Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-12
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Every affine connection is the levi civita connection of a riemannian metric

Statement

Every affine connection is the Levi–Civita connection of some Riemannian metric.

Facts & Assumptions

Given: The universal existence assertion for a compatible metric with Levi–Civita connection.

[F1]

A smooth matrix of one-forms in a global tangent frame defines an affine connection (Local connection forms glue exactly when they obey the transformation law).

[F2]

Torsion is T(X,Y)=∇XY−∇YX−[X,Y] (Torsion tensor of an affine connection).

[F3]

A Levi–Civita connection must be torsion free (Levi civita connection).

Refutation

1.1F1F2given

On R2 use frame (∂x,∂y) and the matrix with sole nonzero entry ωxy=dx. It defines a connection by [F1], with ∇∂x∂y=∂x and all other coordinate derivatives zero. Coordinate fields commute, so [F2] gives T(∂x,∂y)=∂x−0−0=∂x.

2.1F2F3step 1.1∎

This vector is nonzero everywhere. Torsion depends on the connection and bracket, with no metric in its definition; changing a metric cannot change this value. Therefore [F3] excludes this connection from being Levi–Civita for every Riemannian metric on the plane, refuting the assertion.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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