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Smoothness of submetries

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Pith's one-line read A submetry from a smooth Riemannian manifold has a smooth base, and on nonnegatively curved compact manifolds the submetry itself is smooth.

desk verdict A serious, mostly detailed proof of the Berestovskii–Guijarro conjecture; the compact non-negative curvature upgrade rests on an asserted one-sentence transfer from [Wil07] that the referee should require be written out. read the letter →

arxiv 2411.15324 v1 pith:YJG5ZD5N submitted 2024-11-22 math.DG

Alexander Lytchak , Burkhard Wilking This is my paper · ORCID
classification math.DG MSC 53C2053C2153C23
keywords submetryRiemanniansubmersionnonnegativesectionalcurvatureBerestovskii–GuijarroconjecturesingularfoliationdualequifocalityJacobifields
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

This paper proves that the base space of a submetry from a smooth Riemannian manifold is always a smooth Riemannian manifold (Theorem 1.2), and that a Riemannian submersion from a compact nonnegatively curved manifold must itself be smooth (Theorem 1.1). Together these results resolve the Berestovskii–Guijarro conjecture, which asserted exactly this smoothness under nonnegative curvature. The proofs are carried out in the generality of submetries, so they also apply to isometric group actions and singular Riemannian foliations with closed leaves. A sympathetic reader should care because the result says that the notoriously wild metric maps that submetries can be are tamed by curvature assumptions that appear in many geometric rigidity problems.

What carries the argument

The central objects are the holonomy fields of basic normal vector fields along a fiber, and the Lagrangians of $L$-Jacobi fields they generate along horizontal geodesics. For a transnormal submetry these holonomy fields span the vertical spaces along the geodesic, and Proposition 14.1 establishes that the spaces of focalizing Jacobi fields depend Lipschitz continuously on the base point (with $C^{k-1}$ dependence when the submetry is $C^k$). This equifocality lets the paper control the second fundamental form of a fiber and then apply the regularity criterion of Theorem 4.1 to bootstrap smoothness. The transversal Jacobi equation of Wilking links these Jacobi-field spaces to the Jacobi equation on the quotient, which is what identifies horizontal focal multiplicities with conjugate multiplicities along quasi-geodesics in the smooth Riemannian orbifold $Y^m \cup Y^{m-1}$. Finally, the dual foliation—the decomposition into sets connected by piecewise horizontal geodesics—carries the smoothness upgrade from dense open sets to all of $M$ once its leaves are known to be complete.

What would settle it

A single example of a $C^1$ Riemannian submersion from a compact nonnegatively curved smooth Riemannian manifold onto a smooth manifold that is not smooth would refute the second assertion of Theorem 1.1; likewise, a transnormal submetry from a compact nonnegatively curved manifold with an incomplete dual leaf would break the final smoothness upgrade in Section 20.

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Extended reading notes

Core claim

The central discovery is that the $C^{1,1}$ regularity boundary found by Berestovskii and Guijarro—submetries between smooth manifolds need not be $C^2$—cannot be realized when the total space has nonnegative sectional curvature. In the compact Riemannian submersion case, the paper proves that the base metric is smooth and that the map $P$ is smooth; for general submetries it proves smoothness of each stratum of the base's canonical stratification. The mechanism behind the upgrade is that focal behavior along horizontal geodesics is equifocal: the spaces of $L$-Jacobi fields that focalize at a given time depend Lipschitz continuously on the point, and this equifocality feeds a bootstrap that raises the differentiability class of the fibers one derivative at a time. The final step, from $C^2$ and dense smoothness to global smoothness, goes through the dual foliation of the submetry and its completeness.

Load-bearing premise

The conclusion that $C^2$ regularity upgrades to full smoothness rests on the transfer of Wilking's dual-leaf completeness theorem from smooth Riemannian submersions to the $C^2$ submetries produced in Theorem 16.1, and the paper states that the proof 'applies literally' without a full re-derivation.

Editorial extensions

If this is right

  • The base of any submetry from a smooth Riemannian manifold is a smooth Riemannian manifold, so quotient objects such as orbit spaces of isometric group actions carry canonical smooth structures.
  • On compact nonnegatively curved manifolds, isometric group actions and closed-leaf singular Riemannian foliations have smooth quotient maps, not just smooth quotients.
  • The O'Neill inequality $\kappa(E) \le \kappa(E')$ is valid for $C^{1,1}$ Riemannian submersions, and equality characterizes the presence of totally geodesic horizontal sections.
  • All transnormal submetries from Euclidean space are smooth and have basic mean curvature, giving a large supply of smooth singular Riemannian foliations.
  • The equifocality bootstrap provides a general template: under nonnegative curvature, understanding focal Jacobi fields along horizontal geodesics is enough to force smoothness.

Reading between the lines

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

  • If the dual-leaf completeness transfer holds verbatim, the same argument should extend Theorem 1.5 to all complete nonnegatively curved manifolds, closing the gap the paper leaves via Theorem 1.6.
  • One testable extension is to check whether Proposition 14.1 holds for local submetries without transnormality, which would likely imply smoothness of the base stratum in the nontransnormal case as well.
  • The role of the codimension-one stratum theorem suggests that similar slice arguments might prove the whole base is a smooth Riemannian orbifold up to codimension at least two, a question the paper phrases as the polarity conjecture.
  • The paper's Question 1.11—whether non-smooth compact examples exist—narrows to submetries with incomplete dual leaves, so the search for counterexamples should focus on dual-leaf geometry rather than on local fiber singularities.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies regularity of submetries from a smooth Riemannian manifold to a metric space, and applies the results to Riemannian submersions. The main theorem states that a surjective C^1 Riemannian submersion from a smooth Riemannian manifold has smooth base, and that if in addition the total space is compact and non-negatively curved, the submersion itself is smooth. The proofs proceed through a long chain of structural results: smoothness of the strata of the base (Theorem 8.1), a codimension-one slice description (Theorem 9.1), a rigidity theorem forcing transnormality from C^2 regular fibers (Theorem 13.6), a C^2-regularity theorem in non-negative curvature (Theorem 16.1), almost-everywhere smoothness with a rank argument (Theorem 17.1), propagation of smoothness along dual leaves (Theorem 19.1), and a final upgrade to global smoothness once all dual leaves are complete (Theorem 20.3). The completeness of dual leaves in the compact Riemannian submersion case is asserted by a literal transfer of [Wil07, Theorem 3] and, in the symmetric-space case, of [eSS22].

Significance. If the main theorem is correct, it resolves the Berestovskii–Guijarro conjecture in the compact case and closes the C^{1,1}-versus-smooth regularity gap left by [BG00]. The paper contains substantial independent material of lasting value: the smoothness of the base strata, the codimension-one orbifold description, the equifocality statement of Proposition 14.1, the almost-everywhere O'Neill formula, and the C^2-regularity theorem in non-negative curvature. The proofs of Theorems 8.1, 16.1 and 19.1 are detailed and mostly self-contained within the framework of [KL22]. However, the final compact and symmetric-space upgrades depend on an asserted, rather than verified, transfer of smooth-object theorems to the C^2 setting.

major comments (3)
  1. [Section 20.2] The assertion that "The proof of [Wil07, Theorem 3] (given under the assumption that P is smooth) applies literally" is load-bearing for the second statement of Theorem 1.1. At this point in the paper, P is only known to be C^2 by Theorem 16.1, the dual foliation is only C^1 by Proposition 18.4, and the horizontal distribution is Lipschitz but not necessarily smooth. The reference to [Wil07, Theorem 3] is not a proof: the completeness of each dual leaf is exactly the hypothesis needed by Theorem 20.3, and Theorem 20.1 only converts that completeness into Riemannianness of the dual foliation, it does not establish it. The text does not identify which steps in [Wil07, Theorem 3] require smoothness or how they are replaced by the C^2 analogues proved here (for example Proposition 14.1, Corollary 18.7, and Theorem 16.1). Unless such a check is supplied, the compact non-negative curvature part of Theorem 1.1 is not established by the manuscript as written.
  2. [Section 20.3] The same transfer problem occurs in the sentence "the arguments of [eSS22] apply without changes to the present situation and show that the dual leaves of P are complete." This is used to prove the symmetric-space case of Theorem 1.5, which is one of the advertised main results. The regularity category is again C^2, while [eSS22] is cited for smooth submetries; the assertion that every argument transfers is not demonstrated. The authors should either provide the missing verification, or explicitly state this as a conditional result pending the C^2 extension of [eSS22].
  3. [Section 20.2] A related but secondary point is the sentence in the same section that the proof of [Wil07, Theorem 2] "applies literally" to yield Theorem 20.1. This is less severe than the transfer of [Wil07, Theorem 3], because Theorem 20.1 is conditional on the completeness of dual leaves and the surrounding paper has already developed many of the needed tools. Still, the phrase "applies literally" is a substitute for an argument. If the authors retain this style of citation, they should list the specific statements from [Wil07] that are being reused and point to the corresponding lemmas in the present paper that supply the needed regularity.
minor comments (5)
  1. [Section 1.1] The name "Berestovksii" in the sentence "Valeryi Berestovksii and Luis Guijarro verified..." is a typo for "Berestovskii".
  2. [Section 8.1] In Theorem 8.1, "any stratum Y e is isometiric to a smooth Riemannian manifold" contains a typo: "isometiric" should be "isometric".
  3. [Section 5.5] The sentence "We such γ a regular horizontal geodesic" is missing a verb; it should read "We call such γ a regular horizontal geodesic."
  4. [Section 16.1] In the proof of Theorem 16.1, the phrase "the unqiue extension" should read "the unique extension".
  5. [Section 13.2] The cross-reference "Under the assumption of Theorem 13.2" in Proposition 13.3 should be "Under the assumption of Proposition 13.2", since the hypothesis is stated in Proposition 13.2.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proof chain is self-contained except for final external transfers that are gap-like, not circular.

full rationale

The derivation chain does not exhibit a circular step. In the main body (Sections 8–19), the paper proves the smoothness of strata, the O’Neill inequality, transnormality, C^2 regularity, and propagation of smoothness along dual leaves without assuming the conclusion. The only step that delegates to prior work is the invocation in Sections 20.2–20.3 of [Wil07, Theorem 3] and [eSS22] to obtain completeness of dual leaves. These are external theorems about smooth foliations and symmetric spaces, not restatements of the target theorem, and the paper does not define any object in terms of the desired conclusion. The assertion that the proofs ‘apply literally’ to the C^2 submetry setting is a regularity-transfer claim that may require checking, but failing to check it is a gap in rigor, not a circular definition or a fitted-input prediction. Accordingly no circular step is exhibited.

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

No numbers are fitted and no entities are postulated. The proof rests on a chain of prior geometric theorems; the least supported link is the asserted transfer of dual-leaf completeness from smooth to C^2 submetries. The other imported results are standard or established in the cited literature.

assumptions (6)
  • domain assumption Strata of the quotient Y^e are C^{1,1} Riemannian manifolds with Lipschitz metric; fibers and preimages of strata have positive reach.
    Imported from [KL22] and used as the starting point for Theorems 1.2, 1.3, 8.1 and 9.1.
  • standard math Transversal Jacobi equation and Wilking's Lagrangian calculus apply to Lagrangians of normal Jacobi fields.
    Used in Section 3.5 and Section 11.2 to identify horizontal focal multiplicities with conjugate points along quotient geodesics.
  • standard math Focal indices of Lagrangian spaces vary continuously and focalizing subspaces converge to limits.
    Keeps track of focal-generated subspaces in the C^2 and smoothness bootstrap arguments of Sections 16 and 17.
  • standard math Stefan-Sussmann theorem: orbits of a family of C^k flows form a singular foliation.
    Used in Proposition 18.4 to make dual leaves leaves of a C^{k-1} singular foliation.
  • standard math Elliptic regularity in the form of Sabitov-Sheffel and DeTurck-Kazhdan, plus Nash embedding.
    Used in Theorem 4.1 to upgrade submanifold regularity from second fundamental form or mean curvature regularity.
  • domain assumption Dual-leaf completeness theorems of [Wil07, Theorem 3] and [eSS22] transfer to the C^2 submetries obtained here.
    The paper says their proofs apply literally in Sections 20.2 and 20.3 but does not write the adaptation; this is the weakest imported step.

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Pith. "Pith review of Smoothness of submetries." pith.science (2026). https://pith.science/paper/YJG5ZD5N

@misc{pith2026241115324,
  author       = {Pith},
  title        = {Pith review of: Smoothness of submetries},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YJG5ZD5N}},
  note         = {Machine review of arXiv:2411.15324}
}
read the original abstract

We prove that a Riemannian submersion between smooth, compact, non-negatively curved Riemannian manifolds has to be smooth, resolving a conjecture by Berestovskii--Guijarro. We show that without any curvature assumption, the smoothness of the base is implied by the smoothness of the total space. Results are proven in the much more general setting of submetries. These are metric generalizations of Riemannian submersions and isometric actions, which have recently appeared in different areas of geometry.

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