On a Riemannian manifold, there is a unique connection which is torsion-free and compatible with the metric. This connection is called the Levi-Civita connection.

# Fundamental Theorem of Riemannian Geometry

## See also

Covariant Derivative, Levi-Civita Connection, Riemannian Manifold, Riemannian Metric
*This entry contributed by Todd
Rowland*

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## Cite this as:

Rowland, Todd. "Fundamental Theorem of Riemannian Geometry." From *MathWorld*--A Wolfram Web Resource,
created by Eric W. Weisstein. https://mathworld.wolfram.com/FundamentalTheoremofRiemannianGeometry.html