Stratified spaces and synthetic Ricci curvature bounds
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Bibliographic record
Abstract
We prove that a compact stratified space satisfies the Riemannian curvature-dimension condition <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="normal">RCD</mml:mi> <mml:mo>(</mml:mo> <mml:mi>K</mml:mi> <mml:mo>,</mml:mo> <mml:mi>N</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> if and only if its Ricci tensor is bounded below by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>ℝ</mml:mi> </mml:mrow> </mml:math> on the regular set, the cone angle along the stratum of codimension two is smaller than or equal to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>π</mml:mi> </mml:mrow> </mml:math> and its dimension is at most equal to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> . This gives a new wide class of geometric examples of metric measure spaces satisfying the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="normal">RCD</mml:mi> <mml:mo>(</mml:mo> <mml:mi>K</mml:mi> <mml:mo>,</mml:mo> <mml:mi>N</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> curvature-dimension condition, including for instance spherical suspensions, orbifolds, Kähler–Einstein manifolds with a divisor, Einstein manifolds with conical singularities along a curve. We also obtain new analytic and geometric results on stratified spaces, such as Bishop–Gromov volume inequality, Laplacian comparison, Lévy–Gromov isoperimetric inequality. Our result also implies a similar characterization of compact stratified spaces carrying a lower curvature bound in the sense of Alexandrov.
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Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.001 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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