Deriving the shape of the surface from its Gaussian curvature

A global statement about a compact surface with constant Gaussian curvature is derived by elementary differential geometry methods. Surfaces and curves embedded in three-dimensional Euclidian space are introduced, as well as several key properties such as the tangent plane, the first and second fundamental form, and the Weingarten map. Furthermore, intrinsic and extrinsic properties of surfaces are analyzed, and the Gaussian curvature, originally derived as an extrinsic property, is proven to be an intrinsic property in Gauss Theorema Egregium. Lastly, through the aid of umbilical points on a surface, the statement that a compact, connected surface with constant Gaussian curvature is a sphere is proven.