HEXAGONbeta

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 2 / 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.

References

  1. R. L. Bryant, S.-S. Chern, R. B. Gardner, H. L. Goldschmidt, and P. A. Griffiths, Exterior Differential Systems , Mathematical Sciences Research Institute Publications, vol. 18, Springer-Verlag, 1991. doi:10.1007/978-1-4613-9714-4 (https://doi.org/10.1007/978-1-4613-9714-4). Full text (https://sites.math.duke.edu/~bryant/MSRI-v18-Bryant-Chern-et-al.pdf).DOI
  2. 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).DOI
  3. 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
  4. J. J. Duistermaat and L. Hörmander, Fourier integral operators. II , Acta Math. 128 (1972), 183--269. doi:10.1007/BF02392165 (https://doi.org/10.1007/BF02392165).DOI
  5. J. B. Goodman and D. Yang, Local solvability of nonlinear partial differential equations of real principal type , unpublished manuscript, revised October 11, 1988, https://cims.nyu.edu/~yangd/papers/goodman-yang.pdf.link
  6. 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
  7. M. Gromov, Geometric, algebraic, and analytic descendants of Nash isometric embedding theorems , Bull. Amer. Math. Soc. 54 (2017), 173--245. Author manuscript, §3.7 (https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/nash-copy-Oct9.pdf).link
  8. 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
  9. 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
  10. 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
Show all 13 referencesShow fewer references
  1. 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
  2. G. Nakamura and Y. Maeda, Local smooth isometric embeddings of low-dimensional Riemannian manifolds into Euclidean spaces , Trans. Amer. Math. Soc. 313 (1989), no. 1, 1--51. doi:10.1090/S0002-9947-1989-0992597-8 (https://doi.org/10.1090/S0002-9947-1989-0992597-8).DOI
  3. 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. v1Submitted by Wenchang YangInitial depositSupersededOct 19, 2026
  2. v2Submitted by Wenchang YangSubmitted as a new versionCurrentOct 21, 2026