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

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The pullback of a riemannian metric by every smooth map is a riemannian metric

Statement

Every smooth map pulls a Riemannian metric back to a Riemannian metric.

Facts & Assumptions

Given: The proposed universal claim; take $F : R \to R$ , $F (x) = 0$ , with target metric $h = d y^{2}$ .

[F1]

Pullback of a riemannian metric as a tensor: For smooth $F : M \to N$ and a Riemannian metric $h$ on $N$ , its pullback tensor is $(F^{*} h)_{p} (v, w) = h_{F (p)} (d F_{p} v, d F_{p} w)$ . This is def-pullback-of-a-covariant-tensor-field for the tensor in def-riemannian-metric-and-riemannian-manifold. It is always symmetric and positive semidefinite; the name does not assert positive definiteness. Smoothness and the precise immersion criterion are established next.

[F2]

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 $p$ .

Refutation

technique · direct

1.1

The coordinate function of $F$ is constant, hence smooth with $d F_{x} (v) = 0$ for every $x, v \in R$ . The target quadratic form is $h_{y} (w, w) = w^{2} > 0$ for $w \neq 0$ , so the target is Riemannian.

given

2.1

The pullback definition gives $(F^{*} h)_{x} (\partial_{x}, \partial_{x}) = h_{0} (0, 0) = 0$ . Since $\partial_{x} \neq 0$ , positive definiteness fails; equivalently this $F$ is not an immersion. Thus this smooth map refutes the claim.

F1F2step 1.1∎

Source locator

Lee, Introduction to Smooth Manifolds, 2nd ed., pp. 330–331, pullback metrics and Proposition 13.9; the constant-map computation above is explicit.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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)