EVERY CLOSED SMOOTH $n$-MANIFOLD EMBEDS     ISOMETRICALLY IN $\mathbb{R}^{\,n(n+1)/2+28}$

Yi Wang; Wenchang Yang

EVERY CLOSED SMOOTH $n$ -MANIFOLD EMBEDS ISOMETRICALLY IN $\mathbb{R}^{\,n(n+1)/2+28}$

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

Contributed by Yi Wang ↗, Wenchang Yang ↗

Version 1 / Nov 06, 2026 / CC BY 4.0

Abstract

We prove that every closed smooth Riemannian

n

-manifold admits a smooth isometric embedding into

\mathbb{R}^{s_n+28}

, where

s_n=n(n+1)/2

. This improves the previous upper bound

s_n+\max\{n+5,\,2n\}

to a dimension-independent surplus of

28

.

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.

Tools used

python: python

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Version history

v1 · 126 pagesSubmitted by Wenchang YangInitial depositCurrentNov 06, 2026