LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)

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Koszul formula is necessary for a levi civita connection

Statement

Every Levi–Civita connection obeys the Koszul formula $2 g (\nabla_{X} Y, Z) = K (X, Y, Z),$ where $K (X, Y, Z) = X g (Y, Z) + Y g (Z, X) - Z g (X, Y) - g (X, [Y, Z]) + g (Y, [Z, X]) + g (Z, [X, Y]) .$ The right side uses only the supplied metric and vector fields.

Facts & Assumptions

Given: A Levi–Civita connection for $g$ and local fields $X, Y, Z$ .

[F1]

Levi–Civita means metric compatibility and $\nabla_{X} Y - \nabla_{Y} X = [X, Y]$ (Levi civita connection).

Proof

1.1

Expand $S = X g (Y, Z) + Y g (Z, X) - Z g (X, Y)$ by compatibility. Its six terms are $g (\nabla_{X} Y, Z) + g (Y, \nabla_{X} Z) + g (\nabla_{Y} Z, X) + g (Z, \nabla_{Y} X) - g (\nabla_{Z} X, Y) - g (X, \nabla_{Z} Y)$ .

2.1

Substitute $\nabla_{Y} X = \nabla_{X} Y - [X, Y]$ , $\nabla_{Z} X = \nabla_{X} Z + [Z, X]$ and $\nabla_{Z} Y = \nabla_{Y} Z + [Z, Y]$ . Symmetry of $g$ cancels the $\nabla_{X} Z$ and $\nabla_{Y} Z$ pairs, giving $S = 2 g (\nabla_{X} Y, Z) - g (Z, [X, Y]) - g (Y, [Z, X]) + g (X, [Y, Z])$ . Rearrangement yields the formula. Zero fields cause no exception and in dimension zero all terms vanish. The identity is local, including boundary-chart derivatives, and no choice axiom occurs.

F1step 1.1∎

Depends on

Levi civita connection

Used by

Christoffel formula for the levi civita connection Proposition
Fundamental theorem of riemannian geometry Theorem
The koszul formula defines an affine connection Theorem

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Ved Datar, Lectures on Riemannian Geometry (standard reference, not scraped)