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arxiv: 2606.18725 · v1 · pith:BTIPYTBOnew · submitted 2026-06-17 · 🧮 math.DG

Generalized Chen's inequalities for Riemannian submersions and Riemannian maps with Applications

Ravindra Singh This is my paper

Pith reviewed 2026-06-26 20:13 UTC · model grok-4.3

classification 🧮 math.DG
keywords Riemannian submersionsRiemannian mapsgeneralized Chen inequalitiesdelta invariantssecond fundamental formspace formsshape operators
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The pith

Riemannian submersions and maps satisfy optimal generalized Chen inequalities that relate δ-invariants of vertical and horizontal distributions to second fundamental tensor quantities.

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

The paper establishes generalized B.-Y. Chen inequalities that apply specifically to Riemannian submersions and Riemannian maps. It employs Chen's generalized δ-invariants to obtain optimal bounds that combine invariants tied to the vertical and horizontal distributions with extrinsic terms coming from the second fundamental tensors. Equality cases receive explicit algebraic characterizations in terms of the associated shape operators, which carry geometric meaning. The inequalities are then specialized to produce concrete versions when the total space or target manifold is a real or complex space form.

Core claim

For a Riemannian submersion or Riemannian map, the generalized δ-invariants associated with the vertical and horizontal distributions obey optimal inequalities involving the squared norms of the second fundamental forms, and equality holds precisely when the shape operators satisfy stated algebraic conditions that admit a geometric interpretation.

What carries the argument

Generalized δ-invariants of the vertical and horizontal distributions, combined with the second fundamental tensors to form the inequalities.

If this is right

  • Explicit generalized Chen inequalities hold when the total or target manifold is a real space form.
  • The same explicit inequalities hold when the total or target manifold is a complex space form.
  • Equality cases are characterized by precise algebraic conditions on the shape operators.
  • The results extend both the classical Chen inequalities and several recent Chen-type inequalities already known for submersions and maps.

Where Pith is reading between the lines

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

  • The inequalities supply a uniform way to compare distortion of vertical and horizontal parts across different submersions.
  • Direct numerical checks on standard examples such as Hopf fibrations or projections from spheres would test the sharpness of the bounds.
  • The same δ-invariant technique could be applied to maps that are neither submersions nor immersions but still possess well-defined horizontal and vertical distributions.

Load-bearing premise

The vertical and horizontal distributions are smooth and the second fundamental tensors are well-defined on the manifolds.

What would settle it

Compute the generalized δ-invariants and second fundamental form quantities explicitly for the Hopf fibration S^3 to S^2 and check whether the proposed inequality holds with equality only under the stated shape-operator conditions.

read the original abstract

In this paper, we establish generalized B.-Y. Chen inequalities for Riemannian submersions and Riemannian maps between Riemannian manifolds by employing the generalized $\delta$-invariants introduced by Chen. We derive optimal inequalities involving the generalized $\delta$-invariants associated with the vertical and horizontal distributions together with extrinsic invariants determined by the second fundamental tensors of the submersion and the Riemannian map. Furthermore, we characterize the equality cases through precise algebraic conditions on the corresponding shape operators, providing their geometric interpretation. As applications, we obtain explicit generalized Chen inequalities for Riemannian submersions and Riemannian maps whose total or target manifolds are real space forms and complex space forms. These results extend the classical Chen inequalities as well as several recent Chen-type inequalities for Riemannian submersions and Riemannian maps available in the literature.

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.

Referee Report

0 major / 2 minor

Summary. The manuscript claims to derive generalized B.-Y. Chen inequalities for Riemannian submersions and Riemannian maps by applying Chen's generalized δ-invariants separately to the vertical and horizontal distributions, combining them with extrinsic curvature terms from the second fundamental form (or tensors) of the submersion/map, obtaining optimal bounds, characterizing equality cases by algebraic conditions on the shape operators, and specializing the results to total or target manifolds that are real or complex space forms.

Significance. If the derivations are correct, the work supplies a direct, standard extension of Chen-type inequalities to the setting of submersions and maps; the resulting bounds and equality characterizations would be usable for extrinsic geometry questions on these structures, particularly when the ambient spaces are space forms. The approach relies only on the usual smoothness and tensoriality assumptions already present in the definitions of Riemannian submersions and maps.

minor comments (2)
  1. The abstract and introduction should explicitly state the dimension and rank assumptions on the vertical and horizontal distributions (e.g., dim V ≥ 2, dim H ≥ 2) that are needed for the δ-invariants to be non-trivial; this is standard but currently implicit.
  2. Notation for the generalized δ-invariants (vertical vs. horizontal) and for the second fundamental tensors of the submersion versus the map should be unified and introduced once in §2 before being used in the inequality statements.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation of minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper derives generalized Chen inequalities by applying the definitions of the generalized δ-invariants separately to vertical and horizontal distributions, then combining them with the second fundamental form and characterizing equality via algebraic conditions on shape operators. These steps follow directly from the standard definitions and tensorial properties of Riemannian submersions and maps, without any reduction to fitted parameters, self-definitional loops, or load-bearing self-citations. The assumptions invoked are the usual smoothness conditions already built into the objects under study, and the results are presented as extensions of classical techniques rather than tautological renamings or imported uniqueness theorems.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard axioms of Riemannian geometry and the prior definition of generalized δ-invariants; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • standard math Riemannian manifolds are smooth manifolds equipped with a positive-definite metric tensor.
    Foundational assumption required for all definitions of submersions, maps, and curvature invariants.
  • domain assumption The generalized δ-invariants are those previously introduced by Chen.
    The paper employs these invariants without re-deriving them.

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discussion (0)

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Reference graph

Works this paper leans on

42 extracted references · 1 linked inside Pith

  1. [1]

    Alegre, P.; Blair, D.\ E.; Carriazo, A.: Generalized Sasakian-space-forms , Israel J. Math. 141 (2004), 157-183. MR2063031

    2004

  2. [2]

    Alegre, P.; Carriazo, A.; Kim, Y.\ H.; Yoon, D.\ W.: B. Y. Chen's inequality for submanifolds of generalized space forms , Indian J. Pure Appl. Math. 38 (2007), no. 3, 185-201. MR2333966

    2007

  3. [3]

    Global Anal

    Alegre, P.; Chen, B.\ Y.; Munteanu, M.\ I.: Riemannian submersions, -invariants, and optimal inequality , Ann. Global Anal. Geom. 42 (2012), no. 3, 317-331. MR2972616

    2012

  4. [4]

    Arslan, K.; Ezentas, R.; Mihai, I.; Murathan, C.; \" O zg\" u r, C.: Certain inequalities for submanifolds in ( , ) -contact space forms , Bull. Austral. Math. Soc. 64 (2001), no. 2, 201-212. MR1860058

    2001

  5. [5]

    Aytimur, H.; \" O zg\" u r, C.: Sharp inequalities for anti-invariant Riemannian submersions from Sasakian space forms , J. Geom. Phys. 166 (2021), 1-12. MR4249123

    2021

  6. [6]

    MR2682326

    Blair, D.\ E.: Riemannian Geometry of contact and symplectic manifolds , Progress in Mathematics 203 , Birkh \" a user Boston, Ltd., Boston, MA, 2010. MR2682326

    2010

  7. [7]

    Chen, B.\ Y.: Some pinching and classification theorems for minimal submanifolds , Arch. Math. 60 (1993), no. 6, 568-578. MR1216703

    1993

  8. [8]

    Chen, B.\ Y.: A general inequality for submanifolds in complex-space-forms and its applications , Arch. Math. 67 (1996), no. 6, 519-528. MR1418914

    1996

  9. [9]

    27 (1995), no

    Chen, B.\ Y.: A Riemannian invariant and its applications to submanifold theory , Results Math. 27 (1995), no. 1-2, 17-26. MR1317820

    1995

  10. [10]

    Chen, B.\ Y.: Mean curvature and shape operator of isometric immersions in real-space-forms , Glasgow Math. J. 38 (1996), no. 1, 87-97. MR1373963

    1996

  11. [11]

    Chen, B.\ Y.: Some new obstructions to minimal and Lagrangian isometric immersions , Japan. J. Math. 26 (2000), no. 1, 105-127. MR1771434

    2000

  12. [12]

    Chen, B.\ -Y.: Pseudo-Riemannian geometry, -invariants and applications , World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011. MR2799371

    2011

  13. [13]

    11 (2023), no

    Chen, B.\ Y.; Vilcu, G.\ E.: Recent developments on the first Chen inequality in differential geometry , Math. 11 (2023), no. 19, 1-50

    2023

  14. [14]

    MR2110043

    Falcitelli, M.; Ianus, S.; Pastore, A.\ M.: Riemannian submersions and related topics , World Scientific Publishing Co., Inc., River Edge, NJ, 2004. MR2110043

    2004

  15. [15]

    Fischer, A.E.: Riemannian maps between Riemannian manifolds . Contemp. Math., 132 (1992), 331-366

    1992

  16. [16]

    Kluwer, (1999)

    Garc\'ia-R\'io, E., Kupeli, D.N.: Semi-Riemannian Maps and Their Applications . Kluwer, (1999)

    1999

  17. [17]

    Gray, A.: Pseudo-Riemannian almost product manifolds and submersions , J. Math. Mech. 16 (1967), no. 7, 715-737. MR205184

    1967

  18. [18]

    G\" u lbahar, M.; Meri c , S .\ E.; Kili c , E.: Sharp inequalities involving the Ricci curvature for Riemannian submersions , Kragujevac J. Math. 41 (2017), no. 2, 279-293. MR3741764

    2017

  19. [19]

    Hong, S.; Matsumoto, K.; Tripathi, M.\ M.: Certain basic inequalities for submanifolds of locally conformal Kaehler space forms , SUT J. Math. 41 (2005), no. 1, 75-94. MR2168454

    2005

  20. [20]

    Kim, J.\ -S.; Choi, J.: A basic inequality for submanifolds in a cosymplectic space form , Int. J. Math. Math. Sci. (2003), no. 9, 539-547. MR1968341

    2003

  21. [21]

    Kim, Y.\ H.; Kim, D.\ -S.: A basic inequality for submanifolds in Sasakian space forms , Houston J. Math. 25 (1999), no. 2, 247-257. MR1697623

    1999

  22. [22]

    Chen inequalities for submanifolds in generalised complex space forms , Bull

    Kim, J.\ -S.; Song, Y.\ -M.; Tripathi, M.\ M.: B.-Y. Chen inequalities for submanifolds in generalised complex space forms , Bull. Korean Math. Soc. 40 (2003), no. 3, 411-423. MR1996851

    2003

  23. [23]

    MR152974

    Kobayashi, S.; Nomizu, K.: Foundations of differential geometry , Vol I , Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1963. MR152974

    1963

  24. [24]

    Lee, C.\ W.; Lee, J.\ W.; S ahin, B.; V\^ lcu, G.\ E.: Optimal inequalities for Riemannian maps and Riemannian submersions involving Casorati curvatures , Ann. Mat. Pura Appl. 200 (2021), no. 3, 1277-1295. MR4242128

    2021

  25. [25]

    M.: Chen's first inequality for Riemannian maps to complex space forms and -invariants

    Meena, K., S ahin, B., Shah, H. M.: Chen's first inequality for Riemannian maps to complex space forms and -invariants. Period. Math. Hungar., 91 (2) (2025), 588-603

    2025

  26. [26]

    : The imbedding problem for Riemannian manifolds , Ann

    Nash, J.\ F. : The imbedding problem for Riemannian manifolds , Ann. of Math. 63 (1956), no. 2, 20-63. MR75639

    1956

  27. [27]

    O'Neill, B.: The fundamental equations of a submersion , Michigan Math. J. 13 (1966), 459-469. MR200865

    1966

  28. [28]

    Nore, T.: Second fundamental form of a map . Ann. Mat. Pura Appl., 146 (1986), 281-310

    1986

  29. [29]

    Ogiue, K.: n -dimensional complex space forms immersed in n+n(n+1)/2 -dimensional complex space forms , J. Math. Soc. Japan 24 (1972), no. 3, 518-526. MR310809

    1972

  30. [30]

    \" O zg\" u r, C.: B. Y. Chen inequalities for submanifolds of a Riemannian manifold of quasi-constant curvature , Turkish J. Math. 35 (2011), no. 3, 501-509. MR2867335

    2011

  31. [31]

    \" O zg\" u r, C.; De, A.: Chen inequalities for submanifolds of a Riemannian manifold of nearly quasi-constant curvature , Publ. Math. Debrecen 82 (2013), no. 2, 439-450. MR3034357

    2013

  32. [32]

    MR3644540

    S ahin, B.: Riemannian submersions, Riemannian maps in Hermitian geometry, and their applications , Elsevier/Academic Press, London, 2017. MR3644540

    2017

  33. [33]

    S ahin, B.: Invariant and anti-invariant Riemannian maps to K\"ahler manifolds . Int. J. Geom. Methods Mod. Phys., 7 (3) (2010), 337-355

    2010

  34. [34]

    S ahin, B.: Chen's first inequality for Riemannian maps. Ann. Polon. Math., 117 (3) (2016), 249-258

    2016

  35. [35]

    Singh, R., Meena, K., Meena, K.\ C.: General Casorati inequalities and implications for Riemannian maps and Riemannian submersions , J. Math. Anal. Appl. 558 (2026), no. 1, 1-31

    2026

  36. [36]

    arxiv preprint, (2025), arXiv:2510.10505v2 [math.DG]

    Singh, R.; Meena, K.; Meena, K.C.: General Chen's first inequality and applications for Riemannian maps. arxiv preprint, (2025), arXiv:2510.10505v2 [math.DG]

    arXiv 2025

  37. [37]

    arxiv preprint, (2026), arXiv:2602.15301v2 [math.DG]

    Singh, R.; Tripathi, M.M.: B.-Y Chen's inequalities for Riemannian submersions and their applications. arxiv preprint, (2026), arXiv:2602.15301v2 [math.DG]

    Pith/arXiv arXiv 2026

  38. [38]

    Singh, R.; Meena, K.; Meena, K.C.: General Chen-Ricci

  39. [39]

    Takahashi, T.: A note on K\" a hlerian hypersurfaces of spaces of constant curvature , Kumamoto J. Sci. Math. 9 (1972), 21-24. MR301657

    1972

  40. [40]

    Indian Acad

    Tripathi, M.\ M.; Kim, J.\ S.; Kim, S.\ B.: A basic inequality for submanifolds in locally conformal almost cosymplectic manifolds , Proc. Indian Acad. Sci. Math. Sci. 112 (2002), no. 3, 415-423. MR1921790

    2002

  41. [41]

    Geometry, groups and mathematical philosophy

    Tripathi, M.\ M.: Kulkarni-Nomizu tensor fields , In "Geometry, groups and mathematical philosophy", Contemp. Math. Amer. Math. Soc. 811 (2025), 259-278. MR4859379

    2025

  42. [42]

    4, World Scientific, London, United Kingdom, 2026

    Tripathi, M.\ M.: Handbook of the differential geometry of space forms , Series in Algebraic and Differential Geometry, no. 4, World Scientific, London, United Kingdom, 2026. (in press)

    2026