TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)

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Christoffel symbol transformation law

Statement

For coordinates $x^{i}$ and $y^{a}$ , the symbols transform by ${\tilde{Γ}}^{c}_{a b} = \frac{\partial y^{c}}{\partial x^{k}} (\frac{\partial x^{i}}{\partial y^{a}} \frac{\partial x^{j}}{\partial y^{b}} Γ^{k}_{i j} + \frac{\partial^{2} x^{k}}{\partial y^{a} \partial y^{b}}) .$ Repeated indices are summed. The second-derivative term is inhomogeneous.

Facts & Assumptions

Given: An affine connection and two overlapping smooth coordinate charts.

[F1]

The symbols are the coefficients of coordinate-frame derivatives (Christoffel symbols of an affine connection).

[F2]

The connection is function-linear in the direction and has the section Leibniz rule (Connection laws in directional form).

Proof

1.1

Put $J_{a}^{i} = \partial x^{i} / \partial y^{a}$ , so $\partial_{y^{a}} = J_{a}^{i} \partial_{x^{i}}$ . Applying both connection rules gives $\nabla_{\partial_{y^{a}}} \partial_{y^{b}} = J_{a}^{i} (\partial_{x^{i}} J_{b}^{k}) \partial_{x^{k}} + J_{a}^{i} J_{b}^{j} Γ^{k}_{i j} \partial_{x^{k}}$ . The first coefficient is $\partial_{y^{a}} J_{b}^{k} = \partial^{2} x^{k} / (\partial y^{a} \partial y^{b})$ by the chain rule.

F1F2

2.1

Express $\partial_{x^{k}} = (\partial y^{c} / \partial x^{k}) \partial_{y^{c}}$ and compare coefficients to obtain the displayed transformation. Identity changes recover the original symbols, affine coordinate changes have no second-derivative term, and dimension one has the corresponding scalar second derivative. Dimension zero has no indices. Coordinate changes have invertible Jacobians; no singular coordinate substitution is admitted.

F1step 1.1∎

Depends on

Used by

Christoffel symbols are components of a tensor False statement

Dependency tree · two levels

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Sources

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