Differential varieties

Classical differential geometry deals with curves and surfaces in Euclidean space from the point of view of differential calculus. Concepts studied include tangents to curves, tangent planes to surfaces, curvature, lengths and areas, vector fields and their integral curves.

This elementary view of curves and surfaces soon proves inadequate when faced with the need to consider sets of points that depend on any number of parameters. If this idea is properly clarified, we arrive at the notion of differential variety, which forms the basis of modern differential geometry.

In this article, we'll first look at the properties of curves and surfaces, then at general notions linked to the structure of differential varieties.