Result 357, Differential geometry

Bi-Lipschitz coordinates at every regular RCD point

Proves that every regular point of a noncollapsed RCD(K,n)\mathop{\mathrm{RCD}}\nolimits (K,n) space, for K ∈ ℝ and integer n ≥ 2, has an open neighborhood bi-Lipschitz to an open subset of ℝn. Regularity requires all pointed tangents to be Euclidean, the reference measure is exactly Hn\mathcal H^n, and the chart compares ambient distances with a point-dependent finite constant.

Proof

The bigger picture

Why it matters

Looking Euclidean under unlimited magnification does not automatically provide usable coordinates nearby. The manuscript claims that, within a specified class of spaces with lower curvature bounds, every regular point has coordinates that distort distances by only a bounded factor.

What changes?

For every real K and integer n at least two, the manuscript treats noncollapsed RCD(K,n) spaces: generalized spaces with synthetic Ricci curvature bounded below by K and reference measure exactly n-dimensional Hausdorff measure. Regularity means every pointed tangent, a limiting magnified view, is Euclidean. Each regular point reportedly has an open neighborhood bi-Lipschitz to an open subset of n-dimensional Euclidean space, comparing restricted ambient distances. The neighborhood is point-specific; the abstract claims a distortion constant depending only on n.

What does that help mathematicians do?

The claimed advance turns information about all infinitesimal limits at a point into control over an entire open neighborhood. Researchers could deduce that the local topology there is Euclidean and compare distances in both directions through coordinates. Using ambient distances matters: the comparison concerns the space's original metric, not a replacement metric defined by paths constrained to stay inside the neighborhood.

Are there practical applications?

Its immediate value is foundational for studying spaces governed by synthetic curvature bounds. These charts would let researchers translate local metric questions near regular points into Euclidean coordinates while retaining quantitative distance control. The result applies under noncollapse and the stated tangent condition; it does not supply coordinates at arbitrary singular points or an algorithm for constructing them.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Bi-Lipschitz Coordinates at Regular Points of Noncollapsed RCD Spaces

September 25, 2026 43 pages

We resolve the regular-point bi-Lipschitz conjecture in the noncollapsed setting. For every integer n ≥ 2 and every real K, every regular point of a noncollapsed RCD(K,n)\mathrm{RCD}(K,n) space has an open neighborhood bi-Lipschitz homeomorphic to an open subset of ℝn. The bi-Lipschitz constant depends only on n, and the neighborhood uses the restricted ambient distance.

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Cite (BibTeX) 
@misc{OAI:Bi-Lipschitz-Coordinates-at-Regular-Points-of-Noncollapsed-RCD-Spaces-September-25-2026,
  author = {{OpenAI}},
  title = {{Bi-Lipschitz Coordinates at Regular Points of Noncollapsed $\mathrm{RCD}$ Spaces}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Bi-Lipschitz-Coordinates-at-Regular-Points-of-Noncollapsed-RCD-Spaces-September-25-2026/paper.pdf}{OAI:Bi-Lipschitz-Coordinates-at-Regular-Points-of-Noncollapsed-RCD-Spaces-September-25-2026}},
  year = {2026}
}

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