Download Geometry of Surfaces (Universitext) by John Stillwell PDF

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By John Stillwell

The geometry of surfaces is a perfect start line for studying geometry, for, between different purposes, the idea of surfaces of continuing curvature has maximal connectivity with the remainder of arithmetic. this article offers the coed with the information of a geometry of larger scope than the classical geometry taught this present day, that is now not an enough foundation for arithmetic or physics, either one of that are turning into more and more geometric. It comprises workouts and casual discussions.

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Extra resources for Geometry of Surfaces (Universitext)

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The resulting surface cannot lie in ordinary three-dimensional space without intersecting itself, though a fairly repre. sentative part of it can. 5). The twisted cylinder is obtained by prolonging the transverse line segments of M, just as the strip S is obtained by prolonging the transverse line segments of R. This intuitive picture gives a glimpse of some of the curious properties of the twisted cylinder, which the reader may already know from the Mobius band (see Exercises). However, even more so than in the case of the ordinary cylinder, it is formally easier to define the twisted cylinder as a quotient ]R2/r.

But our coverings really are special in a second respect, inasmuch as the covering surface is always the euclidean plane. It is quite possible to define coverings by other surfaces. For example, a torus can be covered by a cylinder. The plane enjoys the distinction of being able to cover any euclidean surface, and for that reason it is called the universal covering surface. The concept of universal covering is usually developed in the topological setting, where it is shown that any reasonable space has a universal covering.

3 gives an intuitive picture of the orbit map by representing an orbit (set of stars in ]R2) belonging to the abstract cylinder ]R2/r by a point (single star) belonging to a concrete cylinder in space. r sends all the stars in ]R2 to the single star on the cylinder and, therefore, "wraps the plane around the cylinder". For this reason, r· is called a covering of]R2 /r by ]R2. The local isometry property of the orbit map means that, within discs of diameter < 1/2, the geometry ofthe cylinder is the same as the geometry of the plane.

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