Differential geometry

The aim of this article is to introduce the basics of the local differential geometry of curves and surfaces in Euclidean space. Initially, we will study the theory of curves, which will then serve as a basis for the theory of surfaces. In both cases, for curves and surfaces, we will follow the same thread. First, we'll introduce the parametric representation of curves and surfaces on which the study of differential geometry is based. It is within this framework that we shall introduce the important notion of the geometric magnitude of a curve or surface, to which we shall dedicate the remainder of this article in order to characterize curves and surfaces. In particular, we'll look at the metric properties and curvature of curves and surfaces. The study will conclude with the presentation of the fundamental theorem of the theory of curves, respectively surfaces, which provides a means of characterizing and distinguishing curves, respectively surfaces, as well as reconstructing them from certain characteristic data.

More specifically, the points addressed in this dossier are as follows:

• parametric representation of curves and surfaces, leading to definitions of regular and singular points, as well as changes in parameters and geometric magnitudes;

• metric properties of curves and surfaces, in particular the notion of curvilinear abscissa for curves and that of the first fundamental form for surfaces;

• notions of curvature, in particular the magnitudes curvature and torsion for curves and the second fundamental form, normal curvature, principal curvatures, Gauss curvature and mean curvature for surfaces.

Many theoretical and practical disciplines use these results, see for example .

So, as far as theory is concerned, there are interactions between differential geometry and other fields of mathematics, such as analysis , the theory of differential equations , variational calculation