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Riemannian Manifolds as Metric Spaces and the Geometric Meaning of Sectional and Ricci Curvature

In: A Panoramic View of Riemannian Geometry

Author

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  • Marcel Berger

    (Institut des Hautes Études Scientifiques IHES)

Abstract

We want to study the metric of a Riemannian manifold. The first tasks to address are: 1. to compute the metric d as defined by equation 4.13 on page 174 (namely itd (p, q) is the infimum of the lengths of curves connecting p to q) 2. to determine if there are curves realizing this distance (called segments or shortest paths or minimal geodesics according to your taste) and 3. to study them.

Suggested Citation

  • Marcel Berger, 2003. "Riemannian Manifolds as Metric Spaces and the Geometric Meaning of Sectional and Ricci Curvature," Springer Books, in: A Panoramic View of Riemannian Geometry, chapter 6, pages 221-297, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-18245-7_6
    DOI: 10.1007/978-3-642-18245-7_6
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