ABSTRACT
Chapter 8 generalizes the differential geometry of plane curves to a theory of curves or any regular surfaces. The Frenet frame gets replaced by the Darboux frame, which includes information from the curve and from the surface at the same time. How the Darboux frame changes leads to concepts of normal curvature, geodesic curvature, and geodesic torsion. A profound application of Green’s Theorem leads to the profound Gauss-Bonnet Theorem that relates the total Gaussian curvature of a region ℛ with the geodesic curvature on the boundary ∂ℛ and the Euler characteristic of the region ℛ. By considering curves with a geodesic curvature that is constantly 0, we arrive at the notion of a geodesic on a surface, the generalization of a straight line in the plane. Finally, we end the chapter with applications of the Gauss-Bonnet Theorem to non-Euclidean geometry. This theory produces analytic models of spherical and hyperbolic geometry, geometries which were first approached from a synthetic direction and which stunned the mathematical community.
