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Статья «A panoramic glimpse of manifolds with sectional curvature bounded from below, "Алгебра и анализ"»

Авторы:
  • GROVE K.1
стр. 7-48
Платно
1 University of Notre Dame
  • SDI: 007.001.0234-0852.2017.029.001.2
Ключевые слова:
  • ower curvature bound
  • Toponogov distance comparison theorem
  • Bishop Gromov volume comparison theorem
  • GromovAHausdorfF topology
Аннотация:
Rather than providing a comprehensive survey on manifolds curved from below, the paper is aimed at exhibiting and discussing some of the main ideas and tools that have been developed over decades. For the same reason, only a relatively small sample of results is presented to illustrate this development and in doing this, simplicity is emphasized over generality. In the same vein, at most a glimpse of an idea or strategy of a proof is given. The search for relations between geometry and topology of Riemannian manifolds has been at the center of global Riemannian geometry since its beginning. On the geometric side, a large part of this line of investigations has been dominated by the role of curvature in various forms. For natural and geometrically appealing reasons, early efforts were focussed on manifolds whose curvature does not change sign, i.e., with lower or upper curvature bound 0, and in particular with positive or negative curvature. It took several decades for ideas and tools to emerge breaking the ground for dealing with manifolds with curvatures of both signs. In this survey we will focus our attention on (closed) Riemannian manifolds with a given lower (sectional) curvature bound. In addition to the basic global Toponogov distance comparison theorem, and the Bishop-Gromov volume comparison theorem, the key tools are critical point theory for distance functions, the Gromov-Hausdorff topology, and Alexandrov geometry. Our aim is to discuss these tools as well as more classical tools such the Morse theory of geodesics, and in the case of nonnegative curvature also convexity. The insights

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