3 - Differential geometry of surfaces in E3

Abstract

This chapter focuses on the surfaces embedded in three-dimensional Euclidean space, which may be represented by a variety of mathematical expressions in terms of rectangular cartesian co-ordinates x, y, z. Intrinsic properties of surfaces are those which can be formulated without reference to the space in which they are embedded, in particular, without reference to vectors normal to the surface. These properties depend essentially on a certain quadratic differential form, the so-called first fundamental form, which contains all the basic information about the metrical properties of the surface. The aspect of the intrinsic geometry of surfaces arises from the problem of determining the curve of minimum length that joins two given points on the surface. For example, when the surface is a plane in E3, the shortest distance between two points will be a straight line. If the general problem is approached through the usual calculus of variations, differential equations are obtained, which any solution must satisfy. Curves which satisfy these equations are called geodesics, although not all solutions are necessarily curves of minimum length.