Download Comparison Theorems in Riemannian Geometry by Jeff Cheeger and David G. Ebin PDF

By Jeff Cheeger and David G. Ebin

The significant topic of this e-book is the interplay among the curvature of an entire Riemannian manifold and its topology and international geometry. the 1st 5 chapters are preparatory in nature. they start with a truly concise advent to Riemannian geometry, via an exposition of Toponogov's theorem--the first such remedy in a e-book in English. subsequent comes an in depth presentation of homogeneous areas within which the most objective is to discover formulation for his or her curvature. a brief bankruptcy of Morse idea is by way of one at the injectivity radius. Chapters 6-9 care for a number of the such a lot correct contributions to the topic within the years 1959 to 1974. those comprise the pinching (or sphere) theorem, Berger's theorem for symmetric areas, the differentiable sphere theorem, the constitution of entire manifolds of non-negative curvature, and at last, effects in regards to the constitution of whole manifolds of non-positive curvature. Emphasis is given to the phenomenon of tension, specifically, the truth that even if the conclusions which carry lower than the idea of a few strict inequality on curvature can fail while the stern inequality on curvature can fail while the stern inequality is comfortable to a susceptible one, the failure can take place merely in a constrained manner, that may frequently be categorized as much as isometry. a lot of the fabric, really the final 4 chapters, was once primarily cutting-edge whilst the ebook first seemed in 1975. on account that then, the topic has exploded, however the fabric lined within the booklet nonetheless represents a necessary prerequisite for someone who desires to paintings within the box.

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From this point of view Diophantine analysis may be regarded as having constituted a hindrance to the development of coordinate geometry ; but in another respect it was a See Max Dehn, "Historische 'Obersicht," in A. Schoenflies and M . , Berlin ( 1931 )), p. 379-393. 11 See , e. , 0. Neugebauer and A. Sachs, "Mathematical Cuneiform Texts" (A merican Oriental Series, v. XXI X ), New Haven, Conn. ( 1945), passim. • See T. L. ntus of A lemndria, Cambridge ( 1910), p. 129. u tlu � 36 HISTORY OF ANALYTIC GEOMETRY step forward.

22 Long afterwards, the in­ sistence on the use of the simplest possible means appropriate to a given geometrical problem was strongly emphasized in the analytic geometry of Descartes and his successors . lack 11 J. H. Weaver, "On Foci of Conics," Bulletin, A merican Matliematical Society, ( 1916-1917), p. 357--365. 11 See T'he Works of A rchimedes (Heath ), p. lxvi i . v. XXIII THE ALEXANDRIAN AGE 37 To Pappus one owes also the clearest statement in antiquity on the nature of analysis. Greek geometry was divided into three parts : the elements (as found in Euclid) ; practical geometry or geodesy (rep­ resented by Heron) ; and higher geometry (illustrated by Apollonius and Archimedes) .

Smith, however, places Perseus In the middle of the second century. See History of Mathematics, v. I , p. 1 18. 34 HISTORY OF ANALYTIC GEOMETRY flexibility and extent of the modem treatment shows surprising nar­ rowness of the ancient point of view. Inspired by the Pythagoreans, they had found number everywhere in nature, but they overlooked much of the geometric beauty which natural phenomena afford. Aesthetically one of the most gifted people of all times, the only curves which they found in the heavens and on the earth were circles and straight lines.

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