Anti-invariant Riemannian submersions from locally conformal Kaehler manifolds

Majid Ali Choudhary

arxiv: 1907.01977 · v1 · pith:7KA4O3MPnew · submitted 2019-07-03 · 🧮 math.DG

Anti-invariant Riemannian submersions from locally conformal Kaehler manifolds

Majid Ali Choudhary This is my paper

Pith reviewed 2026-05-25 09:26 UTC · model grok-4.3

classification 🧮 math.DG

keywords anti-invariant Riemannian submersionlocally conformal Kähler manifoldLagrangian submersionfoliation geometrydecomposition theoremRiemannian submersionalmost complex structure

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The pith

Anti-invariant Riemannian submersions extend to locally conformal Kähler manifolds and induce foliation decompositions on the total space.

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

The paper extends the definition of anti-invariant Riemannian submersions and Lagrangian submersions from almost Hermitian manifolds to locally conformal Kähler manifolds. It examines the foliations produced by these submersions and establishes decomposition theorems that split the tangent bundle of the total manifold according to the vertical and horizontal distributions. A reader would care because the locally conformal Kähler condition is weaker than the Kähler condition, so the extension shows that the same style of geometric conclusions survive when a conformal factor is present. The work treats the interaction between the anti-invariance condition and the Lee form of the manifold to obtain integrability and orthogonality results for the distributions.

Core claim

Anti-invariant Riemannian submersions from locally conformal Kähler manifolds induce foliations on the total manifold whose geometry permits decomposition theorems that split the space into orthogonal integrable distributions, extending the corresponding results known for almost Hermitian manifolds.

What carries the argument

Anti-invariant Riemannian submersion, defined by the condition that the vertical distribution is mapped by the almost complex structure into the horizontal distribution, together with the locally conformal Kähler structure on the total space that governs the foliation.

Load-bearing premise

The anti-invariance condition remains compatible with the conformal factor so that the foliation integrability and decomposition properties carry over without extra correction terms.

What would settle it

An explicit locally conformal Kähler manifold equipped with an anti-invariant Riemannian submersion in which the vertical distribution fails to be integrable or the expected orthogonal decomposition of the tangent bundle does not hold.

read the original abstract

B. Sahin [9] introduced the notion of anti-invariant Riemannian submersions from almost Hermitian manifolds onto Riemannian manifolds. In the present paper we extend the notion of anti-invariant and Lagrangian Riemannian submersions (a special anti-invariant Riemannian submersion) to the case of locally conformal Kaehler manifolds. We discuss the geometry of foliation and obtain some decomposition theorems for the total manifold of such submersions.

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 makes a routine extension of anti-invariant submersions to locally conformal Kaehler manifolds and obtains some foliation decomposition theorems.

read the letter

The paper extends the notion of anti-invariant Riemannian submersions, originally defined by Sahin for almost Hermitian manifolds, to the setting of locally conformal Kaehler manifolds. It also treats the Lagrangian case as a special subcase and derives decomposition theorems for the total space coming from the foliation properties of the vertical and horizontal distributions. What stands out is that the authors carry the definitions over in a direct way and then work out how the geometry of the foliations behaves under the locally conformal condition. The closed Lee form and the conformal factor are incorporated into the calculations without apparent extra assumptions. This produces some statements about the total manifold decomposing in certain ways. The abstract indicates they discuss the geometry of the foliation induced by the submersion. The limitation is that this remains an incremental application rather than a new idea. The results follow the same pattern as the almost Hermitian case, adjusted for the conformal structure. There is no indication of surprising phenomena or obstructions that only appear in the l.c.K. setting. The work stays firmly inside the existing program on submersions from Hermitian manifolds. Since the full text is available, one can check whether the proofs introduce any post-hoc restrictions or if they are straightforward adaptations. A reader who follows papers on Riemannian submersions and foliations in almost Hermitian or Kaehler geometry would find this a natural next step. Someone looking for broader implications or new techniques would not. The citation pattern looks standard, referencing the prior definition and building forward. No free parameters or invented entities are involved. I think this paper should go to peer review. The claims are specific enough to be checked, and the extension is reasonable to document even if the novelty is modest. It shows clear thinking in applying the definitions consistently.

Referee Report

0 major / 3 minor

Summary. The paper extends the notion of anti-invariant Riemannian submersions and the special case of Lagrangian Riemannian submersions from almost Hermitian manifolds to locally conformal Kähler manifolds. It examines the induced foliation geometry and derives decomposition theorems for the total manifold.

Significance. If the extension and theorems hold, the work provides a direct generalization of Sahin's results to the locally conformal Kähler setting. This could enable analogous foliation techniques and structural decompositions in a broader class of manifolds where the Lee form is closed and the structure is conformally Kähler.

minor comments (3)

The abstract states that 'some decomposition theorems' are obtained; the introduction should explicitly list the main theorems (with equation or theorem numbers) to clarify the precise statements being proved.
Ensure that the definition of the anti-invariant condition (adapted from [9]) is restated verbatim in §2 or §3 so that the interaction with the locally conformal Kähler structure (closed Lee form, conformal factor) is immediately visible without requiring the reader to consult the reference.
In the foliation-geometry section, verify that all curvature identities used in the decomposition theorems are derived from the l.c.K. condition rather than assumed from the almost-Hermitian case; a short remark on any additional terms arising from the Lee form would improve clarity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of the manuscript and the recommendation of minor revision. The referee's description accurately reflects the paper's content and goals. No major comments appear in the report, so we have no specific points requiring rebuttal or clarification at this stage.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper extends the definition of anti-invariant and Lagrangian Riemannian submersions (introduced in cited work [9] by a different author) to locally conformal Kähler manifolds, then derives foliation geometry and decomposition theorems. No equations or claims reduce by construction to fitted inputs, self-definitions, or self-citation chains; the central results follow from applying standard submersion conditions to the new ambient structure without circular reduction. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on the standard axioms of Riemannian geometry and the definition introduced in the cited reference [9]; no free parameters, new entities, or ad-hoc axioms are visible from the abstract.

axioms (2)

domain assumption The total space carries a locally conformal Kaehler structure
This is the domain on which the extended submersions are defined.
domain assumption The submersion satisfies the anti-invariant condition with respect to the almost complex structure
This is the defining property carried over from the earlier almost Hermitian setting.

pith-pipeline@v0.9.0 · 5582 in / 1246 out tokens · 28049 ms · 2026-05-25T09:26:59.480255+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

and Wood J.C., Harmonic morphisms between Riemannian man- ifolds, London Mathematical Society Monographs, 29, Oxford University Press, The clarendon Press, Oxford, (2003)

Baird P. and Wood J.C., Harmonic morphisms between Riemannian man- ifolds, London Mathematical Society Monographs, 29, Oxford University Press, The clarendon Press, Oxford, (2003)

work page 2003
[2]

and Ornea, L., Locally Conformal Kahler Geometry , Basel: Birkhauser, (1998)

Dragomir S. and Ornea, L., Locally Conformal Kahler Geometry , Basel: Birkhauser, (1998)

work page 1998
[3]

Gray A., Pseudo-Riemannian almost product manifolds and submersio ns, J. Math. Mech., 16 (1967), 715-737

work page 1967
[4]

and Vilcu G

Ianus S., Mazzocco R. and Vilcu G. E., Riemannian submersions from quaternionic manifolds , Acta Appl. Math., 104(1) (2008), 83-89

work page 2008
[5]

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

Neill B. O’., The fundamental equations of a submersion , Michigan Math. J., 13(1996), 459-469

work page 1996
[6]

S., H-slant submersions , Bull

Park K. S., H-slant submersions , Bull. Korean Math. Soc., 49(2) (2012), 329-338

work page 2012
[7]

S., H-semi-invariant submersions , Taiwanese Journal of Mathe- matics, 16(5) (2012), 1865-1878

Park K. S., H-semi-invariant submersions , Taiwanese Journal of Mathe- matics, 16(5) (2012), 1865-1878

work page 2012
[8]

and Reckziegel H., Twisted products in pseudo-Riemannian ge- ometry, Geom

Ponge R. and Reckziegel H., Twisted products in pseudo-Riemannian ge- ometry, Geom. Dedicata, 48(1) (1993), 15-25

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[9]

18 Majid Ali Choudhary

Sahin B., Anti-invariant Riemannian submersions from almost Hermit ian manifolds, Central European Journal of Mathematics, 8(3) (2010), 437- 447. 18 Majid Ali Choudhary

work page 2010
[10]

Sahin B., Slant submersions from almost Hermitian manifolds , Bull. Math. Soc. Sci. Math. Roumanie Tome, 54(102) No. 1 , (2011), 93-105

work page 2011
[11]

J., 17(2) April 2013, 629-659

Sahin B., Semi-invariant Riemannian submersions from almost Hermit ian manifolds, Taiwanese math. J., 17(2) April 2013, 629-659

work page 2013
[12]

Math., 24 (1976), 338-351

Vaisman I., On Locally Conformal Almost Kaehler Manifolds , Israel J. Math., 24 (1976), 338-351

work page 1976

[1] [1]

and Wood J.C., Harmonic morphisms between Riemannian man- ifolds, London Mathematical Society Monographs, 29, Oxford University Press, The clarendon Press, Oxford, (2003)

Baird P. and Wood J.C., Harmonic morphisms between Riemannian man- ifolds, London Mathematical Society Monographs, 29, Oxford University Press, The clarendon Press, Oxford, (2003)

work page 2003

[2] [2]

and Ornea, L., Locally Conformal Kahler Geometry , Basel: Birkhauser, (1998)

Dragomir S. and Ornea, L., Locally Conformal Kahler Geometry , Basel: Birkhauser, (1998)

work page 1998

[3] [3]

Gray A., Pseudo-Riemannian almost product manifolds and submersio ns, J. Math. Mech., 16 (1967), 715-737

work page 1967

[4] [4]

and Vilcu G

Ianus S., Mazzocco R. and Vilcu G. E., Riemannian submersions from quaternionic manifolds , Acta Appl. Math., 104(1) (2008), 83-89

work page 2008

[5] [5]

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

Neill B. O’., The fundamental equations of a submersion , Michigan Math. J., 13(1996), 459-469

work page 1996

[6] [6]

S., H-slant submersions , Bull

Park K. S., H-slant submersions , Bull. Korean Math. Soc., 49(2) (2012), 329-338

work page 2012

[7] [7]

S., H-semi-invariant submersions , Taiwanese Journal of Mathe- matics, 16(5) (2012), 1865-1878

Park K. S., H-semi-invariant submersions , Taiwanese Journal of Mathe- matics, 16(5) (2012), 1865-1878

work page 2012

[8] [8]

and Reckziegel H., Twisted products in pseudo-Riemannian ge- ometry, Geom

Ponge R. and Reckziegel H., Twisted products in pseudo-Riemannian ge- ometry, Geom. Dedicata, 48(1) (1993), 15-25

work page 1993

[9] [9]

18 Majid Ali Choudhary

Sahin B., Anti-invariant Riemannian submersions from almost Hermit ian manifolds, Central European Journal of Mathematics, 8(3) (2010), 437- 447. 18 Majid Ali Choudhary

work page 2010

[10] [10]

Sahin B., Slant submersions from almost Hermitian manifolds , Bull. Math. Soc. Sci. Math. Roumanie Tome, 54(102) No. 1 , (2011), 93-105

work page 2011

[11] [11]

J., 17(2) April 2013, 629-659

Sahin B., Semi-invariant Riemannian submersions from almost Hermit ian manifolds, Taiwanese math. J., 17(2) April 2013, 629-659

work page 2013

[12] [12]

Math., 24 (1976), 338-351

Vaisman I., On Locally Conformal Almost Kaehler Manifolds , Israel J. Math., 24 (1976), 338-351

work page 1976