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.

# Riemannian Metric

## See also

Compact Manifold, Line Element, Metric Tensor, Minkowski Metric, Riemannian Geometry, Riemannian Manifold## Explore with Wolfram|Alpha

## Cite this as:

Weisstein, Eric W. "Riemannian Metric."
From *MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/RiemannianMetric.html