Riemannian Metrics

Abstract

The intrinsic geometric features of an n-surface S depend only on dot products of vectors tangent to S and derivatives along parametrized curves in S of functions obtained as dot products of vector fields tangent to S along these curves. In other words, given the dot product on each tangent space S _p , p ∈ S, the intrinsic geometry of S can be studied without reference to the way in which S sits in ℝ^{n +1}. If we are told what the dot product is on each S _p then we can compute, for example, the lengths of curves in S, the volume of S, the geodesics in S, parallel transport along curves in S, and the Gauss-Kronecker curvature of S (if n is even) without any knowledge of how S curves around in ℝ^{n +1}. In fact, if we are given a dot product on each tangent space S _p different from the one which comes from ℝ ⁿ⁺¹ _p , we can still do these intrinsic computations but of course the results of our computations will depend on the dot products used, and the geometry we find will in general be quite different from the geometry we are familiar with. The geometry obtained from such dot products is called Riemannian geometry; the collection of dot products on the tangent spaces S _p from which the geometry is derived is called a Riemannian metric.