By Marcel Berger

ISBN-10: 3540653171

ISBN-13: 9783540653172

This publication introduces readers to the dwelling issues of Riemannian Geometry and info the most effects identified so far. the implications are acknowledged with out distinct proofs however the major principles concerned are defined, affording the reader a sweeping panoramic view of virtually the whole lot of the sector.

From the experiences ''The publication has intrinsic worth for a scholar in addition to for an skilled geometer. also, it truly is a compendium in Riemannian Geometry.'' --MATHEMATICAL REVIEWS

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**Extra info for A Panoramic View of Riemannian Geometry**

**Sample text**

This does not mean that α(t) is monotonically increasing: it means that the unit circle is at the end algebraically covered once. Beware that the same point of the unit circle can be obtained many times but that, for example, if you come back a second time at a value previously passed you will have to come again a third time to “erase” it. To make all this mathematically precise one needs the notions of universal covering, simple connectivity, etc. The Umlaufsatz was essentially known to Riemann but a rigorous proof is pretty hard (try one if not convinced).

Try to prove it your own way before dismissing this result. 24 1 Euclidean Geometry Fig. 24. 2 The Four Vertex Theorem On a simple closed plane curve the curvature k(t) is a continuous function (a periodic one). So it admits at least one maximum and one minimum. At such a point the derivative k (t) vanishes. Points where k (t) = 0 are called vertices (think of ellipses if you are interested in the etymology). So any closed plane curve has at least two vertices. Now try to draw a curve with only two vertices: you will have a hard time (this is no reason not to try it seriously) because the four-vertex theorem asserts precisely that every closed plane curve has at least four vertices.

5 The Isoperimetric Inequality for Curves The isoperimetric inequality for closed plane curves tells us that among all such curves bounding regions of ﬁxed area, the circle, and only the circle, is the shortest. Or, among closed curves with ﬁxed length, the circle encloses the greatest area. Explicitly, if A is the enclosed area, and L the perimeter, L2 ≥ 4πA with equality only for circles. This was known to the Greeks, but the history of the proof is fantastic. 3 of the article by Talenti in Gruber and Wills 1993 [661].

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