On Zermelo-Like Problems: Gauss–Bonnet Inequality and E. Hopf Theorem

• DOI: 10.1007/s10883-008-9056-6
• Publication Date: 2009-01-12T07:02:37Z
• Authors: Ulysse Serres
• Keywords: Mathematics - Differential Geometry; [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS]; 53B40; 34K35; 37C10; 37E35; 53B40; 53C22; 93C15; [MATH.MATH-DS] Mathematics [math]/Dynamical Systems [math.DS]; control curvature; 93C15; 01 natural sciences; Zermelo's navigation problem; 34K35; FOS: Mathematics; 0101 mathematics; Conjugate points; Mathematics - Optimization and Control; Riemannian manifold; Gauss-Bonnet formula; [MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC]; 34K35; 53B40; 93C15; feedback transformation; Differential Geometry (math.DG); [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG]; Optimization and Control (math.OC); [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]; [MATH.MATH-DG] Mathematics [math]/Differential Geometry [math.DG]

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.<br/>27 pages, 1 figure<br/>