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

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Christoffel formula for the levi civita connection

Statement

In coordinates for a Riemannian metric with matrix $(g_{i j})$ and inverse $(g^{i j})$ , its Levi–Civita symbols are $Γ^{k}_{i j} = \frac{1}{2} \sum_{ℓ} g^{k ℓ} (\partial_{i} g_{j ℓ} + \partial_{j} g_{i ℓ} - \partial_{ℓ} g_{i j}) .$

Facts & Assumptions

Given: A supplied Riemannian metric and a coordinate chart.

[F1]

The unique Levi–Civita connection exists (Fundamental theorem of riemannian geometry).

[F2]

It obeys the Koszul formula (Koszul formula is necessary for a levi civita connection).

[F3]

Symbols are its coordinate derivative coefficients (Christoffel symbols of an affine connection).

[F4]

Coordinate fields commute (Coordinate formula for the Lie bracket).

Proof

1.1

Insert $X = \partial_{i}, Y = \partial_{j}, Z = \partial_{ℓ}$ in the Koszul identity. All bracket terms vanish, giving $2 \sum_{k} Γ^{k}_{i j} g_{k ℓ} = \partial_{i} g_{j ℓ} + \partial_{j} g_{i ℓ} - \partial_{ℓ} g_{i j}$ .

F1F2F3F4

2.1

Multiply by $g^{m ℓ}$ and sum over $ℓ$ . Since $\sum_{ℓ} g_{k ℓ} g^{m ℓ} = δ_{k}^{m}$ , division by two gives the claimed expression. Constant metric coefficients give zero symbols, dimension one gives $Γ^{1}_{11} = g_{11}^{'} / (2 g_{11})$ , and dimension zero gives an empty formula. Inverting a positive-definite matrix is legitimate at every point, including boundary points.

step 1.1∎

Depends on

Used by

Christoffel symbols in polar coordinates Example
Levi civita connection of a conformal plane metric Example
The euclidean levi civita connection Example
Torsion free means curvature free False statement
Gradient hessian and divergence connection formulas Proposition

Dependency tree · two levels

12 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)