Non-canonical variations of Riemannian submersions with totally geodesic fibers

Tomasz Zawadzki

arxiv: 2604.05118 · v1 · submitted 2026-04-06 · 🧮 math.DG

Non-canonical variations of Riemannian submersions with totally geodesic fibers

Tomasz Zawadzki This is my paper

Pith reviewed 2026-05-10 19:01 UTC · model grok-4.3

classification 🧮 math.DG

keywords Riemannian submersionstotally geodesic fiberssectional curvaturesmetric variations1-formK-contact structuresfat submersionshorizontal distribution

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The pith

A 1-form variation of the horizontal distribution can turn all sectional curvatures positive on a product of a positive-curvature manifold with a circle while keeping the projection a Riemannian submersion with totally geodesic fibers.

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

The paper constructs deformations of the metric on a manifold that is locally a product of a space with positive sectional curvature and a circle. These deformations are defined by a 1-form and change only the horizontal distribution. They keep the projection map a Riemannian submersion whose fibers remain totally geodesic and whose fiber metric stays fixed. When the 1-form satisfies explicit algebraic conditions, every sectional curvature of the deformed metric becomes positive. The same technique yields new contact structures from K-contact ones and produces examples of fat submersions whose vertizontal curvatures vary along each fiber.

Core claim

What carries the argument

The 1-form that defines the variation of the horizontal distribution, which adjusts the metric while preserving the submersion, the totally geodesic property of the fibers, and the fiber metric.

If this is right

Explicit conditions on the 1-form guarantee positive sectional curvatures on the deformed product metric.
K-contact structures admit deformations to weak contact metric structures via the same 1-form variations.
Fat Riemannian submersions with totally geodesic fibers exist whose vertizontal curvatures are non-constant along fibers.
For submersions coming from isometric group actions with fiber dimension greater than one, the variations preserve the group action while changing the horizontal curvatures.

Where Pith is reading between the lines

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

The technique supplies a systematic way to deform product metrics into ones with strictly positive curvature without destroying the submersion structure.
Similar 1-form variations might be used to study curvature properties on other bundles whose fibers are circles or tori.
The non-canonical character of the variations suggests that many standard constructions of positive-curvature metrics can be further deformed while keeping submersion data intact.

Load-bearing premise

The 1-form must preserve the given Riemannian submersion, keep the fibers totally geodesic, and fix the metric on the fibers at least locally.

What would settle it

An explicit example of a positive-curvature manifold times a circle on which no such 1-form exists that makes every sectional curvature positive.

read the original abstract

Using variations of Riemannian metric that preserve a given Riemannian submersion, keep its fibers totally geodesic and the metric restricted to the fibers fixed, but change the horizontal distribution, we examine changes of sectional curvatures in horizontal and vertical directions. We obtain conditions, in terms of a 1-form defining a variation, to locally make all sectional curvatures positive on the product of a manifold with positive curvature and a circle, while preserving the Riemannian submersion with geodesic fibers defined by the projection from the product. We examine conditions for obtaining weak contact metric structures from K-contact structures. We demonstrate existence of fat Riemannian submersions with totally geodesic fibers and vertizontal (i.e., spanned by a horizontal and a vertical vector) curvatures non-constant along a fiber. For a Riemannian submersion defined by an isometric group action, with totally geodesic fibers of dimension higher than one, we find variations that preserve the isometric action, while changing the horizontal distribution and its curvatures.

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.

Desk Editor's Note private letter to a colleague

This paper gives explicit 1-form conditions for deforming the horizontal distribution in Riemannian submersions while keeping fibers totally geodesic, yielding local positive sectional curvature on positive-curvature manifolds times a circle plus some new fat submersion examples.

read the letter

The key move here is using metric variations that preserve a given Riemannian submersion, keep fibers totally geodesic, and fix the fiber metric but shift the horizontal distribution via a 1-form. From that they extract conditions on the 1-form that make all sectional curvatures positive locally on M times S1 when M already has positive curvature. They also get conditions turning K-contact structures into weak contact ones and produce fat submersions with totally geodesic fibers where vertizontal curvature varies along the fiber. For submersions from isometric group actions with fiber dimension bigger than one, the variations preserve the action while changing the horizontal curvatures. The derivations rely on the Koszul formula and O'Neill identities applied directly to the constrained data, which keeps things straightforward. The stress-test note confirms the positivity inequalities are solvable locally without hidden sign problems or dimension cancellations, so the central claims hold up on the stated assumptions. One soft spot is that the positivity and the new examples stay local or perturbative; the paper does not appear to deliver global extensions or many fully explicit coordinate charts that would make the fat submersions immediately reproducible by others. The contact-structure application is a nice side result but feels secondary to the curvature-variation core. This is aimed at people who build or deform examples in Riemannian geometry and contact geometry. It supplies reusable technical tools rather than settling open existence questions, yet the constructions are concrete enough to be worth checking. I would bring it to a reading group focused on submersions or positive curvature. It deserves peer review so referees can verify the curvature formulas and see how flexible the 1-form conditions really are in practice.

Referee Report

0 major / 3 minor

Summary. The manuscript studies non-canonical variations of Riemannian metrics on submersions with totally geodesic fibers that preserve the submersion structure and fiber metric while altering the horizontal distribution through a defining 1-form. It derives explicit conditions on this 1-form to produce positive sectional curvatures on the product of a positive-curvature manifold with a circle, gives criteria for obtaining weak contact metric structures from K-contact structures, establishes existence of fat Riemannian submersions with non-constant vertizontal curvatures along fibers, and constructs variations that preserve isometric group actions while changing horizontal curvatures for fibers of dimension greater than one.

Significance. If the derivations hold, the paper supplies a direct, constructive method for deforming Riemannian submersions to control curvature signs while rigidly preserving fiber geometry and the submersion property. The reduction of positivity conditions to solvable inequalities on the coefficients of the 1-form (derived from the Koszul formula and O'Neill identities) is a clear strength, as is the absence of ad-hoc parameters or circular definitions. These techniques could be useful for constructing examples in positive-curvature geometry and for deforming contact structures.

minor comments (3)

The abstract is dense and enumerates four distinct results without indicating the paper's sectional organization or theorem numbering; a brief roadmap sentence would improve readability.
The term 'vertizontal' is used in the abstract and introduction before its definition (spanned by one horizontal and one vertical vector) appears; a parenthetical clarification on first use would help.
In the discussion of fat submersions, the explicit example manifold realizing non-constant vertizontal curvature along the fiber is referenced only generically; adding a concrete reference or coordinate description would strengthen the existence claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their detailed summary of the manuscript and for the positive assessment of its significance. The recommendation for minor revision is noted. However, the report contains no specific major comments or criticisms to address point by point. We are happy to incorporate any minor clarifications or adjustments if the editor or referee provides further details.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper derives curvature variation formulas directly from the Koszul formula and O'Neill-type identities applied to metric variations constrained to preserve the Riemannian submersion, totally geodesic fibers, and fixed fiber metric. These constraints are imposed by definition on the 1-form variation, and the resulting positivity conditions on M × S¹ reduce to explicit inequalities on the 1-form coefficients that are solved locally when the base curvature is positive. No step renames a fitted parameter as a prediction, imports uniqueness via self-citation, or reduces the claimed results to the input data by construction. The constructions for contact structures and fat submersions are likewise explicit existence arguments without load-bearing self-referential loops.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The work rests on the standard axioms of Riemannian geometry and the definition of Riemannian submersions with totally geodesic fibers; the 1-form is an auxiliary object chosen to satisfy the stated conditions rather than a fitted parameter.

free parameters (1)

1-form defining the variation
The variation is parameterized by a 1-form whose choice determines the new horizontal distribution and curvature signs; it is not fitted to data but selected to meet the positivity conditions.

axioms (2)

domain assumption The base manifold has positive sectional curvature and the total space is its product with a circle.
Invoked when stating the curvature-positivity result for the projection submersion.
domain assumption The variation preserves the Riemannian submersion structure and totally geodesic fibers.
Central hypothesis under which all curvature calculations are performed.

pith-pipeline@v0.9.0 · 5457 in / 1551 out tokens · 63056 ms · 2026-05-10T19:01:52.517465+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Using variations of Riemannian metric that preserve a given Riemannian submersion, keep its fibers totally geodesic and the metric restricted to the fibers fixed, but change the horizontal distribution, we examine changes of sectional curvatures...
IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

we prove that the second derivative of all vertizontal curvatures is positive where the 1-form has non-degenerate exterior differential

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

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Blair, D.: Riemannian Geometry of Contact and Symplectic Manifolds, Birkh¨ auser (2010)

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Boyer, C., Galicki, K.: Sasakian geometry, Oxford Mathematical Monographs. Oxford University Press, Oxford, 2008

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Ehresmann, C.: Les connexions infinit´ esimales dans un espace fibr´ e diff´ erentiable. Colloque de topologie (espaces fibr´ es), Bruxelles, 1950, 29-55. Georges Thone, Li` ege; Masson & Cie, Paris (1951)

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Goertsches, O., Loiudice, E.: How to construct all metric f-K-contact manifolds. Advances in Geometry, vol. 21, no. 4, pp. 591-598 (2021)

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Gray, A.: Pseudo-Riemannian almost product manifolds and submersions. J. Math. Mech., 16, no. 7, 715-737 (1967)

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Gromoll, D., Walschap, G.: Metric foliations and curvature, Birkh¨ auser (2009)

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Z., 238, 829–832 (2001)

Kashiwada, T.: On a contact 3-structure, Math. Z., 238, 829–832 (2001)

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Rovenski, V.: Foliated structure of weak nearly Sasakian manifolds. Annali di Matematica 203, 2641–2652 (2024). https://doi.org/10.1007/s10231-024-01459- 7

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Sharipov, X., Sharipova, S., Salimov, E., Mardiev, I.: Local Equivalence of Rie- mannian Submersions via Differential Invariants, preprint, arXiv:2603.14820v1 (2026) 48

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Zawadzki, T.: Variations of metric that preserve a Riemannian submersion and geometry of its fibers, preprint, arXiv:2412.00969 (2025)

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[20]

Ziller, W.: Examples of Riemannian manifolds with nonnegative sectional cur- vature, in: Metric and Comparison Geometry, Surv. Diff. Geom. 11, ed. K. Grove and J. Cheeger, 63-102 (2007)

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Ziller, W.: Fatness Revisited, lecture notes (2005). 49

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[1] [1]

L.: Einstein manifolds

Besse, A. L.: Einstein manifolds. Springer-Verlag, Berlin (1987) 47

work page 1987

[2] [2]

Blair, D.: Riemannian Geometry of Contact and Symplectic Manifolds, Birkh¨ auser (2010)

work page 2010

[3] [3]

Sentenac, P.: Conjecture de H

Bourguignon, J.P., Deschamps, A. Sentenac, P.: Conjecture de H. Hopf sur les produits des vari´ et´ es. Ann. Sci. Ecole Norm. Super. 5, 277-302 (1972)

work page 1972

[4] [4]

Oxford University Press, Oxford, 2008

Boyer, C., Galicki, K.: Sasakian geometry, Oxford Mathematical Monographs. Oxford University Press, Oxford, 2008

work page 2008

[5] [5]

Cheeger, J.: Some examples of manifolds of nonnegative curvature. J. Diff. Geom. 8 , 623-628 (1973)

work page 1973

[6] [6]

Colloque de topologie (espaces fibr´ es), Bruxelles, 1950, 29-55

Ehresmann, C.: Les connexions infinit´ esimales dans un espace fibr´ e diff´ erentiable. Colloque de topologie (espaces fibr´ es), Bruxelles, 1950, 29-55. Georges Thone, Li` ege; Masson & Cie, Paris (1951)

work page 1950

[7] [7]

Advances in Geometry, vol

Goertsches, O., Loiudice, E.: How to construct all metric f-K-contact manifolds. Advances in Geometry, vol. 21, no. 4, pp. 591-598 (2021)

work page 2021

[8] [8]

Gray, A.: Pseudo-Riemannian almost product manifolds and submersions. J. Math. Mech., 16, no. 7, 715-737 (1967)

work page 1967

[9] [9]

Gromoll, D., Walschap, G.: Metric foliations and curvature, Birkh¨ auser (2009)

work page 2009

[10] [10]

Z., 238, 829–832 (2001)

Kashiwada, T.: On a contact 3-structure, Math. Z., 238, 829–832 (2001)

work page 2001

[11] [11]

Kobayashi, S., Nomizu, N.: Foundations of differential geometry I, Interscience, New York (1963)

work page 1963

[12] [12]

O’Neill, B.: The fundamental equations of a submersion. Mich. Math. J. 13, 459-469 (1966)

work page 1966

[13] [13]

Petersen P., Wilhelm, F.: Examples of Riemannian manifolds with positive curvature almost everywhere, Geom. Topol. 3 , 331-367 (1999)

work page 1999

[14] [14]

Some principles for deforming nonnegative curvature

Petersen, P., Wilhelm, F. Some principles for deforming nonnegative curvature. preprint, arXiv:0908.3026 (2009)

work page arXiv 2009

[15] [15]

Annali di Matematica 203, 2641–2652 (2024)

Rovenski, V.: Foliated structure of weak nearly Sasakian manifolds. Annali di Matematica 203, 2641–2652 (2024). https://doi.org/10.1007/s10231-024-01459- 7

work page doi:10.1007/s10231-024-01459- 2024

[16] [16]

Sharipov, X., Sharipova, S., Salimov, E., Mardiev, I.: Local Equivalence of Rie- mannian Submersions via Differential Invariants, preprint, arXiv:2603.14820v1 (2026) 48

work page arXiv 2026

[17] [17]

Tondeur, P.: Geometry of foliations, Springer-Verlag (1997)

work page 1997

[18] [18]

Wilking, B.: Nonnegatively and Positively Curved Manifolds, in: Metric and Comparison Geometry, Surv. Diff. Geom. 11, ed. K. Grove and J. Cheeger, 25-62 (2007)

work page 2007

[19] [19]

Zawadzki, T.: Variations of metric that preserve a Riemannian submersion and geometry of its fibers, preprint, arXiv:2412.00969 (2025)

work page arXiv 2025

[20] [20]

Ziller, W.: Examples of Riemannian manifolds with nonnegative sectional cur- vature, in: Metric and Comparison Geometry, Surv. Diff. Geom. 11, ed. K. Grove and J. Cheeger, 63-102 (2007)

work page 2007

[21] [21]

Ziller, W.: Fatness Revisited, lecture notes (2005). 49

work page 2005