On functions whose curves of steepest descent are straight lines

Abstract

We consider functions such that their curves of steepest descent are straight lines. We prove that, along these curves, all principal curvatures of level surfaces of such functions satisfy a simple nonlinear ordinary differential equation. This result is used to get quantitative geometrical information on the level surfaces

AMS(MOS):

• 35B50
• 31A15

Keywords:

• Geometry of solutions
• eiconal equations