Surfaces in 3D-Spaces

Abstract

This chapter is devoted to the Differential Geometry of a surface in a 3-space. We need to know basic calculus. All functions which appear from now are smooth, i.e. they are indefinitely differentiable functions in one or several variables at each point of their domain of definition. First, we see surfaces in an Euclidean 3-dimensional space and we understand how the Euclidean inner product induces, via the first fundamental form, a way to measure lengths and angles for vectors belonging to tangent planes to the surface. We can also measure lengths of curves who belong to surfaces, areas of regions and the Gaussian curvature of a surface at each point. If at beginning, the curvature seems to be dependent on the embedding in the ambient Euclidean space, after we prove Gauss’ formulas, we step into the intrinsic theory of surfaces where Gauss’ equations and the Theorema Egregium offer another perspective: the surfaces can be seen as pieces of a plane endowed with a metric, and this metric only determines the curvature. In Minkowski 3-spaces we have the same picture, the Minkowski product determines a non-Euclidean metric of a surface which allows us to conclude about the intrinsic Geometry of it. Therefore, in both cases the surface becomes irrelevant for our study. In fact we study the Geometry of a metric and we obtain relevant geometric aspects about the piece of plane endowed with that metric. This point of view will be continued in the next chapter when we better understand the nature of geometric objects which appear in Differential Geometry. Both chapters regarding Differential Geometry were adapted using ideas from [27–30].

Ab initio res.