By John Stillwell

The geometry of surfaces is a perfect place to begin for studying geometry, for, between different purposes, the speculation of surfaces of continuing curvature has maximal connectivity with the remainder of arithmetic. this article offers the scholar 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 changing into more and more geometric. It contains workouts and casual discussions.

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**Extra info for Geometry of Surfaces (Universitext)**

**Example text**

If r, r' are rotations of §2, under what circumstances does rr' = r'r? 2. Find a product of three reflections with no fixed points. 4) maps all but one point of§2, the "north pole" N = (0,0,1), which is the center of projection. If we let u, v be the x, y coordinates of the image point P' of P = (x, y, z) under this projection, then the line N P' has parametric equations x = 0 + tu, y = 0 + tv, Z = 1 - t. , t 2(u 2 +v2 + 1) = 2t, from which t = 0 corresponding to N) or t = 2/(u 2 + v 2 + 1). Substituting the latter value in (1), we get 2u 2v u2 + v2 (2) (x, y, z) = ( u 2 + v 2 + l' u2 + v2 + l' u2 + v 2 + 1 .

Enlarging C to C U {oo} is actually not difficult. The result is a sphere, as we will see in Chapter 3. "Counting the same complex number more than once" is the more delicate construction of Riemann surfaces or branched coverings oj the sphere, which we will meet in Chapter 8. Doubly periodic functions such as (2) are called elliptic functions because an early example arose from the formula for the arc length of an ellipse. This formula is actually a (multiple-valued) integral rather than a function, and the early development of the subject was laborious because it focused on elliptic integrals rather than on their inverses, which are elliptic functions.

It is the geometry of angle-preserving maps of C onto itself. 1 Euclid on Manifolds The aim of this chapter is to answer the question: which unbounded surfaces look locally like the euclidean plane ]R2? The question arises because ]R2 is intended to model "fiat" surfaces in the real world; yet all physical fiat surfaces are of finite extent and have boundaries. It is not clear that such a surface would resemble ]R2 when extended indefinitely, even if small parts of it matched small parts of]R2 with absolute precision.