PropositionStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-12

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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

1.1F1F2F3F4

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

2.1step 1.1∎

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.

Depends on

Used by

A cut point that is not conjugate because two minimizers arrive Counterexample
Corner terms are required even in the plane Counterexample
Ricci lower bound does not control every sectional curvature in dimension at least three Counterexample
The same intrinsic planar strip can have different extrinsic curvature after bending Counterexample
A deformed sphere has the same total curvature Example
A flat torus showing simple connectedness is needed for global exp injectivity Example
A great sphere is totally geodesic Example
Area defect of a hyperbolic geodesic triangle Example
Area excess of a spherical geodesic triangle Example
Christoffel symbols in polar coordinates Example
Curvature of a Riemannian product Example
Cut locus of a point on a flat circle Example
Cut locus on a flat rectangular torus from the Dirichlet cell Example
Euclidean space has zero curvature Example
Flat torus and zero Euler characteristic Example
Gauss-Bonnet for a spherical cap Example
Gaussian curvature of a surface of revolution Example
Geodesics in the Poincare upper half-plane Example
Geodesics of a Riemannian product Example
Hopf–Rinow on a flat cylinder Example
Hyperbolic space has negative constant sectional curvature Example
Jacobi fields in euclidean space Example
Levi civita connection of a conformal plane metric Example
Principal curvatures of a round sphere Example
Projective-plane curvature via a hemisphere Example
Straight lines as Euclidean geodesics Example
The catenoid has zero mean curvature but is not totally geodesic Example
The cylinder has zero Gaussian curvature but nonzero second fundamental form Example
The euclidean levi civita connection Example
The exponential map of a flat torus is not injective Example
The punctured Euclidean plane is geodesically incomplete Example
The round sphere has positive constant sectional curvature Example
Total curvature of a round sphere Example
A geodesic stops minimizing exactly at its first conjugate point False statement
Cartan hadamard says exp p is injective without simple connectedness False statement
Every affinely reparametrized geodesic remains unit speed False statement
Every geodesic segment is globally length minimizing False statement
Every vector field along a geodesic is a Jacobi field False statement
Geodesic curvature need not equal ambient curve curvature False statement
Ricci curvature and scalar curvature determine the full Riemann tensor in every dimension False statement

…and 13 more results.

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)