By Marcel Berger
Riemannian geometry has this present day develop into an enormous and demanding topic. This new ebook of Marcel Berger units out to introduce readers to many of the residing themes of the sphere and produce them fast to the most effects identified to this point. those effects are said with no unique proofs however the major rules concerned are defined and influenced. this permits the reader to acquire a sweeping panoramic view of just about everything of the sector. in spite of the fact that, on account that a Riemannian manifold is, even at the start, a sophisticated item, beautiful to hugely non-natural options, the 1st 3 chapters dedicate themselves to introducing a few of the strategies and instruments of Riemannian geometry within the so much common and motivating approach, following specifically Gauss and Riemann.
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Additional info for A Panoramic View of Riemannian Geometry
8 we see that Lemma 3 The geodesic curvature of a curve on a surface depends only on the inner metric of the surface, not on how the surface is embedded in E3 . c'(s) c f(s) c"(s) M n(s) Fig. 40. Variation of a curve on a surface As before we deduce from this formula that the curve c can be shortest only if its geodesic curvature vanishes identically: kg (t) = 0 for every t. This amounts exactly to ask the curve c to have acceleration normal to the surface. This condition was known to Jacob Bernoulli I, who taught it to his student Euler in the beginning of the 18th century.
Counting periodic geodesics is diﬀerent from running only once, but it is not geometrically distinct. For example when working on counting functions one should be careful to distinguish between the counting function for geometric periodic geodesics and that for parameterized ones. The question is diﬃcult because the standard ways to prove existence of periodic geodesics consider them as motions. We will devote an entire chapter to this (and more) in the case of a general compact manifold, namely chapter 10.
Try to prove it your own way before dismissing this result. 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.
A Panoramic View of Riemannian Geometry by Marcel Berger