie algebroid is given.