Modern study of varieties
Differential varieties

The previous study of curves and surfaces highlighted two aspects. On the one hand, they are parts of a normed vector space; on the other, they have an internal structure that sometimes allows us to ignore the way in which they are immersed in the surrounding space. For example, when studying arcs traced on a surface, it's best to represent them using coordinates deduced from a surface parameterization. Let's take the case of a sphere at