By Marcel Berger

Riemannian geometry has at the present time develop into an unlimited and demanding topic. This new ebook of Marcel Berger units out to introduce readers to many of the dwelling themes of the sector and bring them speedy to the most effects identified thus far. those effects are acknowledged with no precise proofs however the major rules concerned are defined and stimulated. this permits the reader to procure a sweeping panoramic view of virtually everything of the sector. in spite of the fact that, on the grounds that a Riemannian manifold is, even firstly, a sophisticated item, attractive to hugely non-natural techniques, the 1st 3 chapters dedicate themselves to introducing some of the suggestions and instruments of Riemannian geometry within the so much common and motivating approach, following particularly Gauss and Riemann.

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Gromoll and K. Grove, “The low-dimensional metric foliations of Euclidean spheres”, J. Diff. Geom. 28 (1988), 143–156. [Gromoll and Meyer 1969] D. Gromoll and W. Meyer, “On differentiable functions with isolated critical points”, Topology 8 (1969), 361–369. [Gromoll and Meyer 1974] D. Gromoll and W. Meyer, “An exotic sphere with nonnegative sectional curvature”, Ann. of Math. 100 (1974), 401–408. INJECTIVITY RADIUS ESTIMATES AND SPHERE THEOREMS 45 [Gromov 1981a] M. Gromov, Structures m´ etriques pour les vari´ et´es riemanniennes, edited by J.

3 with some results from algebraic topology. This reduction had already been known to Berger [1962a]. Complete proofs are given in [Abresch and Meyer a]. Here we list only the basic steps. 2 directly. 4 [Berger 1962a, Proposition 2]. 3. Then any compact Riemannian manifold M n satisfying δhs ≤ KM ≤ 1 and π π ≤ inj M n ≤ diam M n ≤ √ 2 δhs admits a continuous, piecewise smooth map f : RPn → M n of degree 1. Here deg f denotes the standard integral mapping degree if M n is odd-dimensional and orientable.

His argument was based on a detailed investigation of the geometry of Dirichlet cells in the universal covering of M n , and relied on the hypothesis π1 (M n ) = 0. 3 is very different. It requires some refined Jacobi field estimates, which might be useful in other contexts as well. Our plan is to explain these Jacobi field estimates in the next section and describe the proof of the horseshoe inequality in Section 7. 3 with some results from algebraic topology. This reduction had already been known to Berger [1962a].