Suppose for every point in a manifold , an inner product is defined on a tangent space of at . 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 alsoCompact Manifold, Line Element, Metric Tensor, Minkowski Metric, Riemannian Geometry, Riemannian Manifold
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Weisstein, Eric W. "Riemannian Metric." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RiemannianMetric.html