ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

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Gaussian curvature of a surface of revolution

Statement

This item assumes ${A C}_{ω}$ , namely countable choice. In the propagated dependency chain, that assumption is required through Sectional curvature; after those interfaces are fixed, the remaining local or finite argument makes no additional countable-family choice.

Let $I, J \subseteq R$ be open intervals, let $r, z : I \to R$ be smooth functions satisfying

$r (u) > 0, r^{'} (u)^{2} + z^{'} (u)^{2} = 1,$

and consider the surface-of-revolution parametrization

$X (u, v) = (r (u) \cos v, r (u) \sin v, z (u)) .$

On each associated surface chart with the metric induced from Euclidean $R^{3}$ , the Gaussian curvature is

$K (u, v) = - \frac{r^{''} (u)}{r (u)} .$

Here Gaussian curvature means the sectional curvature of the unique tangent two-plane of this Riemannian surface. No value at an axis $r = 0$ is asserted, and no further family choice is made beyond the stated inherited assumption.

Facts & Assumptions

Given: ${A C}_{ω}$ , the supplied intervals, smooth unit-speed profile with $r > 0$ , the displayed local surface parametrization, and the standard Euclidean metric.

[A1]

${A C}_{ω}$ is countable choice and is required here through Sectional curvature; after those supplied interfaces are fixed, the remaining local or finite calculation makes no additional countable-family choice.

[F1]

A pullback of a Riemannian metric is Riemannian exactly when the map is an immersion. Pullback of a riemannian metric is riemannian exactly for immersions.

[F2]

The Levi–Civita symbols of a coordinate metric are $Γ^{k}_{i j} = \frac{1}{2} g^{k ℓ} (\partial_{i} g_{j ℓ} + \partial_{j} g_{i ℓ} - \partial_{ℓ} g_{i j})$ . Christoffel formula for the levi civita connection.

[F3]

With $R (\partial_{i}, \partial_{j}) \partial_{k} = R^{ℓ}_{k i j} \partial_{ℓ}$ , the coordinate curvature formula is $R^{ℓ}_{k i j} = \partial_{i} Γ^{ℓ}_{j k} - \partial_{j} Γ^{ℓ}_{i k} + Γ^{m}_{j k} Γ^{ℓ}_{i m} - Γ^{m}_{i k} Γ^{ℓ}_{j m}$ . Coordinate formula for the curvature tensor.

[F4]

The four-tensor is $Rm (A, B, C, D) = g (R (A, B) C, D)$ , and the sectional curvature of the plane spanned by independent $A, B$ is $Rm (A, B, B, A) / (g (A, A) g (B, B) - g (A, B)^{2})$ . Riemann curvature four-tensor, Sectional curvature.

Verification

technique · direct coordinate calculation

1.1

Differentiation gives $X_{u} = (r^{'} \cos v, r^{'} \sin v, z^{'})$ and $X_{v} = (- r \sin v, r \cos v, 0)$ . Their Euclidean inner products are $⟨ X_{u}, X_{u} ⟩ = r^{' 2} + z^{' 2} = 1$ , $⟨ X_{u}, X_{v} ⟩ = 0$ , and $⟨ X_{v}, X_{v} ⟩ = r^{2}$ . Thus $d X$ is injective because $r > 0$ , so [F1] gives the induced Riemannian metric $g = d u^{2} + r (u)^{2} d v^{2}$ , with matrix $diag (1, r^{2})$ and inverse $diag (1, r^{- 2})$ .

F1givenalgebra

2.1

Write $x^{1} = u, x^{2} = v$ . The only nonconstant metric entry is $g_{22} = r^{2}$ , with $\partial_{1} g_{22} = 2 r r^{'}$ and $\partial_{2} g_{22} = 0$ . Substitution in [F2] gives $Γ^{1}_{22} = - r r^{'}$ and $Γ^{2}_{12} = Γ^{2}_{21} = r^{'} / r$ ; every other $Γ^{k}_{i j}$ is zero.

F2step 1.1algebra

3.1

In [F3], the component needed for the coordinate two-plane is $R^{1}_{212} = \partial_{1} Γ^{1}_{22} - \partial_{2} Γ^{1}_{12} + Γ^{m}_{22} Γ^{1}_{1 m} - Γ^{m}_{12} Γ^{1}_{2 m}$ . By step 2.1 the four terms are $- (r^{' 2} + r r^{''})$ , $0$ , $0$ , and $+ r^{' 2}$ , respectively; hence $R^{1}_{212} = - r r^{''}$ .

F3step 2.1algebra

4.1

By [F4] and $g_{11} = 1, g_{12} = 0$ , $Rm (\partial_{u}, \partial_{v}, \partial_{v}, \partial_{u}) = g (R (\partial_{u}, \partial_{v}) \partial_{v}, \partial_{u}) = R^{1}_{212} = - r r^{''}$ . The Gram determinant of $(\partial_{u}, \partial_{v})$ is $g_{11} g_{22} - g_{12}^{2} = r^{2} > 0$ , so the unique tangent two-plane has $K = (- r r^{''}) / r^{2} = - r^{''} / r$ .

A1F4step 1.1step 3.1algebra

5.1

If either parameter interval is empty, there are no points and the claim is vacuous; otherwise the chart is intrinsically two-dimensional, so zero- and one-dimensional curvature cases are inapplicable. The hypothesis $r > 0$ makes the metric and Gram determinant nondegenerate; at $r = 0$ this parametrization loses its angular direction, so the formula asserts neither a value nor a limit there. The intervals are open, so no parameter endpoint or manifold-boundary value is claimed. All functions, coordinates, and tangent vectors are explicitly supplied, and the computation makes no family selection, so no further family choice is made beyond the stated inherited assumption. The claim is an equality, not a biconditional.

F1F2F3F4step 1.1step 2.1step 3.1step 4.1∎

Depends on

Used by

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Dependency tree · two levels

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Sources

John M. Lee, Riemannian Manifolds: An Introduction to Curvature (standard reference, not scraped)