By Marcel Berger
This e-book introduces readers to the dwelling themes of Riemannian Geometry and information the most effects identified to this point. the consequences are said with out special proofs however the major rules concerned are defined, affording the reader a sweeping panoramic view of just about the whole thing of the sector.
From the studies ''The publication has intrinsic worth for a pupil 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
Consequently, all curves having the same length are isometric, namely they are the same as metric spaces. e. preserves distances). Moreover all of them are isometric to an interval of the same length on the Euclidean line E1 . 1. Alternately, look at the triangle inequality. For the inner metric of a plane curve, at least locally, d (p, r) = d (p, q) + d (q, r) for any three points p, q, r with q between p and r. 11. Or think about the word “between”. If we call a curve closed when it is periodic, then such a curve has a total length, and two closed curves are isometric exactly when they have the same length.
Then one can deﬁne an angle α(t) for c (t) which is a real number, deﬁned only up to integer multiples of 2π. The curvature is K(t) = dα/dt. To get rid of the 2π ambiguity, there is an important fact which we are going to use below. For regular curves, when t runs through the interval of deﬁnition, one can follow α(t) by continuity and then deﬁne a map t → α(t) into R. Note that if the curve is closed with period T then α(t + T ) will in general diﬀer from α(t) by an integral multiple of 2π. 3 Plane Curves 15 Fig.
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 . One might also consult chapter 12 of Berger 1987 . For the plane, and arbitrary dimensions as well, there is the excellent reference Burago & Zalgaller 1988 . To our knowledge, it is the only source of not just one but all of the classical proofs of the isoperimetric inequality, handling the cases of equality with minimal regularity assumptions on the boundary.