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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 $T (X, Y) = \nabla_{X} Y - \nabla_{Y} X - [X, Y]$ (Torsion tensor of an affine connection).

[F3]

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

Refutation

1.1

On $R^{2}$ use frame $(\partial_{x}, \partial_{y})$ and the matrix with sole nonzero entry $ω^{x}_{y} = d x$ . It defines a connection by [F1], with $\nabla_{\partial_{x}} \partial_{y} = \partial_{x}$ and all other coordinate derivatives zero. Coordinate fields commute, so [F2] gives $T (\partial_{x}, \partial_{y}) = \partial_{x} - 0 - 0 = \partial_{x}$ .

F1F2given

2.1

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.

F2F3step 1.1∎

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

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