Complete connections on fiber bundles

TL;DR

This paper provides a rigorous proof that every smooth fiber bundle admits a complete connection, addressing gaps in previous proofs, and extends the result to Riemannian submersions with illustrative examples.

Contribution

It offers a definitive proof of the existence of complete connections on smooth fiber bundles and extends the theorem to Riemannian submersions, correcting prior inaccuracies.

Findings

Confirmed that all smooth fiber bundles admit complete connections

Identified and fixed gaps in previous proofs

Extended the theorem to Riemannian submersions

Abstract

Every smooth fiber bundle admits a complete (Ehresmann) connection. This result appears in several references, with a proof on which we have found a gap, that does not seem possible to remedy. In this note we provide a definite proof for this fact, explain the problem with the previous one, and illustrate with examples. We also establish a version of the theorem involving Riemannian submersions.