ABSTRACT
Chapter 7 begins by introducing the Christoffel symbols, functions related to the second order derivatives of the parametrizations of the surface. Though at first brush, someone might suspect that these functions would depend on the normal vector, in fact they are intrinsic functions, which means they can be defined exclusively in terms of the metric tensor. The next section in the chapter presents what Gauss called the Theorema Egregium, or “Excellent Theorem,” which shows that the Gaussian curvature is again an intrinsic property of the surface. Finally, the chapter ends with the Fundamental Theorem of Surface Theory, which states the conditions under which the first and second fundamental form uniquely define a surface up to location and orientation in space.
