Sectional Curvature for Kantorovich-Wasserstein and Hellinger-Kantorovich Geometries

Karl-Theodor Sturm

REVIEW 4 major objections 5 minor 1 cited by

Sectional Curvature for Kantorovich-Wasserstein and Hellinger-Kantorovich Geometries

T0 review · 4 major / 5 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read On the space of finite measures with the Hellinger-Kantorovich metric, sectional curvature exists for smooth directions and decomposes into a negative lifted part and a nonnegative twisted part.

desk verdict Sturm's explicit HK sectional curvature formulas are a real advance, but the unproved geodesic property of his exponential map is load-bearing and needs fixing before the interpretation and the Alexandrov conclusions hold. read the letter →

arxiv 2606.14318 v2 pith:VOAXO2OA submitted 2026-06-12 math.DG math.FAmath.MGmath.PR

Karl-Theodor Sturm This is my paper · ORCID

classification math.DGmath.FAmath.MGmath.PR MSC 53C2353C2149Q22

keywords sectionalcurvatureHellinger-KantorovichmetricKantorovich-WassersteingeodesicspacesunbalancedoptimaltransportconeconstructionRiccimeasure-valuedgeometry

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes an explicit sectional curvature for the metric space of finite measures on a Riemannian manifold carrying the Hellinger-Kantorovich metric. For every measure and every pair of smooth compactly supported functions, the curvature exists and is the sum of two parts: a 'lifted' part coming from the base manifold's own curvature shifted by minus one, and a 'twisted' part that is always nonnegative and reflects the creation and annihilation of mass. In the Euclidean absolutely continuous case the twisted part is given by a closed variational formula, and on the torus all curvature numbers can be computed in a Fourier basis, revealing simultaneously negative directions, positive average curvature, and divergent Ricci curvature. The same geometric framework supplies a rigorous derivation of the sectional curvature of the Kantorovich-Wasserstein space of probability measures, extending previous formal results to arbitrary measures and geometric tangent directions.

What carries the argument

The load-bearing identity is the purely metric formula for sectional curvature in geodesic spaces: for two geodesics α,β from z, Sec_z(α,β) = 3 lim_{t→0} [t² d_z(α,β)² − d²(α_t,β_t)]/(t⁴ Λ). The paper computes this quantity in the Hellinger-Kantorovich space by using the analytic tangent space H¹(μ) (functions with norm ∫(|∇φ|²+4φ²)dμ) and the exponential map exp^Me_μ(tφ), defined by pushing μ forward along geodesics in the metric cone over M. The split into lifted and twisted parts comes from comparing the squared distance in the base cone, integrated over the lifted measure, with the true Hellinger-Kantorovich distance of the pushed-forward measures; their difference is the nonnegative twi

What would settle it

Choose M=R, μ=δ_0, and smooth φ; compute the Hellinger-Kantorovich distance between exp^Me_μ(sφ) and exp^Me_μ(tψ) directly from the cone representation for small s,t and check whether it equals |s−t|·(∫(|∇φ|²+4φ²)dμ)^{1/2} up to O(t⁴). If it does not, the exponential curves are not geodesics and the paper's Secμ is not the metric sectional curvature.

Watch

Extended reading notes

Core claim

Sectional curvature of the Hellinger-Kantorovich space of finite measures on a Riemannian manifold is shown to exist for all measures μ and smooth directions φ,ψ, and to decompose as Secμ(φ,ψ) = (1/Λ) ∫_M (Sec^M_x(∇φ,∇ψ)−1)(|∇φ|²|∇ψ|² − ⟨∇φ,∇ψ⟩²)dμ + Sec∇_μ(φ,ψ), with the twisted part Sec∇_μ always ≥ 0. In the Euclidean absolutely continuous case, Sec∇_μ(φ,ψ) = (3/Λ) inf_η ∫ [|F−∇η|²+4η²]dμ, F = ½(∇²φ∇ψ−∇²ψ∇φ), and on the torus the curvature of Fourier modes is given by an explicit universal formula. For the Wasserstein space P2(M), the same construction recovers the formal curvature formulas in full generality, including arbitrary measures and geometric tangent directions.

Load-bearing premise

The paper's interpretation of the computed limits as the actual metric sectional curvature rests on the unproved claim ('one easily verifies') in Definition 4.21 that, for every measure μ and smooth compactly supported φ, the curve t↦exp^Me_μ(tφ) is an HK-geodesic for small t; if that fails, the quantity Secμ(φ,ψ) is not the curvature of geodesics in the sense of Section 2.

Editorial extensions

If this is right

For M=R^n, directions exist where sectional curvature is strictly negative and directions where it is strictly positive, so (Me(M),HK) has neither nonnegative nor nonpositive Alexandrov curvature.
On the n-torus with normalized volume, the curvature between Fourier modes is known explicitly; super-orthogonal modes have curvature between −1 and −1/25, while averages over large balls are positive and Ricci curvature diverges.
The Wasserstein space P2(M) sectional curvature is now rigorous for arbitrary measures and geometric tangent directions; at a Dirac measure it is nonpositive whenever the base has nonpositive sectional curvature, and identically zero for Euclidean base.
In one dimension the twisted part of HK curvature does not vanish (unlike Wasserstein curvature), so even the line or circle carries nontrivial measure-space curvature.
The zero measure has a well-defined geometric tangent space, and all sectional curvatures there vanish.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same decomposition may hold for other interpolated transport metrics (e.g., Gaussian Hellinger-Kantorovich variants), suggesting a general 'base curvature − 1 plus nonnegative interaction' law.
The explicit Euclidean twisted formula gives a ready-made computational tool: for a density ρ, one can evaluate curvature by solving one weighted Poisson equation, which could be used to test geodesic dispersion properties numerically.
The diverging Ricci curvature on the torus suggests that in high-frequency limits the HK geometry becomes strongly curved; the paper's renormalized Ricci curvature may be the correct quantity for statistical applications.
If the exponential-map geodesicity assumption fails for singular μ, the formulas still describe the curvature of the embedded exponential family, which may be the right object for optimization on measures rather than the full metric space.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

What's actually new: explicit sectional curvature formulas for the Hellinger-Kantorovich space (Me(M), HK) — lifted part coming from the base manifold minus 1, twisted part nonnegative with a variational formula in the Euclidean case. The torus computations (Ricci divergence, renormalized Ricci) are concrete and new. For P2, the paper rigorously extends Lott's formal calculations to general measures and geometric tangent spaces. That is substantial.

The detailed proofs for the Euclidean absolutely continuous case are solid: the four-step transport estimate, the projection onto gradients, the variational characterization. The decomposition theorem 5.2 is clean. The torus Fourier computations are elaborate and, as far as I can tell, correct. The explicit formulas will be widely cited.

Now the soft spots, in proportion. The stress-test note is right: Definition 4.21 states "One easily verifies" that exp^Me_mu(t phi) is an HK-geodesic for any mu and any C_c^infty phi. That is not obvious for arbitrary mu. The sectional curvature Sec_mu(phi, psi) in Def. 5.1 is defined as a limit using these exponential curves. If those curves are not genuine geodesics, Sec_mu is only a curvature-like quantity along exponential paths, not the sectional curvature of (Me, HK). The Alexandrov conclusions (Thm 5.18, Cor 6.16) rely on Theorem 2.4, which assumes geodesics. So this is a load-bearing gap, not a cosmetic one. The assertion is plausible — for absolutely continuous mu it follows from the cone version of Brenier–McCann — but for singular mu or mass at distances ≥ π/2, it needs a real argument. The paper does not supply one.

Two smaller issues: the "easily verified" extension from the one-parameter to the two-parameter limit in Theorem 3.24 and Theorem 5.8 is sketched in a sentence; and the passage from smooth absolutely continuous mu to general mu is waved through. These are lesser because the core computation is already long, but they are nontrivial.

Also minor: the abstract says the lifted part is negative and the twisted part positive; the theorem says Sec^↑ = Sec^M - 1 which is negative for Euclidean, and Sec^∇ ≥ 0. The intro has a typo "Sec^↑ ≥ 0" where it should be Sec^∇. Sloppy but not confusing.

Who this is for: metric geometers and optimal transport people. It deserves a serious referee — the formulas are important and likely correct, but the geodesic-property gap needs to be either proved or explicitly assumed. A good referee could sort that out. Send it to review.

Referee Report

4 major / 5 minor

Summary. This paper develops a synthetic sectional curvature for geodesic metric spaces (Definition 2.3), then applies it to the Hellinger-Kantorovich space (M^e(M),HK) and to the Kantorovich-Wasserstein space (P_2(M),W_2). The central claimed structure is stated in Section 5.1: for every μ in M^e(M) and φ,ψ in C_c^∞(M), the sectional curvature exists and decomposes as Sec_μ(φ,ψ) = Sec^↑_μ(φ,ψ) + Sec^∇_μ(φ,ψ). The lifted part is expressed through the base sectional curvature minus 1; the twisted part is nonnegative and, for Euclidean absolutely continuous μ, is given by the explicit H^1-projection formula (3/Λ) inf_η ∫ [|F−∇η|² + 4η²] dμ, F = ½(∇²φ∇ψ − ∇²ψ∇φ). The torus section gives explicit Fourier formulas, uniform bounds, examples with negative/positive curvature, divergent renormalized Ricci, and a divergence result for the Ricci sum. The P_2 section recovers and extends formulas of Lott. The final conclusions are that (M^e(R^n),HK) and (M^e(T^n),HK) admit no Alexandrov curvature bounds.

Significance. If fully substantiated, this would be a significant contribution. It appears to give the first general curvature formulas for the Hellinger-Kantorovich geometry, reveals a nontrivial competition between a negative lifted part and a positive twisted part, and offers a rigorous optimal-transport route that avoids the Otto-calculus restriction to absolutely continuous measures. The synthetic definition of sectional curvature for geodesic spaces is natural, and the explicit torus calculations are concrete and checkable. The decomposition into lifted and twisted parts, the spatial scaling theorem, and the construction of renormalized Ricci curvature are all valuable. However, several load-bearing assertions are currently unproved: the geodesic property of the exponential map in Definition 4.21, the existence and regularity of the cone optimal map in Theorem 5.8, and the two-parameter limit extension. These gaps must be closed before the advertised interpretation as sectional curvature of the geodesic space is fully justified.

major comments (4)

[§4.4.2, Definition 4.21] The statement 'One easily verifies that for any φ∈C_c^∞(M) and sufficiently small τ>0, the curve t↦exp^{M^e}_μ(tφ) is an HK-geodesic' is asserted without proof. This is load-bearing: Definition 5.1 and Theorem 5.2 define Sec_μ through these curves, and Theorem 5.18 and Corollary 6.16 use Theorem 2.4 to convert the computed values into Alexandrov curvature statements. The claim is not a routine verification for arbitrary μ. Theorem 4.20 covers only absolutely continuous μ; for singular μ one must prove that tΦ = t r²φ is c-concave on the cone and that exp^C(t∇Φ) induces an optimal W_C coupling from the canonical lift of μ. Without this, the quantity in Definition 5.1 is only a curvature-like limit along pseudo-geodesics, not the sectional curvature of the metric space in the sense of Section 2. Please supply a proof or restrict the main theorem to a class of measures for which the geodesi
[§5.3, Theorem 5.8 proof, step (ii)] The proof invokes the Pinning Theorem 4.4 together with the McCann–Brenier theorem on the cone to assert the existence of an optimal map T_t on C transporting a lift of α_t to a lift of β_t, and then asserts the special form T_t = exp^C(∇θ_t, 2rθ_t) and smooth dependence of θ_t on t 'by elliptic regularity theory.' None of these steps is justified in the text. Pinning Theorem 4.4 gives equality of HK distance with a W_C distance for some lift of β_t; it does not by itself produce a Monge map. Moreover, the canonical lifts p_t are supported on an n-dimensional graph in the (n+1)-dimensional cone, not absolutely continuous on C, so the usual McCann–Brenier hypothesis is not satisfied. This is precisely where the H^1-projection formula and the explicit twisted curvature arise. A rigorous derivation of the existence, form, and regularity of T_t is needed.
[Theorem 3.24 (vi) and Theorem 5.8 (vi)] In both proofs, the step labelled 'The final extension to the 2-parameter limit' is dismissed with 'This is easily verified.' This is not a harmless omission. Definition 5.1 (and Definition 3.8) require the limit over all sequences (s_ℓ,t_ℓ) with bounded ratio, whereas the detailed computations establish only the one-parameter limit s=t→0. Since existence of the two-parameter limit is part of the claimed theorem, a uniform-in-ratio estimate is needed. The current assertion leaves the existence of Sec_μ incomplete.
[§3.2, Definition 3.8 and §3.1.2] Definition 3.8 defines Sec_μ(P,Q) for P,Q in the geometric tangent space using the curves exp(tP), exp(tQ). However, Section 3.1.2 calls these 'pseudo geodesics' and the membership condition in Definition 3.4 only requires W_2(μ,exp(tP)) = t|P|_μ for some t>0, not that exp(tP) is a genuine geodesic for all small t. The synthetic definition in Section 2 and the implications in Theorem 2.4 are formulated for genuine geodesics. If the P_2 results are to be interpreted as sectional curvature of the geodesic space (P_2(M),W_2), this mismatch must be resolved; otherwise the corollaries using Theorem 2.4 for P_2 should state the additional geodesic hypothesis explicitly.

minor comments (5)

[§1(c)] The displayed claim 'Sec^↑_μ(φ,ψ) ≥ 0' should read 'Sec^∇_μ(φ,ψ) ≥ 0'. The same symbol swap appears in the abstract/introduction summary of the twisted part.
[§5.3, proof of Theorem 5.8, step (v)] The sentence 'Finally, observe that F = ∇f for f := (Δ_ρ)^{-1} div_μ F' is false as stated for a general vector field F and is not used in the subsequent self-adjointness identity. It should be removed or corrected.
[§6.2, Theorem 6.11] In the upper bound, '≤ 3/2 4^n (|k|²+|ℓ|²)' is missing a multiplication sign; it should read '(3/2)·4^n' or similar.
[§5.6, Theorem 5.18] The sentence 'Now assume that (M^e,HK) where NPC' appears to be missing a word; should be 'satisfies NPC' or 'is NPC'.
[§6.4, proof of Theorem 6.15] The inequality chain near the end has a typo: it should state 1/800 L² C² − 10L² ≥ 1/1000 L² C² for sufficiently large C, not '≥ -10L² ≥ ...'.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation; central HK curvature formulas are computed from independent cone-geometry and optimal-transport expansions, with one non-load-bearing self-citation and an unproved geodesic assertion noted as a correctness risk.

full rationale

The central derivation is self-contained against external benchmarks. The lifted part Sec^↑_μ(φ,ψ) is obtained by expanding the cone distance d_C^2 around exp^C via Lemma 2.1 and then using the standard O'Neill curvature formula for the cone (Lemma 4.12); this is an independent computation, not an input. The twisted part Sec^∇_μ is derived from a fourth-order transport cost estimate, with the optimal map T_t obtained from the Pinning Theorem and the Brenier–McCann theorem on the cone (Theorem 4.20, citing external work [LMS23]); the final inf over η is the projection of (F,0) onto H^1(µ), with no fitted parameter. The torus evaluations in Section 6 are explicit Fourier calculations from the formulas already derived. The only self-citation is [GS18] in Remark 6.25: 'Similar calculations have been carried out and analogous observations have been made in an unpublished work [GS18]...' This remark is not used as evidence for any theorem and is not load-bearing. The unproved assertion in Definition 4.21, 'One easily verifies that for any φ∈C_c^∞(M) and sufficiently small τ>0, the curve t↦exp^Me_μ(tφ) for t∈[0,τ] will be a HK-geodesic in M^e(M)', is load-bearing for applying Theorem 2.4 in Theorem 5.18 and Corollary 6.16, but it is a missing proof rather than a circular identification: the limit defining Sec_μ is defined via exp^Me_μ, and the geodesic claim is a separate assertion connecting that quantity to Section 2. That is a correctness risk, not a circular reduction. Hence no significant circularity; score 2 reflects the minor non-load-bearing self-citation.

Assumptions & free parameters 0 free parameters · 6 assumptions · 1 invented entities

The derivation is essentially self-contained given standard Riemannian and optimal-transport results, with no fitted constants. It relies on several cited theorems and one unproved geodesic assertion; the extended tangent space is a new object but not a free parameter.

assumptions (6)

domain assumption M has uniformly bounded sectional curvature (|Sec| ≤ K)
Standing assumption at start of Sec. 3; used for dominated convergence and finiteness of curvature integrals.
ad hoc to paper exp^Me_μ(tφ) is an HK-geodesic for small t for arbitrary μ and φ∈C_c^∞
Definition 4.21, 'One easily verifies'; unproved and load-bearing.
domain assumption Existence and t-smoothness of H K-optimal transport maps on the cone
Proof of Thm 5.8 uses McCann-Brenier on cone and elliptic regularity to get T_t = exp^C(∇θ_t,2rθ_t) with θ_t smooth in t; requires μ≪m and smooth data.
domain assumption HK metric structure from Liero-Mielke-Savaré [LMS16,18,23]
Cone picture, Reduction Lemma, geodesics, dynamic formulation are cited and used throughout Sec. 4.
domain assumption Theorem 4.20 from [LMS23]: uniqueness of HK geodesics for a.c. measures on R^n
Invoked to justify exponential map form of geodesics in Euclidean twisted-part computation.
standard math O'Neill's warped-product curvature formula
Used in Lemma 4.12 for cone sectional curvature.

invented entities (1)

Extended tangent space T^ext_μ M^e = H^1(μ) ⊗ √M^e(M \ B_{π/2}(supp μ))
purpose: Formalizes tangent directions that combine H^1 perturbations with creation/annihilation of mass outside the cut-off ball, enabling curvature for non-geodesic extended directions.
Defined in Def. 5.14; a mathematical construction within the paper, not an independently testable entity.

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Pith. "Pith review of Sectional Curvature for Kantorovich-Wasserstein and Hellinger-Kantorovich Geometries." pith.science (2026). https://pith.science/paper/VOAXO2OA

@misc{pith2026260614318,
  author       = {Pith},
  title        = {Pith review of: Sectional Curvature for Kantorovich-Wasserstein and Hellinger-Kantorovich Geometries},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VOAXO2OA}},
  note         = {Machine review of arXiv:2606.14318}
}

abstract

We derive an explicit formula for the sectional curvature of the space ${\cal M}(M)$ of finite measures on a Riemannian manifold M. The space ${\cal M}(M)$ is equipped with the Hellinger-Kantorovich metric $HK$. Even in the case M=R^n, the curvature is comprised of two parts: the `lifted part' is negative, and the `twisted part' is positive. It will be analyzed in detail for the multidimensional torus. Our general approach to sectional curvature in geodesic spaces also leads to new insights into the curvature of the space $P_2(M)$ of probability measures on M equipped with the Kantorovich-Wasserstein metric $W_2$.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

A 'global' Perspective on the Differential Geometry of Wasserstein Spaces
math.DG 2026-07 accept novelty 6.0 of 10

On the extended Otto metric, the Levi-Civita connection and Riemann tensor of Wasserstein space are exact lifts of the base manifold's, so Lott-type correction terms are extrinsic projection artifacts.

Reference graph

Works this paper leans on

16 extracted references · 2 linked inside Pith · cited by 1 Pith paper

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Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

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Burago, I.U.D

D. Burago, I.U.D. Burago, and S. Ivanov, A course in metric geometry, Graduate Studies in Mathematics, American Mathematical Society, 2001

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Angelina Bisson, An exposition of the curvature of warped product manifolds, CSUSB Scholar works, https://scholarworks.lib.csusb.edu/etd/1810 (2023)

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1, 1--44

Lenaic Chizat, Gabriel Peyr \'e , Bernhard Schmitzer, and Fran c ois-Xavier Vialard, An interpolating distance between optimal transport and F isher-- R ao metrics , Foundations of Computational Mathematics 18 (2018), no. 1, 1--44

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[5] [5]

Nicol \`o De Ponti, Giacomo Enrico Sodini, and Luca Tamanini, The infimal convolution structure of the H ellinger- K antorovich distance , arXiv preprint arXiv:2503.12939 (2025)

arXiv 2025

[6] [6]

Lorenzo Dello Schiavo and Giacomo Enrico Sodini, The H ellinger- K antorovich metric measure geometry on spaces of measures , arXiv preprint arXiv:2503.07802 (2025)

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Nicola Gigli, On the inverse implication of B renier- M c C ann theorems and the structure of ( P_2(M),W_2) , Methods Appl. Anal. 18 (2011), no. 2, 127--158. 2847481 (2012h:49090)

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[8] [8]

Maria Gordina and Karl-Theodor Sturm, Sectional and R icci curvature on the W asserstein space , Unpublished preprint (2018)

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