Riemannian Metric

Suppose for every point $x$ in a Compact Manifold $M$, an Inner Product $\left\langle{\cdot,\cdot}\right\rangle{}_x$ is defined on a Tangent Space $T_xM$ 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.

See also Compact Manifold, Line Element, Metric Tensor