Riemannian metric

Summary

A Riemannian metric on a smooth manifold is a smoothly varying inner product on each tangent space. It lets one measure lengths of tangent vectors and angles between them, and from these the lengths of curves, distances, areas and volumes. It is a symmetric positive definite tensor field of type $(0, 2)$. Every smooth manifold admits Riemannian metrics, and the choice of one is the starting point of Riemannian geometry.