Curvature: Differential Geometry

Jost-Hinrich Eschenburg
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Jost-Hinrich Eschenburg: Universität Augsburg, Institut für Mathematik

Chapter 5 in Geometry - Intuition and Concepts, 2022, pp 85-93 from Springer

Abstract: Abstract Differential geometry deals with objects that are no longer “straight”, such as curved lines and surfaces. The curvature, which measures the deviation from a straight line or a plane, is the central concept. While the curvature of a curve is given by a single number at each point, a surface (or hypersurface) requires a symmetric matrix whose eigenvalues are the “principal curvatures” of the surface; the eigenvectors are called principal curvature directions. We will unfold only a small part of this geometry, and only with a view to the following chapter, in which the simplest curved surfaces, the spheres, will play a central role. These will be characterized among all curved surfaces by the property that all tangential directions are principal curvature directions. For this we will study a class of curvilinear coordinate systems in space preserved by the angle-preserving (isogonal) mappings of the following chapter, namely, those in which all coordinate surfaces intersect perpendicularly. The tangents of the intersecting lines are then principal curvature lines for both intersecting coordinate surfaces.

Date: 2022
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DOI: 10.1007/978-3-658-38640-5_5

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