Curvature

Abstract

The local geometry of a general Riemannian manifold differs from the flat geometry of the Euclidean space $\mathbb {R}^n$: for example, the internal angles of a geodesic triangle in the $2$-sphere $S^2$ (with the standard metric) always add up to more than $\pi$. A measure of this difference is provided by the notion of curvature, introduced by Gauss in his 1827 paper “General investigations of curved surfaces”, and generalized to arbitrary Riemannian manifolds by Riemann himself (in 1854). It can appear under many guises: the rate of deviation of geodesics, the degree of non-commutativity of covariant derivatives along different vector fields, the difference between the sum of the internal angles of a geodesic triangle and $\pi$, or the angle by which a vector is rotated when parallel-transported along a closed curve. This chapter addresses the various characterizations and properties of curvature.