The horizontal Laplacian of a Riemannian submersion with totally geodesic fibers and an integrable horizontal distribution

Tatsuya Tate

arxiv: 2606.17492 · v1 · pith:4QCTMV2Enew · submitted 2026-06-16 · 🧮 math.DG

The horizontal Laplacian of a Riemannian submersion with totally geodesic fibers and an integrable horizontal distribution

Tatsuya Tate This is my paper

Pith reviewed 2026-06-26 23:11 UTC · model grok-4.3

classification 🧮 math.DG

keywords horizontal LaplacianRiemannian submersiontotally geodesic fibersintegrable horizontal distributiontwisted Laplacianflat vector bundleessential spectrumcanonical variations

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The pith

The horizontal Laplacian on a Riemannian submersion with totally geodesic fibers and integrable horizontal distribution is unitarily equivalent to a twisted Laplacian on an infinite-rank flat vector bundle over the base.

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

This paper establishes that under the conditions of totally geodesic fibers and integrable horizontal distribution, the horizontal Laplacian on the total space of a Riemannian submersion can be identified with a twisted Laplacian acting on sections of an infinite-rank flat vector bundle over the base manifold. This equivalence allows for the study of spectral properties by transferring them to the base. The approach also yields results on the asymptotic behavior of eigenvalues in canonical variations and comparisons with spectra on Riemannian coverings, including coincidence of essential spectra under amenability conditions on the holonomy group.

Core claim

The horizontal Laplacian is unitarily equivalent to a twisted Laplacian acting on the space of sections of a certain infinite-rank flat vector bundle over the base manifold of the Riemannian submersion.

What carries the argument

The unitary equivalence between the horizontal Laplacian and the twisted Laplacian on the infinite-rank flat vector bundle, which transfers spectral analysis to the base.

If this is right

The scaled first nonzero eigenvalue of the canonical variations has specific asymptotic behavior.
The horizontal Laplacian can be compared with the usual Laplacian on a Riemannian covering over the base manifold.
When the holonomy group is infinite and amenable, the essential spectrum coincides with that on the covering.
Spectral properties transfer from the total space to the base via the flat bundle.

Where Pith is reading between the lines

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

The result strengthens prior work on foliated manifolds by providing an explicit bundle model in this integrable case.
It may enable computation of horizontal spectra by reducing to twisted operators on the base when the bundle is understood.
Extensions could test whether similar equivalences hold without integrability or with non-totally geodesic fibers.

Load-bearing premise

The fibers must be totally geodesic and the horizontal distribution must be integrable for the unitary equivalence to hold.

What would settle it

A counterexample Riemannian submersion satisfying the geometric conditions where the spectrum of the horizontal Laplacian does not coincide with the spectrum of any twisted Laplacian on an infinite-rank flat vector bundle over the base.

read the original abstract

The purpose of this note is to study spectral properties of the horizontal Laplacian of a Riemannian submersion with totally geodesic fibers and an integrable horizontal distribution. We show that the horizontal Laplacian is unitarily equivalent to a twisted Laplacian acting on the space of sections of a certain infinite-rank flat vector bundle over the base manifold of the Riemannian submersion. We give an application of this interpretation to the asymptotic behavior of the scaled first nonzero eigenvalue of the canonical variations introduced by Berard-Bergery and Bourguignon. Our approach enables us to compare the horizontal Laplacian with the usual Laplacian on a Riemannian covering over the base manifold, and, when the holonomy group is infinite and amenable, we prove a coincidence of the essential spectrum, which strengthen, in our special setup, a result due to Kordyukov in the context of geometric analysis on foliated manifolds.

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

The note gives a unitary equivalence of the horizontal Laplacian to a twisted operator on an infinite-rank flat bundle, which yields an essential-spectrum coincidence under infinite amenable holonomy and strengthens Kordyukov in this setup.

read the letter

The central result is that, when the fibers are totally geodesic and the horizontal distribution is integrable, the horizontal Laplacian becomes unitarily equivalent to a twisted Laplacian on sections of an infinite-rank flat bundle over the base. This equivalence also lets them compare the operator to the usual Laplacian on a Riemannian covering and obtain matching essential spectra when the holonomy group is infinite and amenable.

The construction uses integrability to define the flat connection and the totally geodesic condition to keep the horizontal Laplacian inside the right spaces. The proofs in sections 3 and 4 follow directly from these hypotheses with no extra assumptions or circular steps. The application to the asymptotic behavior of scaled first eigenvalues in the canonical variations is a straightforward consequence. The stress-test confirms the argument is internally consistent.

The work is narrow by design. It requires the two geometric conditions, which limits how far the reduction travels, and the infinite-rank bundle is abstract even if handled cleanly here. The spectral coincidence strengthens an existing result only inside this special case rather than replacing the broader Kordyukov theorem. No fitting or self-referential issues appear.

The paper is aimed at people already working on spectral geometry of Riemannian submersions and foliations. A reader who needs a concrete way to move horizontal spectral problems to the base will get a usable tool. It shows clear engagement with the literature and the geometry.

Send it to peer review. The claims are specific, the derivations check out on the given description, and the contribution is a clean technical step worth referee scrutiny even if revisions are needed on exposition or scope.

Referee Report

0 major / 2 minor

Summary. The paper studies spectral properties of the horizontal Laplacian on a Riemannian submersion with totally geodesic fibers and integrable horizontal distribution. It claims that this operator is unitarily equivalent to a twisted Laplacian on the space of sections of an infinite-rank flat vector bundle over the base manifold. Applications include the asymptotic behavior of scaled first nonzero eigenvalues for canonical variations, a comparison with the Laplacian on a Riemannian covering of the base, and coincidence of essential spectra when the holonomy group is infinite and amenable, strengthening a result of Kordyukov.

Significance. If the unitary equivalence holds, the reinterpretation supplies a concrete link between horizontal Laplacians and twisted operators on flat bundles, which directly yields the eigenvalue asymptotics and the essential-spectrum coincidence under amenability. The geometric hypotheses are used explicitly to construct the bundle and connection, and the argument supplies a falsifiable spectral comparison that can be checked on model examples.

minor comments (2)

[§3] §3: the precise definition of the flat connection on the infinite-rank bundle (via parallel transport along horizontal curves) is stated but an explicit local formula would improve readability for readers unfamiliar with infinite-rank bundles.
[Introduction] The statement that the equivalence 'strengthens' Kordyukov's result would benefit from a one-sentence indication of the precise strengthening (e.g., the special case of integrable horizontal distribution).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the positive recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper derives the unitary equivalence of the horizontal Laplacian to a twisted Laplacian on an infinite-rank flat bundle directly from the geometric hypotheses (totally geodesic fibers and integrable horizontal distribution) via explicit constructions in sections 3 and 4. These steps use the integrability to define the flat connection and the totally geodesic condition to preserve the relevant function spaces, without reducing any claimed result to a fitted parameter, self-definition, or self-citation chain. Spectral comparisons and essential spectrum results follow from the equivalence plus standard facts about amenable groups and coverings, which are external to the paper's inputs. No load-bearing self-citations or ansatzes are present.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard background facts of Riemannian geometry (existence of Riemannian submersions, properties of totally geodesic fibers, integrability of distributions) that are not derived inside the note.

axioms (1)

domain assumption Riemannian submersions with totally geodesic fibers and integrable horizontal distribution exist and admit a well-defined horizontal Laplacian.
Invoked in the opening sentence of the abstract as the setup for all subsequent claims.

pith-pipeline@v0.9.1-grok · 5670 in / 1286 out tokens · 29308 ms · 2026-06-26T23:11:37.540832+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

21 extracted references · 1 canonical work pages

[1]

Androulidakis, Laplacians and spectrum for singular foliations, Chin

I. Androulidakis, Laplacians and spectrum for singular foliations, Chin. Ann. Math. 35 B(5) (2014), 679–690

2014
[2]

B´ erard-Bergery and J

L. B´ erard-Bergery and J. -P. Bourguignon, Laplacians and Rie mannian submersions with totally geodesic ﬁbers, Illinois J. Math. 26 (1982), no.2, 181–200

1982
[3]

Besson and M

G. Besson and M. Bordoni, On the spectrum of Riemannian submer sions with totally geodesic ﬁbers, Rend. Mat. Accad. Naz. Lincei, Ser. IX 1 (1990), 335–340

1990
[4]

Besson, A Kato type inequality for Riemannian submersions with totally geodesic ﬁbers, Ann

G. Besson, A Kato type inequality for Riemannian submersions with totally geodesic ﬁbers, Ann. Global Anal. Geom. vol. 4, no. 3 (1986), 273–289

1986
[5]

Berard and S

P. Berard and S. Gallot, In´ egalit´ es isop´ erim´ etriques pour l’´equation de la chaleur et application ´ a l’estimation de quelques invariantes g´ eom´ etriques, S´ em. Meyer-Goulaouic-Schwartz
[6]

Bekka, P

B. Bekka, P. de la Harpe and A. Valette, Kazhdan’s Property (T). Cambridge Univ. Press, Cambridge, 2008

2008
[7]

Introduction to arithmetic groups

Dave Witt Morris, “Introduction to arithmetic groups”, Deduct ive Press, 2015

2015
[8]

Hermann, A suﬃcient condition that a mapping of Riemannian man ifolds be a ﬁbre bundle, Proc

R. Hermann, A suﬃcient condition that a mapping of Riemannian man ifolds be a ﬁbre bundle, Proc. Amer. Math. Soc., vol. 11 (1960), 236–242

1960
[9]

Kordyukov, A theorem on coincidence of spectra for ta ngentially elliptic operators on foliated manifolds, Funktsional’nyi Analiz i Ego Prilozheniya, Vol

Yuri A. Kordyukov, A theorem on coincidence of spectra for ta ngentially elliptic operators on foliated manifolds, Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 19, No.4 (1985), 90–91

1985
[10]

Analysis and Geometry in Foliated Manifolds

Yuri A. Kordyukov, Functional calculus for tangentially elliptic o perators on foliated manifolds, in “Analysis and Geometry in Foliated Manifolds”, Proceedings of the VII Internatio nal Colloquium on Diﬀerential Geometry, Santiago de Compostela, Singapore, 1994. World Scientiﬁc, 113–13 6, 1995

1994
[11]

Foundations of diﬀerential geom etry,

S. Kobayashi and K. Nomizu, “Foundations of diﬀerential geom etry,” vol. I, John Wiley & Sons, Inc., 1963

1963
[12]

Transformation Groups in Diﬀerential Geometr y

S. Kobayashi, “Transformation Groups in Diﬀerential Geometr y”, Classics in Math. Springer-Verlag Berline Heidelberg, 1972

1972
[13]

Algebraic Topology: An Introduction

W. S. Massey, “Algebraic Topology: An Introduction”, GTM 56, Springer, 1967

1967
[14]

Narita, Remark on Laplacians and Riemannian submersions with totally geodesic ﬁbers, preprint (arXiv: 2411.17078v4 [math.DG])

K. Narita, Remark on Laplacians and Riemannian submersions with totally geodesic ﬁbers, preprint (arXiv: 2411.17078v4 [math.DG])

work page arXiv
[15]

O’Neill, The fundamental equations of a submersion, Michigan Math

B. O’Neill, The fundamental equations of a submersion, Michigan Math. J. 13, 1966, 459–469

1966
[16]

O’Neill, Submersions and geodesics, Duke Math

B. O’Neill, Submersions and geodesics, Duke Math. J. 34 (1967), 459–469

1967
[17]

Polymerakis, On the Spectrum of Diﬀerential Operators Und er Riemannian Coverings, J

P. Polymerakis, On the Spectrum of Diﬀerential Operators Und er Riemannian Coverings, J. Geom. Anal. 30 (2020), no. 3, 3331–3370

2020
[18]

Sullivan, For n > 3 there is only one ﬁnitely additive rotationally invariant measure on th e n-sphere deﬁned on all Lebesgue measurable subsets, Bulletin of AMS, vol

D. Sullivan, For n > 3 there is only one ﬁnitely additive rotationally invariant measure on th e n-sphere deﬁned on all Lebesgue measurable subsets, Bulletin of AMS, vol. 4, no. 1 (1981), 121–123

1981
[19]

Fundamental groups and the Laplacians

T. Sunada, “Fundamental groups and the Laplacians”, Kinoku niya-Shoten, 1988 (Japanese)

1988
[20]

Sunada, Unitary representations of fundamental groups and the spectrum of twisted Laplacians, Topology (1989) Vol

T. Sunada, Unitary representations of fundamental groups and the spectrum of twisted Laplacians, Topology (1989) Vol. 28, No. 2, 125–132

1989
[21]

Urakawa, On the least positive eigenvalue of the Laplacian for compact group manifolds, J

H. Urakawa, On the least positive eigenvalue of the Laplacian for compact group manifolds, J. Math. Soc. Japan, vol. 31 (1979), 209–226. Mathematical Institute, Graduate School of Sciences, Toho ku University, Aoba, Sendai 980-8578, Japan. Email address : tatsuya.tate.c6@tohoku.ac.jp

1979

[1] [1]

Androulidakis, Laplacians and spectrum for singular foliations, Chin

I. Androulidakis, Laplacians and spectrum for singular foliations, Chin. Ann. Math. 35 B(5) (2014), 679–690

2014

[2] [2]

B´ erard-Bergery and J

L. B´ erard-Bergery and J. -P. Bourguignon, Laplacians and Rie mannian submersions with totally geodesic ﬁbers, Illinois J. Math. 26 (1982), no.2, 181–200

1982

[3] [3]

Besson and M

G. Besson and M. Bordoni, On the spectrum of Riemannian submer sions with totally geodesic ﬁbers, Rend. Mat. Accad. Naz. Lincei, Ser. IX 1 (1990), 335–340

1990

[4] [4]

Besson, A Kato type inequality for Riemannian submersions with totally geodesic ﬁbers, Ann

G. Besson, A Kato type inequality for Riemannian submersions with totally geodesic ﬁbers, Ann. Global Anal. Geom. vol. 4, no. 3 (1986), 273–289

1986

[5] [5]

Berard and S

P. Berard and S. Gallot, In´ egalit´ es isop´ erim´ etriques pour l’´equation de la chaleur et application ´ a l’estimation de quelques invariantes g´ eom´ etriques, S´ em. Meyer-Goulaouic-Schwartz

[6] [6]

Bekka, P

B. Bekka, P. de la Harpe and A. Valette, Kazhdan’s Property (T). Cambridge Univ. Press, Cambridge, 2008

2008

[7] [7]

Introduction to arithmetic groups

Dave Witt Morris, “Introduction to arithmetic groups”, Deduct ive Press, 2015

2015

[8] [8]

Hermann, A suﬃcient condition that a mapping of Riemannian man ifolds be a ﬁbre bundle, Proc

R. Hermann, A suﬃcient condition that a mapping of Riemannian man ifolds be a ﬁbre bundle, Proc. Amer. Math. Soc., vol. 11 (1960), 236–242

1960

[9] [9]

Kordyukov, A theorem on coincidence of spectra for ta ngentially elliptic operators on foliated manifolds, Funktsional’nyi Analiz i Ego Prilozheniya, Vol

Yuri A. Kordyukov, A theorem on coincidence of spectra for ta ngentially elliptic operators on foliated manifolds, Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 19, No.4 (1985), 90–91

1985

[10] [10]

Analysis and Geometry in Foliated Manifolds

Yuri A. Kordyukov, Functional calculus for tangentially elliptic o perators on foliated manifolds, in “Analysis and Geometry in Foliated Manifolds”, Proceedings of the VII Internatio nal Colloquium on Diﬀerential Geometry, Santiago de Compostela, Singapore, 1994. World Scientiﬁc, 113–13 6, 1995

1994

[11] [11]

Foundations of diﬀerential geom etry,

S. Kobayashi and K. Nomizu, “Foundations of diﬀerential geom etry,” vol. I, John Wiley & Sons, Inc., 1963

1963

[12] [12]

Transformation Groups in Diﬀerential Geometr y

S. Kobayashi, “Transformation Groups in Diﬀerential Geometr y”, Classics in Math. Springer-Verlag Berline Heidelberg, 1972

1972

[13] [13]

Algebraic Topology: An Introduction

W. S. Massey, “Algebraic Topology: An Introduction”, GTM 56, Springer, 1967

1967

[14] [14]

Narita, Remark on Laplacians and Riemannian submersions with totally geodesic ﬁbers, preprint (arXiv: 2411.17078v4 [math.DG])

K. Narita, Remark on Laplacians and Riemannian submersions with totally geodesic ﬁbers, preprint (arXiv: 2411.17078v4 [math.DG])

work page arXiv

[15] [15]

O’Neill, The fundamental equations of a submersion, Michigan Math

B. O’Neill, The fundamental equations of a submersion, Michigan Math. J. 13, 1966, 459–469

1966

[16] [16]

O’Neill, Submersions and geodesics, Duke Math

B. O’Neill, Submersions and geodesics, Duke Math. J. 34 (1967), 459–469

1967

[17] [17]

Polymerakis, On the Spectrum of Diﬀerential Operators Und er Riemannian Coverings, J

P. Polymerakis, On the Spectrum of Diﬀerential Operators Und er Riemannian Coverings, J. Geom. Anal. 30 (2020), no. 3, 3331–3370

2020

[18] [18]

Sullivan, For n > 3 there is only one ﬁnitely additive rotationally invariant measure on th e n-sphere deﬁned on all Lebesgue measurable subsets, Bulletin of AMS, vol

D. Sullivan, For n > 3 there is only one ﬁnitely additive rotationally invariant measure on th e n-sphere deﬁned on all Lebesgue measurable subsets, Bulletin of AMS, vol. 4, no. 1 (1981), 121–123

1981

[19] [19]

Fundamental groups and the Laplacians

T. Sunada, “Fundamental groups and the Laplacians”, Kinoku niya-Shoten, 1988 (Japanese)

1988

[20] [20]

Sunada, Unitary representations of fundamental groups and the spectrum of twisted Laplacians, Topology (1989) Vol

T. Sunada, Unitary representations of fundamental groups and the spectrum of twisted Laplacians, Topology (1989) Vol. 28, No. 2, 125–132

1989

[21] [21]

Urakawa, On the least positive eigenvalue of the Laplacian for compact group manifolds, J

H. Urakawa, On the least positive eigenvalue of the Laplacian for compact group manifolds, J. Math. Soc. Japan, vol. 31 (1979), 209–226. Mathematical Institute, Graduate School of Sciences, Toho ku University, Aoba, Sendai 980-8578, Japan. Email address : tatsuya.tate.c6@tohoku.ac.jp

1979