# The Riemann curvature tensor

# The Riemann curvature tensor

This chapter first examines some fundamental examples where the Riemann curvature tensor has an especially simple form. It then discusses the ways in which the Riemann and Ricci curvatures are used to study questions in geometry and differential topology. Topics covered include spherical metrics, flat metrics, and hyperbolic metrics; the Schwarzchild metric; curvature conditions; the Gauss-Bonnet formula; metrics on manifolds of dimension 2; conformal changes; sectional curvatures and universal covering spaces; the Jacobi field equation; constant sectional curvature and the Jacobi field equation; manifolds of dimension 3; and the Riemannian curvature of a compact matrix group.

*Keywords:*
Ricci curvature, Schwarzchild metric, Gauss-Bonnet formula, Jacobi field equation

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