Suppose for every point x in a manifold M, an inner product 〈·, ·〉_x is defined on a tangent space T_x M of M at x. Then the collection of all these inner products is called the Riemannian metric. In 1870, Christoffel and Lipschitz showed how to decide when two Riemannian metrics differ by only a coordinate transformation.

compact manifold | line element | metric tensor | Minkowski metric | Riemannian geometry | Riemannian manifold

Bernhard Riemann