On Zermelo'-like problems: a Gauss-Bonnet inequality and a E. Hopf theorem

TL;DR

This paper reformulates Zermelo's navigation problem as a time-optimal control problem on Riemannian manifolds, deriving a control curvature expression that leads to a Gauss-Bonnet inequality and a generalized Hopf theorem.

Contribution

It introduces an efficient method to evaluate control curvature in Zermelo's problem and generalizes classical geometric theorems to this control setting.

Findings

Derived a simple expression for control curvature in Zermelo's problem.

Established a Gauss-Bonnet inequality for Zermelo's navigation problem.

Generalized Hopf's theorem on flatness of Riemannian tori without conjugate points.

Abstract

The goal of this paper is to describe Zermelo's navigation problem on Riemannian manifolds as a time-optimal control problem and give an efficient method in order to evaluate its control curvature. We will show that up to change the Riemannian metric on the manifold the control curvature of Zermelo's problem has a simple to handle expression which naturally leads to a generalization of the classical Gauss-Bonnet formula in an inequality. This Gauss-Bonnet inequality enables to generalize for Zermelo's problems the E. Hopf theorem on flatness of Riemannian tori without conjugate points.