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Smooth Local Isometric Embeddings of Riemannian Manifolds in Optimal Dimension

A mix of human-written and AI-generated textHuman understanding: some partsmath.DG — Differential Geometrymath.AP — Analysis of PDEs

Contributed by Wenchang Yang, Yi Wang ↗

Version 3 / Oct 20, 2026 / CC BY 4.0

Abstract

We prove that every smooth Riemannian manifold of dimension n≥5n\geq5 admits, near each point, a smooth isometric embedding into Rn(n+1)/2\mathbb{R}^{n(n+1)/2}. This extends the local theorem of Janet and Cartan from real-analytic to arbitrary smooth metrics in these dimensions, and attains the optimal universal ambient dimension without any curvature or genericity assumption. The proof starts from a flat immersion whose second fundamental form has full normal rank. The linearized metric equation then reduces to a determined first-order system, whose characteristic variety is singular. We do not invert this system on its singular set. Instead, the metric is corrected by the quadratic interaction of characteristic waves: their phase-averaged quadratic terms generate the directions missing from the image of the reduced symbol, while the remaining oscillatory products and the residual linear errors are removed by parametrices on regular characteristic and elliptic patches. Exact Fourier localization of the increments yields an iteration that converges in C∞C^\infty on a fixed neighborhood.

Provenance statement

Generative AI tools were extensively used in the development, refinement, and checking of the mathematical proofs. The human contributors have reviewed the main statements and aspects of the proof strategy, but the detailed technical arguments have not yet been fully verified or understood by humans. Theorem 1.1 has also been formalized in Lean 4 by Tianjiao Nie. The formalization is available at https://github.com/ivwumupy/JanetCartan.

Tools used

Lean
Lean

References

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Version history

  1. v1 · 96 pagesSubmitted by Wenchang YangInitial depositSupersededOct 19, 2026
  2. v2 · 105 pagesSubmitted by Wenchang YangSubmitted as a new versionSupersededOct 21, 2026
  3. v3 · 105 pagesSubmitted by Wenchang YangSubmitted as a new versionCurrentNov 05, 2026