By Dmitri Burago, Yuri Burago, Sergei Ivanov
"Metric geometry" is an method of geometry in accordance with the concept of size on a topological house. This procedure skilled a really speedy improvement within the previous few many years and penetrated into many different mathematical disciplines, equivalent to staff concept, dynamical platforms, and partial differential equations. the target of this graduate textbook is twofold: to provide an in depth exposition of simple notions and strategies utilized in the idea of size areas, and, extra regularly, to supply an simple creation right into a huge number of geometrical issues relating to the suggestion of distance, together with Riemannian and Carnot-Caratheodory metrics, the hyperbolic aircraft, distance-volume inequalities, asymptotic geometry (large scale, coarse), Gromov hyperbolic areas, convergence of metric areas, and Alexandrov areas (non-positively and non-negatively curved spaces). The authors are inclined to paintings with "easy-to-touch" mathematical gadgets utilizing "easy-to-visualize" equipment. The authors set a demanding aim of creating the center elements of the ebook available to first-year graduate scholars. so much new options and techniques are brought and illustrated utilizing easiest circumstances and keeping off technicalities. The e-book comprises many routines, which shape an integral part of exposition.
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Additional info for A Course in Metric Geometry (Graduate Studies in Mathematics, Volume 33)
Nevertheless, the idea was to try, by actual ruler and compass constructions in the plane, to recapture one by one the various constructions involved in the three dimensional picture. Greek geometers could do this for many other three dimensional problems, so why not for this one? They did not know it was hopeless . . However, this attempt of Archytas provided an opportunity to focus on important surfaces: the cone, the cylinder and the torus. A more elegant “solution” was proposed by Menaechmus around 350 BC.
This supposes implicitly (in modern language) that the ratio of the measures of two given segments is always a rational number: a fact which we know today to be false. It was perhaps the Pythagorean Hippasus, around 420 BC, who discovered the existence of incommensurable magnitudes: two magnitudes that you cannot possibly measure with the same unit segment. This discovery destroyed a large part of Greek geometry. In any case all results depending on Thales’ theorem, and in particular on the widely used theory of similar triangles, were to be called into question again.
Thus in contemporary terms 1 πR 2 = circumference × R 2 38 2 Some Pioneers of Greek Geometry Fig. 24 from which we get the formula 2πR for the length of the circumference. We shall come back to this problem in Sect. 1, since the first deductive proof of this result is generally attributed to Archimedes. Notice that this last observation proves in particular that the same “number” (or “ratio”) π is to be used for computing both the area and the length of every circle, whatever its diameter. Eudoxus was certainly aware of this fact.
A Course in Metric Geometry (Graduate Studies in Mathematics, Volume 33) by Dmitri Burago, Yuri Burago, Sergei Ivanov