LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6.1-sol)

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Compact curved hypersurfaces admit a finite curved graph cover

Statement

Assume Countable Choice. Let $S \subseteq R^{n}$ ( $n \geq 2$ ) be a compact embedded $C^{\infty}$ hypersurface with a continuous unit normal field $ν$ and everywhere nonvanishing extrinsic Gaussian curvature $K = \det S_{ν} \neq 0$ . Then there exist finitely many open sets $U_{j} \subseteq R^{n - 1}$ , smooth $h_{j} : U_{j} \to R$ with $\det D^{2} h_{j} \neq 0$ on $U_{j}$ , embeddings $X_{j} (y) = (y, h_{j} (y))$ of $graph h_{j}$ onto relatively open pieces $S_{j} \subseteq S$ covering $S$ (after ambient rigid motions), and nonnegative $C^{\infty}$ functions $χ_{j}$ on $S$ with $\sum_{j} χ_{j} = 1$ and $supp χ_{j}$ compactly contained in $S_{j}$ .

Facts & Assumptions

Given: The compact embedded smooth hypersurface, continuous unit normal and nonzero curvature in the statement, with Countable Choice.

[F1]

Smooth graph charts, compact smooth localization, normal independence and the Euclidean curvature convention are established locally. (Smooth Euclidean hypersurface graphs and compact localization, Euclidean hypersurface normals, shape operators and curvature)

[F2]

The graph curvature is $\det D^{2} h / (1 + ∣ \nabla h ∣^{2})^{(n + 1) / 2}$ . (Shape operator and Gauss-Kronecker curvature of a graph)

[A1]

Countable Choice is assumed. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

Proof

technique · direct; apply the local graph and compact localization constructions

1.1givenF1F2

By [F1], every point has a smooth graph chart after a rigid motion. The given continuous normal is locally smooth and equals either the graph normal or its negative with constant sign on a connected smaller chart. Its shape operator therefore differs by that sign; nonvanishing curvature is unchanged. Formula [F2] implies $\det D^{2} h \neq 0$ throughout the smaller graph chart. This argument also works with local normals only, without a global orientation.

2.1F1A1step 1.1∎

Apply the compact localization part of [F1] with $K = S$ to the graph neighbourhoods of step 1.1. Its ambient-ball bumps give finitely many pieces covering $S$ and nonnegative smooth $χ_{j}$ with compact support inside their pieces and sum one. Their graph functions retain their nondegenerate Hessians on the whole chart. These are all the asserted data. The construction needs only finite choices; the assumed Countable Choice remains available to surface-measure consumers.

Depends on

Used by

Stein-Tomas for compact hypersurfaces with nonzero curvature Corollary

Dependency tree · two levels

27 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

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