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