ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Levi civita connection of a conformal plane metric

Example

For a smooth real function $u$ on an open subset of $R^{2}$ , the metric $g = e^{2 u} (d x^{2} + d y^{2})$ has $Γ^{k}_{i j} = δ_{j}^{k} u_{i} + δ_{i}^{k} u_{j} - δ_{i j} u_{k},$ where $u_{i} = \partial_{i} u$ and the last index uses the Cartesian Euclidean convention. For $u = x^{2}$ , the nonzero entries are $Γ^{x}_{x x} = 2 x$ , $Γ^{x}_{y y} = - 2 x$ , $Γ^{y}_{x y} = Γ^{y}_{y x} = 2 x$ .

Facts & Assumptions

Given: The smooth function $u$ and the positive conformal metric on its domain.

[F1]

The Levi–Civita coefficient formula contracts first metric derivatives with half the inverse metric (Christoffel formula for the levi civita connection).

Verification

1.1

The metric and inverse matrices are $g_{i j} = e^{2 u} δ_{i j}$ and $g^{i j} = e^{- 2 u} δ^{i j}$ . Since $\partial_{i} g_{j ℓ} = 2 e^{2 u} u_{i} δ_{j ℓ}$ , substitution in [F1] cancels the factors $2$ , $e^{2 u}$ and $e^{- 2 u}$ and gives $\sum_{ℓ} δ^{k ℓ} (u_{i} δ_{j ℓ} + u_{j} δ_{i ℓ} - u_{ℓ} δ_{i j})$ , exactly the asserted formula.

F1given

2.1

For $u = x^{2}$ , one has $u_{x} = 2 x, u_{y} = 0$ . The formula gives the four listed entries and $Γ^{x}_{x y} = Γ^{x}_{y x} = Γ^{y}_{x x} = Γ^{y}_{y y} = 0$ . At $x = 0$ all entries vanish, whereas at $x = 1$ the four listed entries are $2, - 2, 2, 2$ . Constant $u$ gives zero coefficients everywhere. The conformal factor is strictly positive for every real $u$ , so this calculation never inverts a degenerate metric.

step 1.1∎

Depends on

Christoffel formula for the levi civita connection

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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)