ON ZERMELO-LIKE PROBLEMS: GAUSS-BONNET INEQUALITY AND E. HOPF THEOREM

摘要

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