Download Comparison geometry by Karsten Grove, Peter Petersen PDF

By Karsten Grove, Peter Petersen

This ebook records the new specialise in a department of Riemannian geometry referred to as comparability Geometry. the easy suggestion of evaluating the geometry of an arbitrary Riemannian manifold with the geometries of continuing curvature areas has obvious a huge evolution lately. This quantity is an up to date mirrored image of the hot improvement concerning areas with reduce (or two-sided) curvature bounds. The content material displays probably the most fascinating actions compared geometry in the course of the yr and particularly of the Mathematical Sciences learn Institute's workshop dedicated to the topic. This quantity positive aspects either survey and examine articles. It additionally offers whole proofs: in a single case, a brand new, unified technique is gifted and new proofs are provided. This quantity might be a precious resource for complex researchers and people who desire to know about and give a contribution to this gorgeous topic.

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74 (1961), 391–466. [Synge 1936] J. Synge, “On the connectivity of spaces of positive curvature”, Quart. J. Math. ) 7 (1936), 316–320. [Taimanov 1996] I. Taimanov, “A remark on positively curved manifolds of dimensions 7 and 13”, preprint, Novosibirsk, 1996. INJECTIVITY RADIUS ESTIMATES AND SPHERE THEOREMS 47 [Valiev 1979] F. M. Valiev, “Sharp estimates of sectional curvatures of homogeneous Riemannian metrics on Wallach spaces” (Russian), Sibirsk. Mat. Zh. 20:2 (1979), 248–262, 457. [Wallach 1972] N.

The Geometrization Conjecture may be viewed as a question about the existence of canonical or distinguished Riemannian metrics on 3-manifolds that satisfy certain topological conditions. This type of question has long been of fundamental interest to workers in Riemannian geometry and analysis on manifolds. For instance, it is common folkore that Yamabe viewed his work on what is now known as the Yamabe problem [1960] as a step towards the resolution of the Poincar´e conjecture. Further, it has been a longstanding open problem to understand the existence and moduli space of Einstein metrics (that is, metrics of constant Ricci curvature) on closed n-manifolds.

There are two basic topological decompositions of M , obtained by examining the structure of the simplest types of surfaces embedded in M , namely spheres and tori. 1 (Sphere Decomposition [Kneser 1929; Milnor 1962]). Let M be a closed, oriented 3-manifold. Then M has a finite decomposition as a connected sum M = M1 # M2 # · · · # Mk , where each Mi is prime. The collection {Mi } is unique, up to permutation of the factors. ) This sphere decomposition (or prime decomposition) is obtained by taking a suitable maximal family of disjoint embedded two-spheres in M , none of which bounds a three-ball, and cutting M along those spheres.

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