HEXAGONbeta

Smooth Local Isometric and Conformal Embeddings 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 4 / Oct 20, 2026 / CC BY 4.0

Abstract

Let sn=n(n+1)/2s_n=n(n+1)/2. For every n≥5n\geq5, the smallest Euclidean dimension in which every smooth nn-dimensional Riemannian metric admits a local isometric embedding is sns_n. The corresponding dimension for smooth local conformal embedding is sn−1s_n-1. We construct two flat reference embeddings with the same reduced principal symbol. In the conformal reference, the metric line takes the place of one normal direction. The proof uses quadratic products of characteristic waves to correct the metric and gives a sequence converging in every derivative order on a fixed neighborhood. Polynomial metric germs give both lower bounds.

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

  1. R. L. Bryant, P. A. Griffiths, and D. Yang, Characteristics and existence of isometric embeddings , Duke Math. J. 50 (1983), no. 4, 893--994. doi:10.1215/S0012-7094-83-05040-8 (https://doi.org/10.1215/S0012-7094-83-05040-8). We would like to express our sincere gratitude to Tianjiao Nie for carrying out the Lean formalization of Theorem 1.1. The formalization is available at blue https://github.com/ivwumupy/JanetCartan (https://github.com/ivwumupy/JanetCartan).DOI
  2. É. Cartan, Sur la possibilité de plonger un espace riemannien donné dans un espace euclidien , Ann. Soc. Polon. Math. 6 (1927), 1--7. Original volume, pp. 1--7 (https://mccuan.math.gatech.edu/courses/6455/Cartan1927.pdf).link
  3. R. E. Greene, Isometric embeddings of Riemannian and pseudo-Riemannian manifolds , Mem. Amer. Math. Soc., no. 97 (1970). doi:10.1090/memo/0097 (https://doi.org/10.1090/memo/0097).DOI
  4. Q. Han and M. Khuri, The linearized system for isometric embeddings and its characteristic variety , Adv. Math. 230 (2012), no. 1, 263--293. doi:10.1016/j.aim.2011.11.011 (https://doi.org/10.1016/j.aim.2011.11.011). arXiv:1111.4722 (https://arxiv.org/abs/1111.4722).arXivDOI
  5. M. Janet, Sur la possibilité de plonger un espace riemannien donné dans un espace euclidien , Ann. Soc. Polon. Math. 5 (1926), 38--43. Original article (https://rcin.org.pl/Content/15034/PDF/WA35_31746_cz159-r1926-T5_Annales-art-4.pdf).link
  6. H. Jacobowitz and J. D. Moore, The Cartan--Janet theorem for conformal embeddings (https://doi.org/10.1512/iumj.1974.23.23015), Indiana Univ. Math. J. 23 (1973), no. 2, 187--203.DOI
  7. S. Minsker, On some extensions of Bernstein's inequality for self-adjoint operators , Statist. Probab. Lett. 127 (2017), 111--119. Proof locators refer to arXiv:1112.5448v3 (https://arxiv.org/abs/1112.5448v3).arXiv
  8. J. Nash, The imbedding problem for Riemannian manifolds , Ann. of Math. (2) 63 (1956), no. 1, 20--63. doi:10.2307/1969989 (https://doi.org/10.2307/1969989).DOI
  9. J. A. Tropp, An introduction to matrix concentration inequalities , Found. Trends Mach. Learn. 8 (2015), no. 1--2, 1--230. doi:10.1561/2200000048 (https://doi.org/10.1561/2200000048).DOI

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 versionSupersededNov 05, 2026
  4. v4 · 106 pagesSubmitted by Wenchang YangSubmitted as a new versionCurrentNov 11, 2026