chapter  Chapter 14
34 Pages

Differential geometry

ByJames H. Luscombe

This chapter establishes the mathematics of curvature. Our intuitive notions of curvature are based on experience with two-dimensional surfaces embedded in three-dimensional space. The extrinsic curvature is described in terms of quantities available in the space in which the surface is embedded, such as radius of curvature. The intrinsic curvature of a manifold is specified in terms of its attributes without reference to an embedding space. Intrinsic curvature is defined in terms of two interrelated mathematical concepts: parallel transport and the covariant derivative. Derivative operators on manifolds are never unique: One can choose any type tensor field and get another derivative. The collection of derivative operators on a manifold almost has the structure of a vector space. Geodesics are a class of curves of particular importance to general relativity. Manifolds have plenty of derivative operators, with any one easily obtainable from another.