By Marcel Berger

ISBN-10: 3540653171

ISBN-13: 9783540653172

This booklet introduces readers to the residing subject matters of Riemannian Geometry and info the most effects recognized thus far. the implications are said with out certain proofs however the major rules concerned are defined, affording the reader a sweeping panoramic view of virtually the whole thing of the sphere.

From the stories ''The booklet has intrinsic price 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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**Additional info for A Panoramic View of Riemannian Geometry**

**Example text**

4. 1 Euclidean Geometry 32 Fig. 31. 6 The Geometry of Surfaces Before and After Gauß 33 Fig. 32. 6 The Geometry of Surfaces Before and After Gauß Unlike for curves, we will only deﬁne geometric surfaces in the space E3 (see why for yourself). Geometric curves are everywhere locally diﬀerentiable nice one-dimensional subsets in E2 , while surfaces are the nice subsets of E3 , nice in the sense that there is a local map from E3 to itself sending the local piece of the surface onto an open subset of a plane.

27 on the facing page). 4 Global Theory of Closed Plane Curves 27 Fig. 27. 4 Umlaufsatz with Corners An important remark is in order, which might have already occured to the reader. Consider a triangle in the plane and think of it “` a la Umlaufsatz ”, that is to say: drag the unit tangent vector along as you traverse each side. As a vector it keeps being constant but when you arrive at a vertex of angle A you have to turn by exactly π − A. Coming back to the origin you have ﬁnally turned from (π − A) + (π − B) + (π − C) = 3π − (A + B + C) = 3π − π = 2π As you might guess there is a formula covering both cases, namely for curves with reasonable singularities.

A famous problem is to prove that in any plane closed curve, one can inscribe a square. For this problem, and other open ones, see the book Croft, Falconer & Guy 1994 [410]. e. points so that all chords through them have constant length. 3, chapter 6 of Gardner 1995 [548]. 4. 2 Curvature We now look at the outer geometry of plane curves. The inner geometry of curves does not diﬀer from that of straight lines, but the geometry is radically diﬀerent if we look at the way a curve sits in the plane.

### A Panoramic View of Riemannian Geometry by Marcel Berger

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