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

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Christoffel symbols are components of a tensor

Statement

The Christoffel symbols of any affine connection are the components of a tensor of type $(1, 2)$ .

Facts & Assumptions

Given: The proposed tensorial interpretation.

[F1]

Under coordinate change, Christoffel symbols include a second-derivative term (Christoffel symbol transformation law).

[F2]

Smooth matrices in a global frame define connections (Local connection forms glue exactly when they obey the transformation law).

Refutation

1.1

On $M = (0, \infty)$ take the connection with $\nabla_{\partial_{x}} \partial_{x} = 0$ , supplied by zero matrix in the global tangent frame using [F2]. The coordinate $y = \log x$ is smooth with smooth inverse $x = e^{y}$ . The one-dimensional formula in [F1] gives $\tilde{Γ} = (d y / d x) (d^{2} x / d y^{2}) = e^{- y} e^{y} = 1$ . Equivalently $\partial_{y} = x \partial_{x}$ and $\nabla_{x \partial_{x}} (x \partial_{x}) = x \partial_{x} = \partial_{y}$ .

F1F2given

2.1

If these were components of a fibrewise bilinear map $T : T M \times T M \to T M$ , the original zero component would force $T (\partial_{x}, \partial_{x}) = 0$ , and bilinearity would give $T (\partial_{y}, \partial_{y}) = x^{2} T (\partial_{x}, \partial_{x}) = 0$ . This contradicts the new component $1$ in step 1.1 at every point. The Jacobian is never zero; the failure is not a singular-coordinate artifact.

step 1.1∎

Depends on

Used by

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Dependency tree · two levels

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Sources

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