Curvature in Riemannian Manifolds

Abstract

Since the notion of curvature can be defined for curves and surfaces, it is natural to wonder whether it can be generalized to manifolds of dimension n ≥ 3. Such a generalization does exist and was first proposed by Riemann. However, Riemann’s seminal paper published in 1868 two years after his death only introduced the sectional curvature, and did not contain any proofs or any general methods for computing the sectional curvature. Fifty years or so later, the idea emerged that the curvature of a Riemannian manifold M should be viewed as a measure R(X, Y )Z of the extent to which the operator (X, Y )↦∇X∇YZ is symmetric, where ∇ is a connection on M (where X, Y, Z are vector fields, with Z fixed). It turns out that the operator R(X, Y )Z is C∞(M)-linear in all of its three arguments, so for all p ∈ M, it defines a trilinear map $$\displaystyle R_p{\colon } T_p M \times T_p M \times T_p M \longrightarrow T_p M. $$ The curvature operator R is a rather complicated object, so it is natural to seek a simpler object. Fortunately, there is a simpler object, namely the sectional curvature K(u, v), which arises from R through the formula $$\displaystyle K(u, v) = \langle R(u, v)u, v{\rangle }, $$ for linearly independent unit vectors u, v. When ∇ is the Levi-Civita connection induced by a Riemannian metric on M, it turns out that the curvature operator R can be recovered from the sectional curvature. Another important notion of curvature is the Ricci curvature, Ric(x, y), which arises as the trace of the linear map v↦R(x, v)y. The curvature operator R, sectional curvature, and Ricci curvature are introduced in the first three sections of this chapter.