ABSTRACT
This chapter deals with the intrinsic geometry of surfaces. This means the geometric properties of surfaces that depend on measurements on the surface itself, that is, on the first fundamental coefficients and their derivatives only. The most important topics and subjects in this chapter are the Christoffel symbols of first and second kind and the geodesic curvature, Liouville's theorem geodesic lines on surfaces with orthogonal parameters and on surfaces of revolution, the minimum property of geodesic lines, orthogonal and geodesic parameters, geodesic parallel and geodesic polar coordinates, Levi–Cività parallelism, the derivation formulae by Gauss, and Mainardi and Codazzi, Theorema egregium, conformal, isometric and area preserving maps of surfaces, the Gauss–Bonnet theorem, minimal surfaces and the Weierstrass formulae.
