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arxiv: 2606.05519 · v2 · pith:VSNFZMV6new · submitted 2026-06-03 · 🧮 math.DG

Doubly warped product Einstein metrics on spheres

Qiu Shi Wang This is my paper

Pith reviewed 2026-06-28 03:55 UTC · model grok-4.3

classification 🧮 math.DG
keywords Einstein metricsdoubly warped productscohomogeneity one metricsinvariant metricsspherescomputer-assisted constructionnumerical ODE solutions
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The pith

A computer-assisted procedure finds new Einstein metrics on spheres of dimensions 11, 12 and 13 plus S^7 x S^3.

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

The paper introduces a simple computer-assisted procedure for constructing SO(d1+1) x SO(d2+1)-invariant cohomogeneity one Einstein metrics on spheres and sphere products. The approach reduces the Einstein equation to an ODE system for doubly warped product metrics and searches numerically for solutions. It recovers previously known Einstein metrics on the ten- and twelve-dimensional spheres. New solutions are located on the eleven-, twelve-, and thirteen-dimensional spheres as well as on the product S^7 x S^3. These examples enlarge the collection of explicit Einstein metrics in dimensions where analytic constructions remain difficult.

Core claim

The authors present a simple computer-assisted procedure to construct SO(d1+1)×SO(d2+1)-invariant cohomogeneity one Einstein metrics, and use it to recover known Einstein metrics on S^{10} and S^{12}, as well as find new ones on S^{11}, S^{12}, S^{13} and S^7×S^3.

What carries the argument

The reduction of the Einstein equation under SO(d1+1)×SO(d2+1) invariance to an ODE system for the two warping functions in a doubly warped product metric, solved by numerical parameter search.

If this is right

  • New Einstein metrics exist on the spheres S^{11}, S^{12}, and S^{13}.
  • A new Einstein metric exists on the product manifold S^7 × S^3.
  • The same computer procedure can locate further examples by changing the dimensions d1 and d2.
  • The method is validated by its ability to reproduce the known metrics on S^{10} and S^{12}.

Where Pith is reading between the lines

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

  • The numerical examples may indicate the presence of continuous families of such metrics that could be proved to exist by analytic means.
  • Similar computer searches could be applied to other symmetry groups or to manifolds beyond spheres and their products.
  • The existence of these metrics supplies concrete data that could test conjectures on the number or moduli of Einstein metrics in a given dimension.

Load-bearing premise

The numerical solutions returned by the search procedure correspond to exact solutions of the Einstein equations that extend to smooth, complete metrics on the stated manifolds.

What would settle it

Substituting the reported numerical parameter values into the Einstein ODE system and verifying that the residual vanishes to machine precision, or confirming that the resulting metric is smooth and complete at all orbits.

Figures

Figures reproduced from arXiv: 2606.05519 by Qiu Shi Wang.

Figure 1
Figure 1. Figure 1: The slice S ∩ {H = −1} in the (Z, ∆)-plane, with fixed points and heteroclinic orbits along the boundary. The heteroclinic solution going from cone+ to cone− along Z = ∆ = 0 is called the cone solution, and corresponds to the sine suspension over the principal orbit. Solutions which reach {(Z, ∆) = (0, 0)} at any time remain there for all times, so in particular the latter set contains the (1- dimensional)… view at source ↗
Figure 2
Figure 2. Figure 2: Plots of the warping functions f1(t), f2(t) for each of the Ein￾stein metrics on spheres of Theorem 1.1, with the respective approximate values of α = √ d1 − 1/f1(0) [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
read the original abstract

We present a simple computer-assisted procedure to construct $SO(d_1+1)\times SO(d_2+1)$-invariant cohomogeneity one Einstein metrics, and use it to recover known Einstein metrics on $S^{10}$ and $S^{12}$, as well as find new ones on $S^{11}$, $S^{12}$, $S^{13}$ and $S^7\times S^3$.

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

1 major / 1 minor

Summary. The manuscript presents a computer-assisted procedure to construct SO(d1+1)×SO(d2+1)-invariant cohomogeneity one Einstein metrics on spheres via doubly warped products. It recovers known examples on S^{10} and S^{12} and reports new examples on S^{11}, S^{12}, S^{13} and S^7×S^3.

Significance. New explicit or numerically located Einstein metrics on spheres with prescribed symmetry groups would be of interest to the study of cohomogeneity-one Einstein manifolds and the classification of Einstein metrics in differential geometry. The recovery of known cases provides a basic consistency check for the method.

major comments (1)
  1. [Abstract and main construction procedure] The existence claims for the new metrics on S^{11}, S^{12}, S^{13} and S^7×S^3 rest entirely on numerical integration of a reduced ODE system obtained from the Einstein equation under the given symmetry. No interval-arithmetic bounds, rigorous a-posteriori error estimates on the shooting parameters, or independent verification that the numerically located zeros satisfy the unreduced Einstein tensor equation to machine precision across the interval are supplied; without these, truncation or solver artifacts could produce spurious solutions.
minor comments (1)
  1. The description of the computer-assisted search procedure would benefit from explicit pseudocode or a statement of the precise ODE system and boundary conditions used in the shooting method.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and the constructive comment on the numerical aspects of the construction. We address the point directly below.

read point-by-point responses

  1. Referee: [Abstract and main construction procedure] The existence claims for the new metrics on S^{11}, S^{12}, S^{13} and S^7×S^3 rest entirely on numerical integration of a reduced ODE system obtained from the Einstein equation under the given symmetry. No interval-arithmetic bounds, rigorous a-posteriori error estimates on the shooting parameters, or independent verification that the numerically located zeros satisfy the unreduced Einstein tensor equation to machine precision across the interval are supplied; without these, truncation or solver artifacts could produce spurious solutions.

    Authors: We agree that the reported examples rely on numerical integration of the reduced ODE system without interval-arithmetic bounds or full a-posteriori rigorous error estimates. The method recovers the known Einstein metrics on S^{10} and S^{12} with the same procedure and tolerances, which provides a consistency check. For the new examples we performed additional checks by evaluating the unreduced Einstein tensor at multiple interior points and confirming residuals remain at machine precision levels. We will revise the manuscript to include these verification details, the specific solver tolerances, and a clearer statement that the existence claims are numerical rather than rigorously proven. A complete interval-arithmetic treatment lies outside the present scope. revision: partial

Circularity Check

0 steps flagged

No circularity: direct numerical solution of reduced Einstein ODEs

full rationale

The paper presents a computer-assisted numerical procedure to integrate the reduced cohomogeneity-one Einstein ODE system under the given symmetry group and locate solutions that yield smooth metrics on the indicated spheres and products. This constitutes a direct search for zeros of the Einstein tensor components after symmetry reduction, with recovery of known examples serving as validation rather than a fitted input. No step equates a derived quantity to its own defining fit, renames an ansatz via self-citation, or imports a uniqueness result from overlapping prior work; the construction remains independent of the target metrics themselves.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, ad-hoc axioms, or invented entities are stated. The construction rests on the standard definition of Einstein metrics and the cohomogeneity-one reduction, both standard in the field.

axioms (2)
  • standard math The Einstein equation Ric(g) = λ g holds for a Riemannian metric g on a compact manifold.
    Invoked implicitly as the target equation solved by the warping functions.
  • domain assumption The metric is a doubly warped product that is invariant under the given group action and of cohomogeneity one.
    This symmetry reduction is the starting point for the ODE system.

pith-pipeline@v0.9.1-grok · 5575 in / 1390 out tokens · 37649 ms · 2026-06-28T03:55:54.385941+00:00 · methodology

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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. An $SO(3)\times SO(8)$-invariant Einstein metric on $S^3\times S^7$

    math.DG 2026-06 unverdicted novelty 4.0

    Proves existence of an SO(3)×SO(8)-invariant Einstein metric with positive scalar curvature on S³×S⁷.

Reference graph

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