ABSTRACT
This chapter explains what kind of information about a surface one can know from just the first fundamental form versus that which can be determined from knowledge of both the first and the second fundamental forms. It introduces the Christoffel symbols and studies relations between the coefficients of the first and second fundamental forms. The chapter presents the famous Theorema Egregium, which proves that the Gaussian curvature of a surface is in fact an intrinsic property. From an intuitive perspective, one could understand tensors as objects that are related to the tangent space to a surface at a point. The Gaussian curvature of a surface is an intrinsic property of the surface, that is, it depends only on the coefficients of the metric tensor and higher derivatives thereof. Any bijective function between two regular surfaces preserving the metric tensor defines an isometry between the two surfaces.
