ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

How statement and proof provenance work

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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The round metric on the sphere as an induced metric

Example

The Euclidean inclusion of $S^{n}$ induces its round metric. In spherical coordinates on $S^{2}$ , $g = d θ^{2} + \sin^{2} θ d φ^{2}$ .

Facts & Assumptions

Given: $S^{n} = {p \in R^{n + 1} : ∥ p ∥ = 1}$ and its usual smooth structure; $i$ is inclusion.

[F1]

Pullback of a riemannian metric is riemannian exactly for immersions: $F^{*} h$ is Riemannian if and only if $F$ is an immersion. In general it is positive semidefinite, with radical $\ker d F_{p}$ at $p$ .

[F2]

Coordinate criterion for a riemannian metric: A tensor $g = \sum_{i, j} g_{i j} d x^{i} \otimes d x^{j}$ is Riemannian exactly when its coordinate matrix $G = (g_{i j})$ has smooth entries and is symmetric positive definite. Under $J = \partial x / \partial y$ it transforms by $G_{y} = J^{T} G_{x} J$ .

Verification

technique · direct

1.1

A tangent vector $v$ to $S^{n}$ satisfies $p \cdot v = 0$ by differentiating $∥ p ∥^{2} = 1$ . The inclusion differential sends this vector to the identical Euclidean vector, so is injective. Hence $i^{*} g_{E}$ is Riemannian, and its value is $⟨ v, w ⟩ = v \cdot w$ .

F1given

2.1

For $X (θ, φ) = (\sin θ \cos φ, \sin θ \sin φ, \cos θ)$ , one has $X_{θ} = (\cos θ \cos φ, \cos θ \sin φ, - \sin θ)$ and $X_{φ} = (- \sin θ \sin φ, \sin θ \cos φ, 0)$ . Their dot products are $1$ , $0$ , and $\sin^{2} θ$ , respectively. Thus the coordinate matrix is $diag (1, \sin^{2} θ)$ on $0 < θ < π$ and an angular interval of length less than $2 π$ .

F2step 1.1

3.1

At either pole the spherical parametrization is not a chart, since $X_{φ} = 0$ . The intrinsic quadratic form remains $v \cdot v > 0$ on every nonzero tangent vector by step 1.1; the vanishing coordinate coefficient at a pole therefore does not signify tensor degeneracy.

step 1.1step 2.1∎

Source locator

Lee, Proposition 13.9, p.331, and Example 13.16, p.333, round metric. The spherical-coordinate dot products are displayed above.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

John M. Lee, Introduction to Smooth Manifolds, second edition (standard reference, not scraped)