Characteristic Classes

Abstract

In 1827 Gauss published his classic book Disquisitiones generales circa superficies curvas. He defined the total curvature (now called the Gaussian curvature) $\kappa$ as a function on the surface. In his famous theorema egregium Gauss proved that the total curvature $\kappa$ of a surface S depends only on the first fundamental form (i.e., the metric) of S. Gauss defined the integral curvature $\kappa (\Sigma )$ of a bounded surface Σ to be ${\int\nolimits \nolimits }_{\Sigma }\kappa \ d\sigma$. He computed $\kappa (\Sigma )$ when Σ is a geodesic triangle to prove his celebrated theorem

$$\kappa (\Sigma ) :={ \int\nolimits \nolimits }_{\Sigma }\kappa \ d\sigma= A + B + C - \pi, $$

(5.1)

where A, B, C are the angles of the geodesic triangle Σ. Gauss was aware of the significance of equation (5.1) in the investigation of the Euclidean parallel postulate (see Appendix B for more information). He was interested in surfaces of constant curvature and mentions a surface of revolution of constant negative curvature, namely, a pseudosphere. The geometry of the pseudosphere turns out to be the non-Euclidean geometry of Lobačevski–Bolyai.