= Fundamental theorem of Riemannian geometry
A <Riemannian metric> determines a unique <torsion-free connection> that is a <metric connection>. Metric compatibility and the torsion identity imply the <Koszul formula>, which proves uniqueness. Conversely the right-hand side of the <Koszul formula> is a smooth one-form in its last vector-field argument. The <musical isomorphism> defines $D_XY$ from it; expanding the <Lie bracket of vector fields> verifies the connection rules, metric compatibility and zero torsion. Thus it constructs the <Levi-Civita connection> intrinsically.
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