A Hadamard-type open map theorem for submersions and applications to completeness results in Control Theory

TL;DR

This paper establishes a quantitative openness theorem for $C^1$ submersions and applies it to exponential maps in Carnot-Carathéodory spaces, leading to improved completeness results in Control Theory.

Contribution

It introduces a new quantitative openness theorem for $C^1$ submersions and applies it to enhance classical completeness results in Control Theory.

Findings

Proved a quantitative openness theorem for $C^1$ submersions.

Applied the theorem to exponential maps in Carnot-Carathéodory spaces.

Improved classical completeness results by Palais.

Abstract

We prove a quantitative openness theorem for $C^1$ submersions under suitable assumptions on the differential. We then apply our result to a class of exponential maps appearing in Carnot-Carath\'eodory spaces and we improve a classical completeness result by Palais.