April 2016
Intermediate to advanced
430 pages
8h 58m
English
Content preview from Differential Geometry of Curves and Surfaces, 2nd Edition
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CHAPTER 4
Curves in Space: Global Properties
Paralleling our presentation of curves in the plane, we now turn from local properties of space curves to global properties. As before, global properties of curves are properties that involve the curve as a whole as opposed to properties that are defined in the neighborhood of a point on the curve.
The Jordan Curve Theorem does not apply to curves in ℝ3, so Green’s Theorem, the isoperimetric inequality, and theorems connecting curvature and convexity do not have an equivalent for space curves. On the other hand, curves in space exhibit new types of global properties, in particular, knottedness and linking.
4.1 Basic Properties
Definition 4.1.1. A parametrized space curve C is called closed ...
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