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arxiv: 2602.15301 · v2 · pith:UPHDR2YKnew · submitted 2026-02-17 · 🧮 math.DG

B.-Y. Chen's inequalities for Riemannian submersion and their applications

Ravindra Singh , Mukut Mani Tripathi This is my paper

Pith reviewed 2026-05-21 12:43 UTC · model grok-4.3

classification 🧮 math.DG
keywords Riemannian submersionB.-Y. Chen inequalitiesvertical distributionhorizontal distributionmixed distributionsSasakian space formintrinsic invariantsextrinsic invariants
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The pith

B.-Y. Chen inequalities extend to Riemannian submersions by bounding intrinsic curvatures of vertical, horizontal, and mixed distributions against extrinsic terms.

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

The paper introduces B.-Y. Chen inequalities adapted specifically to Riemannian submersions between Riemannian manifolds. It derives versions of these inequalities for the vertical distribution, the horizontal distribution, and mixed cases, linking intrinsic invariants such as scalar curvature to extrinsic invariants like the squared mean curvature. A sympathetic reader would care because Riemannian submersions model fibrations and projections that preserve lengths and angles, so these relations give new ways to compare curvatures on the total space, the base, and the fibers. The authors also classify the equality cases and obtain explicit results when the total space belongs to real space forms, complex space forms, or generalized Sasakian space forms, with examples showing both equality and strict inequality.

Core claim

For a Riemannian submersion the authors prove that the intrinsic curvature invariants of the vertical distribution, of the horizontal distribution, and of their mixed interactions satisfy B.-Y. Chen-type inequalities expressed in terms of the extrinsic curvature quantities of the submersion; equality holds precisely when certain distributions are totally geodesic or satisfy analogous minimality conditions, and the same inequalities specialize to submersions whose total space is a real, complex, or generalized Sasakian space form.

What carries the argument

The decomposition of the tangent bundle into orthogonal vertical and horizontal distributions together with the induced curvature relations that convert the classical B.-Y. Chen inequality into separate statements for each distribution and for mixed pairs.

If this is right

  • When the total space is a real space form the inequalities reduce to explicit numerical relations between the curvatures of the fibers and the base.
  • Analogous explicit forms hold when the total space is a complex space form or a generalized Sasakian space form.
  • Equality cases characterize submersions in which the vertical or horizontal distributions are totally geodesic.
  • Concrete examples confirm both the equality case and the strict-inequality case for standard projections.

Where Pith is reading between the lines

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

  • The same technique of splitting into vertical and horizontal parts could be tested on other maps that admit an orthogonal decomposition, such as Riemannian foliations.
  • Equality cases may single out homogeneous or symmetric spaces among all possible total spaces.
  • The inequalities supply a possible obstruction to the existence of Riemannian submersions with prescribed curvature on the base or on the fibers.

Load-bearing premise

The standard orthogonal decomposition of the tangent space into vertical and horizontal distributions in a Riemannian submersion remains valid and the usual curvature identities derived from the Levi-Civita connection continue to hold.

What would settle it

A single explicit Riemannian submersion in which the scalar curvature of the vertical distribution exceeds the upper bound supplied by the inequality involving the norm of the mean curvature vector of that distribution would falsify the claim.

read the original abstract

In this paper, we introduce B.-Y. Chen inequalities for Riemannian submersions between Riemannian manifolds. We derive these inequalities for vertical, horizontal, and mixed distributions, establishing relationships between intrinsic invariants and extrinsic invariants. We also investigate the corresponding equality cases. As applications, the results are obtained for submersions whose total space is a real, complex, generalized Sasakian space form. Several examples are provided to illustrate both equality and strict inequality cases.

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 paper introduces B.-Y. Chen inequalities for Riemannian submersions between Riemannian manifolds. It derives these inequalities for vertical, horizontal, and mixed distributions by decomposing the curvature tensor using the O'Neill A-tensor and standard Gauss-Codazzi identities for the distributions, establishing relationships between intrinsic invariants (such as scalar curvature) and extrinsic invariants. Equality cases are characterized when relevant second-fundamental-form or A-tensor terms vanish. Applications are obtained by direct substitution of the curvature tensors for submersions whose total space is a real, complex, or generalized Sasakian space form, with examples illustrating both equality and strict inequality.

Significance. If the derivations hold, the work extends classical B.-Y. Chen inequalities to Riemannian submersions, providing new relations between the intrinsic geometry of the base manifold and the extrinsic geometry of the fibration. The explicit equality conditions and the concrete applications to Sasakian space forms are strengths, as they enable direct computations and potential rigidity results in contact and Sasakian geometry. The approach relies on standard tools of submersion geometry without introducing ad-hoc assumptions.

minor comments (2)
  1. [§2] §2 (Preliminaries): The orthogonal decomposition into vertical and horizontal distributions and the precise role of the O'Neill A-tensor in the curvature identities should be recalled with a short self-contained paragraph for readers less familiar with submersion geometry.
  2. [Applications] Applications section: When substituting the curvature tensors of real/complex/generalized Sasakian space forms, explicitly list the constant sectional curvatures or holomorphic sectional curvatures used, to make the resulting inequalities fully explicit.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and constructive report, which accurately summarizes the main contributions of the paper. We appreciate the recommendation for minor revision and will incorporate all suggested improvements in the revised manuscript.

Circularity Check

0 steps flagged

Derivation self-contained from standard submersion geometry

full rationale

The paper adapts B.-Y. Chen inequalities to Riemannian submersions via the standard orthogonal vertical/horizontal splitting and O'Neill A-tensor decomposition of the curvature tensor. Vertical, horizontal, and mixed cases are obtained directly from Gauss-Codazzi identities for the distributions; equality cases occur precisely when second-fundamental-form or A-tensor terms vanish. Applications to real/complex/generalized Sasakian space forms consist of direct substitution of the known curvature tensors of the total space. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the central relations are independent consequences of established Riemannian submersion theory and are externally falsifiable via the curvature formulas.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only abstract available; no explicit free parameters, axioms, or invented entities listed. Standard Riemannian geometry background is presupposed.

axioms (1)
  • domain assumption Riemannian submersions admit orthogonal vertical-horizontal distribution splitting with standard curvature relations
    Invoked implicitly when deriving inequalities for vertical, horizontal, and mixed cases.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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

    math.DG 2026-06 unverdicted novelty 5.0

    Derives optimal generalized δ-invariant inequalities for Riemannian submersions and maps, characterizes equality cases via shape operators, and applies results to real and complex space forms.

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