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Elementary Differential Geometry by Christian Bär

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2    Curve theory

We analyse curves in n-dimensional space with a special focus on plane curves and space curves. Length, curvature and torsion are introduced. We prove Hopf’s Umlaufsatz for simple closed curves, characterise convex curves and derive the four-vertex theorem. The isoperimetric inequality, which compares the length of a simple closed plane curve with the enclosed area, is proved using the Fourier series. We show that for a given curvature and torsion the resulting space curve is unique up to a Euclidean motion. We investigate how much a space curve needs to curve if it is closed and make the result even stronger in the case that the space curve is knotted.

2.1    Curves in n

We now want to use the tools of differentiation and integration ...

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