Differentialgeometrie von Kurven und Flächen

In differential geometry of curves and surfaces, there are two perspectives. One, which could be called classical differential geometry, emerged alongside the beginnings of differential and integral calculus. Broadly speaking, classical differential geometry studies the local properties of curves and surfaces. Here, we understand local properties as those that depend only on the behavior of the curve or surface in the vicinity of a point. The methods that have proven suitable for studying such properties are the methods of differential calculus. For this reason, the curves and surfaces examined in differential geometry are defined by functions that belong to a certain class of differentiability. The other perspective is known as global differential geometry. In this approach, one investigates the influence of local properties on the behavior of the entire curve or surface. The most interesting and representative part of classical differential geometry is likely the study of surfaces. However, when studying surfaces, some local properties of curves naturally arise. Therefore, we use this first chapter to briefly address curves.