By Marcel Berger

This ebook introduces readers to the dwelling themes of Riemannian Geometry and information the most effects recognized so far. the consequences are said with out unique proofs however the major rules concerned are defined, affording the reader a sweeping panoramic view of just about everything of the sphere.

From the reports ''The publication 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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Mathematically, we consider a curve drawn on M —it is forbidden to leave M and to go, even for one second, into the outside world. Because such a curve c lies in E3 it has a length, and of course the surface’s inhabitants want to ﬁnd the shortest path from p to q. We deﬁne the distance from p to q in M as the inﬁmum of the lengths of all curves going from p to q in M , and denote it by dM (p, q) (or d(p, q) if there is no need to be more precise): d(p, q) = dM (p, q) = inf {length(c) | c is a curve in M from p to q} .

1 A 3 2 B θA θB 1 π-2θC a 3 2 1 2 A θA 3 C B b θB Fig. 26. (a) Angles of contact. (b) A spherical particle and convex particle at a ﬂuid–ﬂuid interface. 3 Convexity with Respect to Arc Length Very recently an interesting local condition on a curve appeared in the literature, namely that the radius of curvature should be a concave function of the arc length: d2 r ≤ 0. ds2 As an exercise compute which parts of an ellipse obey it. 3 above, and Wojtkowski 1986 [1274]) for which almost every nonperiodic trajectory is everywhere dense in space as well as in phase (they are called ergodic).

37. 6 The Geometry of Surfaces Before and After Gauß 37 38 1 Euclidean Geometry cartography is a whole world in itself; if interested see chapter 18 of Berger 1987 [164]. Before leaving the sphere, recall an old formula whose importance for the sequel is not to be underestimated. The formula says that for a spherical triangle T with angles A, B, C its area is given by Area(T ) = A + B + C − π. 7) Fig. 38. Spherical areas and angle sums The real history of this formula seems to have come to light only very recently.