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arxiv: 2507.00471 · v2 · submitted 2025-07-01 · 🧮 math.DG

Universal non-CD of sub-Riemannian manifolds

Dimitri Navarro , Jiayin Pan This is my paper

Pith reviewed 2026-05-19 07:08 UTC · model grok-4.3

classification 🧮 math.DG
keywords sub-Riemannian manifoldscurvature-dimension conditionRCD spacestangent conesGrushin spacesRadon measuressynthetic curvature
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The pith

Sub-Riemannian manifolds equipped with full-support measures are never CD(K,N) unless they are Riemannian.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that a sub-Riemannian manifold with a Radon measure of full support cannot satisfy the curvature-dimension condition CD(K,N) for any real K and any finite N greater than 1, except when the structure is actually Riemannian. This extends earlier non-CD results that required the measure to have smooth positive density. The argument proceeds by analyzing the tangent cones and the geodesics to produce a direct contradiction with the CD inequality. The authors also construct cone-Grushin spaces on Euclidean space that satisfy the RCD condition yet are not sub-Riemannian, because they lack a scalar product along certain curves while retaining horizontal directions, higher Hausdorff dimension, and inhomogeneous dilations.

Core claim

A sub-Riemannian manifold equipped with a full-support Radon measure is never CD(K,N) for any K in R and N in (1, infinity) unless it is Riemannian. The proof rests on the analysis of tangent cones and geodesics to reach a contradiction with the CD condition.

What carries the argument

Tangent cones and geodesics of the sub-Riemannian structure, whose properties yield the contradiction with the CD inequality.

If this is right

  • Only Riemannian manifolds can be CD spaces among sub-Riemannian structures when the measure has full support.
  • Synthetic curvature bounds of CD type are incompatible with genuinely sub-Riemannian geometry.
  • Cone-Grushin spaces give new RCD examples that share some sub-Riemannian traits but fail to be sub-Riemannian.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Different synthetic curvature notions may be needed to capture curvature in genuinely sub-Riemannian settings.
  • The cone-Grushin construction suggests a route to metric spaces with mixed horizontal and vertical scaling that still obey RCD.
  • These spaces could serve as test models for hypoelliptic diffusion or control problems outside classical sub-Riemannian theory.

Load-bearing premise

The tangent cones and geodesics in a non-Riemannian sub-Riemannian manifold produce a contradiction with the curvature-dimension inequality when the measure has full support.

What would settle it

An explicit non-Riemannian sub-Riemannian manifold equipped with a full-support Radon measure that satisfies CD(K,N) for some K and finite N would refute the claim.

read the original abstract

We prove that a sub-Riemannian manifold equipped with a full-support Radon measure is never $\mathrm{CD}(K,N)$ for any $K\in \mathbb{R}$ and $N\in (1,\infty)$ unless it is Riemannian. This generalizes previous non-CD results for sub-Riemannian manifolds, where a measure with smooth and positive density is considered. Our proof is based on the analysis of the tangent cones and the geodesics within. Secondly, we construct new $\mathrm{RCD}$ structures on $\mathbb{R}^n$, named cone-Grushin spaces, that fail to be sub-Riemannian due to the lack of a scalar product along a curve, yet exhibit characteristic features of sub-Riemannian geometry, such as horizontal directions, large Hausdorff dimension, and inhomogeneous metric dilations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper proves that a sub-Riemannian manifold equipped with any full-support Radon measure fails to be CD(K,N) for all real K and all N in (1,∞) unless the structure is Riemannian. The argument proceeds by contradiction using tangent-cone analysis and the behavior of geodesics in the blow-up; it removes the smooth-positive-density assumption from earlier non-CD results. The manuscript also constructs cone-Grushin spaces on R^n that satisfy RCD but are not sub-Riemannian, exhibiting horizontal directions, inhomogeneous dilations, and Hausdorff dimension larger than the topological dimension.

Significance. If the central claim holds, the result shows that the CD condition is incompatible with genuine sub-Riemannian geometry even when the measure is allowed to be singular, provided only that it has full support. This strengthens the separation between Riemannian and sub-Riemannian settings under synthetic curvature bounds. The cone-Grushin examples supply new RCD spaces that mimic several sub-Riemannian features without being sub-Riemannian, which may serve as test cases for future work on the boundary of the CD class. The tangent-cone approach is a natural and potentially robust method once the measure-theoretic details are secured.

major comments (1)
  1. [§3, proof of Theorem 1.1] §3, proof of Theorem 1.1: the step asserting that every tangent measure at a point must reproduce a Carnot-group structure whose geodesics violate entropy convexity relies on the measure charging all horizontal directions uniformly after blow-up. For a general full-support Radon measure (not necessarily absolutely continuous with respect to Popp or Hausdorff measure), it is not immediate that the tangent measure cannot concentrate on a lower-dimensional subset of the tangent cone while still having full support in the original space; this would potentially preserve the CD inequality. A precise statement of how full support passes to the tangent measure and rules out such concentration is needed.
minor comments (2)
  1. [§4] The notation for the cone-Grushin metric and its dilations should be introduced with an explicit formula before the statement of Theorem 4.2.
  2. [Figure 2] Figure 2 caption refers to 'horizontal curves' but the figure itself lacks labels distinguishing horizontal from vertical directions.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, the positive evaluation of the paper's significance, and the constructive comment on the tangent-measure step in the proof of Theorem 1.1. We have revised the manuscript to supply the requested precise statement on how full support passes to tangent measures.

read point-by-point responses

  1. Referee: [§3, proof of Theorem 1.1] §3, proof of Theorem 1.1: the step asserting that every tangent measure at a point must reproduce a Carnot-group structure whose geodesics violate entropy convexity relies on the measure charging all horizontal directions uniformly after blow-up. For a general full-support Radon measure (not necessarily absolutely continuous with respect to Popp or Hausdorff measure), it is not immediate that the tangent measure cannot concentrate on a lower-dimensional subset of the tangent cone while still having full support in the original space; this would potentially preserve the CD inequality. A precise statement of how full support passes to the tangent measure and rules out such concentration is needed.

    Authors: We thank the referee for this observation. The original argument already uses that the CD assumption implies the measure is doubling, so tangent measures exist in the sense of measured Gromov-Hausdorff convergence. To make the inheritance of full support explicit, we have added Lemma 3.2 in the revised §3: if μ has full support on the sub-Riemannian manifold, then any tangent measure ν at p (obtained as a weak-* limit of the rescaled measures μ_{p,r} as r→0) must itself have full support on the entire tangent cone. The proof is by contradiction: if ν(U)=0 for a nonempty open set U in the tangent cone, then for a sequence r_k→0 the rescaled measures would eventually assign zero mass to a corresponding open set in the manifold; lifting back yields a nonempty open set in the original space with μ-measure zero, contradicting full support. Consequently the tangent measure charges every horizontal direction in the Carnot group, and the geodesic entropy-convexity violation proceeds unchanged. This clarification applies directly to arbitrary full-support Radon measures and does not require absolute continuity with respect to Popp or Hausdorff measure. revision: yes

Circularity Check

0 steps flagged

No significant circularity in tangent cone and geodesic analysis

full rationale

The paper establishes the non-CD(K,N) property for non-Riemannian sub-Riemannian manifolds with full-support Radon measures via direct analysis of tangent cones and geodesics to obtain a contradiction with entropy convexity. This structural argument does not reduce by construction to any fitted parameters, self-defined quantities, or load-bearing self-citations within the paper. The generalization from prior smooth-density cases relies on the same geometric features without importing uniqueness theorems or ansatzes from overlapping author work as the sole justification. The derivation chain remains independent of the target result.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The paper relies on standard axioms of the field and introduces a new class of spaces as examples.

axioms (1)
  • standard math Properties of sub-Riemannian structures and Radon measures on manifolds
    These are background assumptions from differential geometry and measure theory.
invented entities (1)
  • cone-Grushin spaces no independent evidence
    purpose: New RCD structures on R^n that fail to be sub-Riemannian
    Constructed in the paper to exhibit sub-Riemannian-like features without being sub-Riemannian.

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Forward citations

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