Riemann Curvature and Ricci Curvature

  • Xinyue Cheng
  • Zhongmin Shen

Abstract

Curvatures are the central concept in geometry. The notion of curvature introduced by B. Riemann faithfully reveals the local geometric properties of a Riemann metric. This curvature is called the Riemann curvature in Riemannian geometry. The Riemann curvature can be extended to Finsler metrics as well as the sectional curvature. In this chapter, we will give a local formula for the Riemann curvature of a Randers metric. Then we shall also study the relationship between the flag curvature and some non-Riemannian geometric quantities.

Keywords

Sectional Curvature Riemannian Geometry Ricci Curvature Riemann Curvature Finsler Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Science Press Beijing and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Xinyue Cheng
    • 1
  • Zhongmin Shen
    • 2
  1. 1.School of Mathematics and StatisticsChongqing University of TechnologyLijiatuo, ChongqingChina
  2. 2.Department of Mathematical SciencesIndiana University-Purdue University Indianapolis (IUPUI)IndianapolisUSA

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