{"id":"b8dac43c-bf52-4aa1-b40a-b578be391dfa","arxiv_id":"2411.15324","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A Riemannian submersion from a compact non-negatively curved manifold is necessarily smooth, resolving a conjecture of Berestovskii-Guijarro.","lead":"This paper proves a conjecture from 2000: every Riemannian submersion from a compact non-negatively curved Riemannian manifold is automatically a smooth map. It also shows that, in the broader setting of submetries, smoothness of the total space forces smoothness of the base.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The final upgrade in Theorem 1.1 hinges on an unproved transfer: the proof of [Wil07, Theorem 3] is asserted to apply literally to the C^2 submetries of Theorem 16.1, but the smoothness-dependent steps are not written out.","rationale":"The reader's verdict identifies exactly the same load-bearing transfer, and I agree with that assessment. The bulk of the paper—Theorems 8.1, 9.1, 12.1, 13.6, 14.1, 16.1, 17.1, and 19.1—forms a coherent, technically detailed development, and the local regularity arguments are substantial and plausible. However, the central claim of Theorem 1.1 for compact nonnegatively curved submersions is completed only by Section 20.2, where the decisive completeness of dual leaves is imported by the phrase 'applies literally' rather than proved. Since Theorem 20.3's hypothesis is exactly this completeness, the conclusion does not follow until the transfer is verified. The gap is not a contradiction with anything proved in the paper, but it is a missing derivation in the central argument; a CONDITIONAL verdict is therefore appropriate. I also considered whether the C^2 regularity from Theorem 16.1 makes the transfer automatic: it gives C^2 fibers, hence C^1 dual leaves and a C^1 horizontal distribution via Proposition 18.4, but the arguments of [Wil07] use smoothness of holonomy and of families of Jacobi fields beyond what is explicitly established here. The symmetric-space transfer in Section 20.3 carries a similar concern, though it does not affect the compact Riemannian submersion case directly. The issue is one of unverified transfer, not of internal inconsistency or authorial intent; a line-by-line adaptation would settle it.","tokens_in":42219,"tokens_out":3537,"duration_ms":36378,"concrete_test":"Write out a complete adaptation of the proof of [Wil07, Theorem 3] in the present setting where P is only known to be C^2 (Theorem 16.1). In particular, verify each use of smoothness of P or of the dual foliation, and reproduce the argument that a maximal horizontal broken geodesic has an endpoint in its dual leaf using only the C^1 regularity of dual leaves from Proposition 18.4 and the conclusions of Corollary 18.7. If any step requires P to be C^3, or requires dual leaves to be smooth rather than C^1, then the transfer fails and Theorem 1.1 is not established.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The decisive step for the compact nonnegatively curved part of Theorem 1.1 is Section 20.2: after Theorem 16.1 gives only C^2 regularity, the text states that the proof of [Wil07, Theorem 3] 'applies literally and shows that all leaves of the dual foliation of P are complete.' This completeness is exactly the hypothesis needed by Theorem 20.3 to promote almost-everywhere smoothness to global smoothness. The assertion is not derived in the paper, and it is not a formal corollary of the earlier theorems: Theorem 20.1 only converts completeness of dual leaves into the dual foliation being Riemannian; it does not produce that completeness. The cited result was proved for smooth Riemannian submersions, where the horizontal distribution, the holonomy maps, and the dual foliation are smooth. Here only C^2 fibers, hence C^1 dual leaves and a Lipschitz/C^1 horizontal distribution, are available. Unless every step of [Wil07, Theorem 3] that uses smoothness is checked in the C^2 category—especially the Jacobi-field comparison and the conclusion that each dual leaf is complete—the final implication of Theorem 1.1 remains unsupported. Section 20.3 makes an analogous transfer from [eSS22] for symmetric spaces, but that is secondary for the compact Riemannian submersion statement.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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].","tokens_in":42443,"tokens_out":12984,"duration_ms":130462,"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":[{"comment":"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.","section":"Section 20.2"},{"comment":"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].","section":"Section 20.3"},{"comment":"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.","section":"Section 20.2"}],"minor_comments":[{"comment":"The name \"Berestovksii\" in the sentence \"Valeryi Berestovksii and Luis Guijarro verified...\" is a typo for \"Berestovskii\".","section":"Section 1.1"},{"comment":"In Theorem 8.1, \"any stratum Y e is isometiric to a smooth Riemannian manifold\" contains a typo: \"isometiric\" should be \"isometric\".","section":"Section 8.1"},{"comment":"The sentence \"We such γ a regular horizontal geodesic\" is missing a verb; it should read \"We call such γ a regular horizontal geodesic.\"","section":"Section 5.5"},{"comment":"In the proof of Theorem 16.1, the phrase \"the unqiue extension\" should read \"the unique extension\".","section":"Section 16.1"},{"comment":"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.","section":"Section 13.2"}],"recommendation":"major_revision","confidential_remarks":"The paper is strong and the bulk of the technical development appears sound and substantial. The sole blocking issue is the unverified transfer of [Wil07, Theorem 3] and [eSS22] to the C^2 setting in Sections 20.2 and 20.3. This is a central logical step for the advertised main theorems, but it is localized and could plausibly be repaired by writing out the transfer in detail. I recommend major revision rather than rejection, and I would be willing to look at a revised version with a complete treatment of the dual-leaf completeness in the C^2 category."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is the real thing—a serious, mostly detailed proof of the Berestovskii–Guijarro conjecture plus a stack of new regularity theorems for submetries. Read Sections 8, 14, 16, and 19 for the substance. The one place the paper leans on an unproved transfer is Section 20, and the referee should not let that slide.\n\nWhat is genuinely new: Theorem 8.1, smoothness of the strata of the base with no curvature assumption, answers Lempert's question and sharpens [BG00]'s C^{1,1} result. Theorem 1.4/13.6 settles Radeschi's transnormality question. Theorem 9.1, the codimension-one slice with reflection quotient, is new input. Equifocality (Proposition 14.1) is the technical heart. The C^2 theorem (16.1) and the dense-open smoothness theorem (17.1) are major results proved in the text, and in positive curvature the full smoothness (Corollary 17.2) is derived rather than imported. The citation pattern is fine: [KL22], [Wil07], and [eSS22] are genuine prior results about different statements, and the authors flag their dependence on [Wil07] openly in Section 1.4. No circularity that I can see.\n\nThe soft spot is exactly where your report puts it, and the stress-test note holds up on reading. Theorem 16.1 yields only C^2 fibers, hence C^1 dual leaves and a Lipschitz/C^1 horizontal distribution. Section 20.2 asserts that the proof of [Wil07, Theorem 3] 'applies literally' to prove completeness of all dual leaves—the hypothesis Theorem 20.3 needs to promote almost-everywhere smoothness to global smoothness. Section 20.3 makes a parallel transfer from [eSS22]. Neither is derived. Wilking's theorem was proved for smooth submersions, where holonomy maps and the dual foliation are smooth; here they are not. The transfer may well go through, since the authors have rebuilt most of the Jacobi-field machinery in lower regularity, but 'applies literally' is doing real work and it is not in the text. That is a load-bearing assertion, not a minor omission.\n\nIt does not sink the rest. The local theorems, the base smoothness, the C^2 statement, and the dense-open smoothness all have real proofs. If the transfer fails, the compact case of Theorem 1.1 falls back to 'C^2 plus almost everywhere smooth'—still substantial, but not the conjecture.\n\nWho benefits: anyone in submetries, singular Riemannian foliations, or Alexandrov quotients. It deserves a serious referee. Recommend conditional acceptance, with expansion of Section 20 as the explicit condition.","headline":"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.","tokens_in":43030,"tokens_out":5650,"would_cite":true,"duration_ms":47696,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C20","53C21","53C23"],"pacs":[],"model":"deepseek-v4-flash","headline":"A submetry from a smooth Riemannian manifold has a smooth base, and on nonnegatively curved compact manifolds the submetry itself is smooth.","keywords":["submetry","Riemannian submersion","nonnegative sectional curvature","Berestovskii–Guijarro conjecture","singular Riemannian foliation","dual foliation","equifocality","Jacobi fields"],"falsifier":"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.","tokens_in":41958,"feed_emoji":"📐","tokens_out":7017,"duration_ms":59260,"temperature":0.7,"pith_summary":"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.","feed_headline":"Nonnegative curvature forces submetries to be smooth","feed_subtitle":"Resolves the Berestovskii–Guijarro conjecture: smooth total metric implies smooth base and smooth map under nonnegative curvature.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Established that submetries between smooth manifolds are $C^{1,1}$ and formulated the conjecture that nonnegative curvature forces smoothness.","marker":"[BG00]"},{"why":"Supplied the duality theorem for Riemannian foliations in nonnegative curvature and the dual-leaf completeness theorem transferred in Section 20.2.","marker":"[Wil07]"},{"why":"Provided the structural theory of submetries used throughout: stratification, positive reach, holonomy maps, and basic normal fields.","marker":"[KL22]"},{"why":"Verified completeness of dual foliations on nonnegatively curved symmetric spaces, the input for Theorem 1.5.","marker":"[eSS22]"},{"why":"Gave the codimension-one orbifold structure and curvature-explosion results applied in the proof of Theorem 9.1.","marker":"[LT10]"},{"why":"Linked focal distances to eigenvalues of the second fundamental form in Euclidean space, used in Theorem 15.2.","marker":"[PT88]"},{"why":"Provided the orbit-foliation theorem used to show dual leaves form a singular $C^{k-1}$ foliation (Proposition 18.4).","marker":"[Ste74]"},{"why":"Posed the smoothness-of-the-base question for Riemannian submersions and proved it in the real-analytic case.","marker":"[Lem19]"},{"why":"Supplied regularity facts about submetries, including holonomy properties of horizontal geodesics and typical directions.","marker":"[Lyt24b]"}],"fun_headline_variants":["Submetries smooth under nonnegative curvature","Curvature forces submetries to be smooth","Nonnegative curvature smooths submetries","Smooth submetries from curvature"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Submetries smooth under nonnegative curvature","Curvature forces submetries to be smooth","Nonnegative curvature smooths submetries","Smooth submetries from curvature"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000226,"raw_usage":{"total_tokens":1402,"prompt_tokens":813,"completion_tokens":589,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":429,"completion_tokens_details":{"reasoning_tokens":533}},"tokens_in":429,"tokens_out":589,"duration_ms":5313,"temperature":1.0,"reasoning_tokens":533,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:25:32.867291+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}