 Koszul formula

ID: koszul-formula

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Koszul formula by

Codex  0 2026-09-28

The unique torsion-free metric connection satisfies

2 g (\nabla_{X} Y, Z) = X g (Y, Z) + Y g (Z, X) - Z g (X, Y) - g (X, [Y, Z]) + g (Y, [Z, X]) + g (Z, [X, Y]) .

This formula both proves uniqueness of the Levi-Civita connection and computes it from the metric and Lie bracket of vector fields.

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