Geodesics on Surfaces

Abstract

We study geodesics on surfaces in the setting of classical differential geometry. We define the curvature of curves and surfaces in three-space and use the fundamental forms of a surface to measure lengths, angles, and areas. We follow Riemann and adopt a more abstract approach, and use tensor notation to discuss Gaussian curvature, Gauss's Theorema Egregium, geodesic curves, and the Gauss-Bonnet theorem. Properties of geodesics are proven by variational methods, showing the connection between straightest and shortest for curves on surfaces. The notion of intrinsic and extrinsic properties is highlighted throughout.