By Marcel Berger

Riemannian geometry has at the present time turn into an unlimited and significant topic. This new booklet of Marcel Berger units out to introduce readers to lots of the residing themes of the sector and bring them fast to the most effects recognized up to now. those effects are said with out targeted proofs however the major principles concerned are defined and influenced. this allows the reader to acquire a sweeping panoramic view of virtually the whole lot of the sphere. notwithstanding, due to the fact that a Riemannian manifold is, even at the beginning, a sophisticated item, attractive to hugely non-natural options, the 1st 3 chapters dedicate themselves to introducing a number of the innovations and instruments of Riemannian geometry within the so much typical and motivating manner, following particularly Gauss and Riemann.

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**Example text**

Every geodesic from one umbilic goes after a time T to the other point of the pair. 4). Caution: if you take a second lap for time T you will come back to your original point, but in general with a diﬀerent direction, unlike on the sphere. This is of course completely proven. 9. 6 The Geometry of Surfaces Before and After Gauß 39 (a) (b) Fig. 44. 5 for the general deﬁnition of the cut locus and details). The cut locus of a point p is the closure of the set of points which can be joined to p by more than one segment.

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). Simplicity of the curve is necessary 20 1 Euclidean Geometry here.

29. A curve undergoing heat shrinking that two immersed (not embedded in general) plane curves with the same turning number can be deformed one into the other through proper immersions. 8 of Berger & Gostiaux 1988 [175]). Very recently, Vladimir I. Arnol d started a revolution when studying plane curves, hammering out a general frame to encompass these results. This is a very active ﬁeld of research today. There is a price to pay of course. It is quite reasonably expensive. The curve has to be considered together with its tangent lines, so that the object to study is the set of all oriented tangent directions to the Euclidean plane, an object of dimension three (not two) and inside it the curve consisting of the tangents of a given plane curve.