ABSTRACT
The Bonnet Theorem is often used to show that two surface are isometric or that a surface having certain fundamental forms can exists, as the reader is invited to convince her/himself by solving the following exercises. However, given their essential role in formulating the equivalent for surfaces of the Serret-Frènet Equations and the Fundamental Theorem of curve theory, as well as their importance in the theory of geodesics, it is also an undertaking which we can evade. In consequence, the Christoffel symbols can be expresses solely in terms of the coefficients E,F,G of the first fundamental form. However, the value of such a, endeavor would be restricted, as it is much more easy and rewarding to solve it for some concrete surfaces, like the important class of surfaces of revolution. Perhaps surprisingly, even such a theoretical and technical aspect of Differential Geometry has its importance in applications.
