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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 1 / Oct 20, 2026 / CC BY 4.0

Abstract

We prove that every smooth Riemannian manifold of dimension n≥5n\ge 5 admits a smooth local isometric embedding into Rn(n+1)/2\mathbb{R}^{n(n+1)/2}. This extends the classical Janet–Cartan theorem from real-analytic to arbitrary smooth metrics in these dimensions, achieving the optimal universal ambient dimension without curvature or genericity assumptions. The proof constructs characteristic waves around a flat immersion whose second fundamental form has full normal rank. Their Hermitian quadratic mean generates the required metric correction, while linear responses cancel oscillatory quadratic interactions and the remaining linear strain. Exact Fourier localization yields an iteration converging in C∞C^\infty on a fixed neighborhood.The detailed geometric and microlocal constructions are proved in the appendices.

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.

References

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Version history

  1. v1Submitted by Wenchang YangInitial depositCurrentOct 19, 2026