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Isometric embeddings of parallelizable and stably parallelizable Riemannian manifolds

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 31, 2026 / CC BY 4.0

Abstract

Let sn=n(n+1)/2s_n=n(n+1)/2. We prove that every smooth parallelizable Riemannian nn-manifold, n≥3n\ge 3, admits a smooth isometric embedding into Rsn+1\mathbb{R}^{s_n+1} whose image lies in an arbitrarily small ball. For closed stably parallelizable manifolds, each of whose connected components has vanishing Euler characteristic, we obtain small-image isometric embeddings into Rsn+5\mathbb{R}^{s_n+5}. The proof combines explicit partial isometries along commuting Hopf flows, an iterative scheme for exact metric correction using oscillatory perturbations constructed from the second fundamental form, and a transversality argument for metric-preserving perturbations.

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.

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Version history

  1. v1Submitted by Wenchang YangInitial depositCurrentOct 30, 2026