ABSTRACT
Study of the differential geometry of curved surfaces starts from the work of Gauss, who studied surfaces embedded in three-dimensional space. One starts by studying a curve in space, parametrizing it by curve length. Then ,one obtains the tangent vector by differentiation, and the curvature vector by differentiating again. Then, for a parametrization of a surface, the first fundamental form (or, as it is nowadays called, the metric) is derived. Then for these embedded surfaces, also a second fundamental form can be derived. Then, a matrix called the shape operator is derived, giving us the principal curvatures and directions of curvature on the surface as the solution to an eigenvalue problem. Both the Gauss total and Germain average curvatures are introduced. Then, for a curve in a surface, it is shown how the curvature vector decomposes into interior and exterior parts. The so-called Christoffel symbols pop up here naturally. They return when discussing geodesics, curves without interior curvature. Geodetic co-ordinates on the ellipsoid of revolution are used for illustration. Finally, Riemann surfaces are discussed, and an attempt is made to move from the local viewpoint of differential geometry to a global perspective, where surfaces may be mapped piecewise by so-called charts, together forming an atlas --- thus avoiding the problem with poles.
