Classical differential geometry is that approach to geometry that takes full advantage of the introduction of numerical coordinates into a geometric space. This use of coordinates in geometry was the essential insight of Rene Descartes”s which allowed the invention of analytic geometry and paved the way for mod- ern differential geometry. A differential geometric space is firstly a topological space on which are defined a sufficiently nice family of coordinate systems. These spaces are called differentiable manifolds and examples are abundant. Depend- ing on what type of geometry is to be studied, extra structure is assumed which may take the form of a distinguished group of symmetries, or the presence of a distinguished tensor such as a metric tensor or symplectic form.
Despite the necessary presence of coordinates, modern differential geometers have learned to present much of the subject without direct reference to any specific (or even generic) coordinate system. This is called the “invariant” or “coordinate free” formulation of differential geometry.