Differential Geometry

Abstract

Differential geometry is the study of curves (both plane and space curves) and surfaces by means of the calculus. When investigating geometric configurations (on the basis of their equations) in differential geometry, we aim mostly at the study of invariant properties, i.e., properties independent of the choice of the coordinate system and so belonging directly to the curve or surface (e.g. the points of inflexion, the curvature and so on). But we also study those properties of geometric configurations that depend on the choice of the coordinate system (e.g. the sections of a surface by the coordinate planes, the slope of the tangent and so on). Differential geometry studies mostly the local properties of curves and surfaces, i.e. those which pertain to sufficiently small portions of the curve or the surface; so it is essentially “geometry in the small”. But differential geometry also investigates those properties of curves and surfaces which pertain to the configuration as a whole (e.g. the length of a curve, the number of vertices and so on).