ABSTRACT
Chapter 6 delves into the local theory of regular surfaces in space. The first fundamental form, also called the metric tensor, allows us to define many geometric concepts such as arclength, angles between curves, and areas of regions. As an application, we consider the science of cartography and implications for the metric of a flat map of a region of a sphere to preserve areas or angles. After introducing the Gauss map of a surface, also called the shape operator, we introduce the second fundamental form that gives a measure of how much a curve on a surface bends at a point p, when it points in a tangent direction https://www.w3.org/1998/Math/MathML"> v → https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003295341/9b6c05fb-09a1-4e60-8c67-4ab3235254ec/content/unequ01-01a.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> From the differential of the Gauss map, we introduce the concept of principal curvature, principal directions, mean curvature H, and Gaussian curvature K of a surface at a point. These concepts encode all the information about how a surface bends and in what directions. Finally, considering the particular cases of surfaces with Gaussian curvature constantly 0 or mean curvature constantly 0, we introduce developable surfaces and the deep topic of minimal surfaces.
