Examples of Curvature Inhomogeneous Submanifolds with Constant Ricci Eigenvalues

Jianquan Ge; Yuyang Zhao

REVIEW 2 major objections 4 minor 16 references

Examples of Curvature Inhomogeneous Submanifolds with Constant Ricci Eigenvalues

T0 review · 2 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read The paper constructs two families of curvature-inhomogeneous manifolds with constant Ricci eigenvalues and proves their Euclidean immersion codimension is the smallest possible in the relevant class.

desk verdict The constructions are real and the adapted-codimension bound is self-contained, but the 'minimum codimension two' headline for the first family rests on an unproved same-author preprint. read the letter →

arxiv 2608.03519 v1 pith:S2S6Q5CC submitted 2026-08-04 math.DG

Jianquan Ge , Yuyang Zhao This is my paper · ORCID

classification math.DG MSC 53C4253C4053C2553B25

keywords constantRiccieigenvaluescurvatureinhomogeneousisometricimmersionsEinsteinwarpedproductSchwarzschild–TangherlinimetricKretschmannscalar

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper establishes that a Riemannian manifold can have constant Ricci eigenvalues—the same list of eigenvalues of the Ricci operator at every point—even though its curvature is not homogeneous, meaning the curvature tensor cannot be carried from one tangent space to another by a linear isometry. The construction produces two explicit families: Einstein warped products crossed with flat Euclidean factors, which have exactly two distinct Ricci eigenvalues and admit a local isometric immersion into Euclidean space of codimension two, the smallest possible; and the complete Ricci-flat Riemannian Schwarzschild–Tangherlini manifold crossed with round spheres, which has k+1 distinct Ricci eigenvalues and admits a global isometric embedding into R^(m+k+2) with flat normal bundle. A companion theorem shows that within the natural product-adapted class the codimension k+2 cannot be lowered, and any immersion into lower codimension must have both a non-adapted second fundamental form and a nonflat normal bundle. The examples therefore separate 'constant Ricci eigenvalues' from 'curvature homogeneous' in dimension at least four, while showing that the hypersurface obstruction from an earlier cited result disappears exactly at codimension two.

What carries the argument

The central machinery is the Einstein warped product N^n = L^2 ×_φ S^{n-2} with profile metric dt^2 + φ'^2 du^2 and warping function φ satisfying φ'^2 = 1 − ρ/(n−1)φ^2 + c/φ^{n−3}; the complete Ricci-flat case is the Riemannian Schwarzschild–Tangherlini manifold. The paper's key observation is that the Kretschmann scalar is a constant plus a term c^2 φ^{2−2n}, so it varies with the warping function and is nonconstant on every open set; the Ricci operator, meanwhile, has constant eigenvalues. The rotational immersion f(x,y) = (h(x), φ(x)y) realizes these metrics in Euclidean space with flat normal bundle, and the Gauss equation supplies the codimension lower bounds.

What would settle it

Try to build a local isometric immersion of W_m^k into R^(m+k+1) with product-adapted second fundamental form; Theorem 1.4(a) rules this out, so any valid construction would refute the sharp-codimension claim, and the proof identifies the obstruction as forcing N_0^n to be flat, contradicting its nonconstant Kretschmann scalar.

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Extended reading notes

Core claim

The paper's strongest claim: for each k, the product W_m^k = N_0^n × S^{n_1}(r_1) × ... × S^{n_k}(r_k), where N_0^n is the complete Riemannian Schwarzschild–Tangherlini manifold, is complete and curvature inhomogeneous while having exactly k+1 distinct constant Ricci eigenvalues; it embeds globally into R^(m+k+2) with flat normal bundle, and no product-adapted immersion uses fewer than k+2 codimensions. The two-eigenvalue family N^n × R^ℓ attains the analogous sharp codimension two.

Load-bearing premise

The minimal-codimension claims rest on the cited corollary that no connected curvature-inhomogeneous manifold of dimension at least four with constant Ricci eigenvalues admits a local hypersurface immersion into a real space form; the paper does not prove this corollary.

Editorial extensions

If this is right

In dimension at least four, constant Ricci eigenvalues does not imply curvature homogeneity; the constructions are explicit counterexamples.
The obstruction to codimension-one immersion obtained from the cited hypersurface result is exactly bypassed by adding one more dimension: both families achieve codimension two or higher.
For W_m^k with pairwise distinct ρ_j, any immersion into codimension k+1 or less must have non-adapted second fundamental form and nonflat normal bundle (Theorem 1.4(b)).
The Riemannian Schwarzschild–Tangherlini manifold itself embeds properly in Euclidean space of codimension two with flat normal bundle.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Inference: the same Kretschmann-scalar test should force curvature inhomogeneity for any Einstein warped product with nonconstant warping, giving a flexible recipe for constant-Ricci-eigenvalue submanifolds at controlled codimension.
Inference: Theorem 1.4(a)'s 'adapted class' restriction suggests a natural project: decide whether k+2 is the absolute minimum codimension for any immersion of W_m^k; Theorem 1.4(b) says any would-be counterexample must be non-adapted with nonflat normal bundle.
Inference: the appendix's formal comparison identifies the Euclidean embedding as the Wick-rotated version of a known horizon-regular embedding of the Schwarzschild metric, a link that might carry over to Euclidean-signature quantum-gravity models, though the paper does not pursue this.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The constructions are real and the paper's main geometric claims hold up, with one important split. The new material is the Kretschmann computation for the warped products (Prop. 2.1), the global embedding of the Riemannian Schwarzschild-Tangherlini manifold (Prop. 2.3), and the sharp bound on adapted codimension for the product with spheres (Thm. 1.4(a)). I checked the algebra in Prop. 2.1 and the Gauss-equation argument in Thm. 1.4(a); both are explicit and correct. The proof that f_N0 extends smoothly across the bolt is also handled properly, which is the kind of detail that usually gets waved away.

The soft spot is narrower than it first appears. The immersion constructions themselves are solid: the first family gives local codimension-two immersions, the second gives global codimension-(k+2) embeddings. What is not self-contained is the 'minimum' part for the first family and for N_0^n: Theorem 1.2(iii) and Proposition 2.3 invoke Corollary 1.1 of the same authors' preprint [5] to rule out codimension one. That corollary is not proved or even stated beyond a one-line reference. If [5] has a gap, those minimality claims fail, while the immersions would still exist. Theorem 1.4(a), by contrast, is independent of [5] and proves what it claims within the adapted product class. The paper is honest about this for the second family (the abstract says 'smallest within the adapted product class'), but the first family's abstract says 'minimum codimension two' without qualification.

I don't see internal circularity. The constructions don't assume what they prove. The citation pattern is heavily self-referential—[3] and [5] do a lot of work—but [3] is published and [5] is flagged as a preprint. A referee should ask the authors to either include the needed corollary or mark those claims as conditional.

Bottom line: this is a useful paper for people in submanifold theory and curvature homogeneity. It deserves peer review. Send it out, but ask the referee to focus on the dependence on [5].

Referee Report

2 major / 4 minor

Summary. The paper constructs two families of curvature-inhomogeneous Riemannian manifolds with constant Ricci eigenvalues. The first family is M^{n+\ell}=N^n\times\mathbb{R}^\ell, where N^n is a non-Ricci-flat Einstein warped product of Dajczer\textendash{}Onti\textendash{}Vlachos; the paper proves it has exactly two constant Ricci eigenvalues, is nowhere locally curvature homogeneous, and admits a local isometric immersion of codimension two, which it claims is minimal. The second family is W_m^k=N_0^n\times\prod S^{n_j}(r_j), where N_0^n is the complete Riemannian Schwarzschild\textendash{}Tangherlini manifold; it has k+1 distinct constant Ricci eigenvalues, is curvature inhomogeneous, and admits a global isometric embedding of codimension k+2, with flat normal bundle. The paper also proves a sharpness theorem for the second family: any isometric immersion adapted to the product structure has codimension at least k+2, and any immersion of codimension at most k+1 with pairwise distinct spherical Ricci eigenvalues must have non-adapted second fundamental form and nonflat normal bundle.

Significance. The constructions are explicit and the main curvature computations are checkable. Proposition 2.1 gives a closed-form Kretschmann scalar for the warped products, and the proof of Theorem 1.4 is self-contained: the adapted-codimension lower bound follows from the Gauss equation plus Ricci-flatness, without appeal to any external result. If Corollary 1.1 of the same-authors' preprint [5] is accepted, the first family also provides the claimed minimal-codimension examples. The paper therefore gives a useful, concrete negative answer to the question whether constant Ricci eigenvalues force curvature homogeneity in higher codimension. Its strongest self-contained contribution is the second family and the sharpness Theorem 1.4, which does not depend on [5]. The main weakness is that the unqualified minimality assertions for the first family and for N_0^n rest on an unpublished same-author preprint.

major comments (2)

[§3.1, Theorem 1.2(iii) and Proposition 2.3] The claim that codimension two is the smallest for the first family is load-bearing, and it depends on Corollary 1.1, imported from the same-authors' unpublished preprint [5, Cor. 1.2]. The proof in §3.1 explicitly invokes Corollary 1.1 to rule out local codimension-one immersions; Proposition 2.3 does the same for N_0^n. If that corollary were false or unproved, the immersions would still exist but the minimality assertions would fail. Please either include a proof of Corollary 1.1 in this paper, or, if that is infeasible, revise the statements and abstract to say 'minimal modulo [5, Cor. 1.1]' or 'minimal subject to the conjecture/preprint [5]'. For N_0^n a direct proof from Ricci-flatness and the Gauss equation is available and would remove part of the dependence; the same is not shown for the product M^{n+\ell}.
[Abstract and Introduction] The abstract states without qualification that the first family 'admits a local isometric immersion of minimum codimension two'. This is stronger than what is proved inside this paper, because the lower-bound half relies on [5]. The reader should be able to identify exactly which parts of the paper are conditional on the preprint. I recommend changing the abstract and Theorem 1.2(iii) so that the dependence on [5, Cor. 1.1] is explicit.

minor comments (4)

[Title/header] The title in the arXiv rendering contains broken spacing: 'CUR V ATURE' and 'CONST ANT'. This should be fixed in the final version.
[Proposition 2.3, proof of injectivity] The injectivity argument handles t>0 and t=0, but the recovery of y at t=0 is only implicit. It is true: the last block has norm φ(t), which equals μ only at t=0, and then division by μ recovers y. A sentence making this explicit would help.
[Theorem 1.4(a)] The phrase 'polarizing first in X and then in Y' is correct but terse. Since this step is central to the adapted-codimension bound, one or two displayed equations showing the polarization would improve readability without changing the argument.
[Remark 2.4 and Appendix A] Formula (14) is stated as a rewrite of (13), and the comparison with Fronsdal is detailed in Appendix A, but (14) is not used in the body of the paper. Consider either moving Appendix A to a separate note or adding a brief statement that it is verification of a historical comparison and not needed for the main theorems.

Circularity Check

1 steps flagged · score 4.0 of 10

Constructions are self-contained, but the 'minimum codimension two' claims for the first family and for N_0^n reduce to a load-bearing same-author preprint [5, Cor 1.2] that is not proved here.

self citation load bearing [Section 3.1, proof of Theorem 1.2(iii); see also Proposition 2.3]
"By part (ii), every nonempty open subset of M is curvature inhomogeneous, while its Ricci eigenvalues remain constant. Consequently, Corollary 1.1, applied to any sufficiently small connected open subset, rules out a local codimension-one immersion, so the codimension two attained by F is the minimum local Euclidean codimension."

Corollary 1.1 is stated in the introduction as '[5, Corollary 1.2]' and is not proved or formalized in this paper; it is a preprint by the same two authors. The minimal-codimension conclusion in Theorem 1.2(iii) is exactly the negation of a codimension-one immersion, and that negation is supplied entirely by this cited corollary. The same self-citation is used in Proposition 2.3 to conclude that codimension two is smallest for N_0^n. Thus those sharpness claims reduce to an unverified self-citation rather than to the paper's own computations. However, the immersions themselves, the Ricci eigenvalue computations, the curvature-inhomogeneity proofs, and the sharpness proof for the spherical family in Theorem 1.4(a) are independently established, so the circularity is localized.

full rationale

The paper's main constructions are self-contained: the warped-product metrics, the explicit immersions/embeddings, the Kretschmann-scalar computations, and the Ricci eigenvalue decompositions are all derived from standard formulas and explicit maps. Theorem 1.4(a) proves the adapted-codimension lower bound k+2 for the spherical family using only the Gauss equation, Ricci-flatness of N_0^n, and the independently established nonvanishing of its curvature tensor. The first family's existence and curvature-inhomogeneity claims are also proved without any input from [5]. The only genuine circular dependency is the unqualified 'minimum codimension two' claim for the first family and for N_0^n, which relies on Corollary 1.1 imported from the same authors' preprint [5] and not proved here. Since this is a load-bearing self-citation for an advertised part of the results, but not for the central construction, a score of 4 is appropriate rather than the lower score that would be used for a minor self-citation with no load-bearing role.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

No new physical entities, forces, or dimensions are postulated. The manifolds are formed from known warped products, Euclidean factors, and round spheres. The genuine imports are the cited theorems from [3] and [5].

free parameters (3)

c = arbitrary nonzero real
Constant in ODE (1) for the warping function. c not equal to 0 excludes the space form case, and curvature inhomogeneity in equation (5) depends on c squared being positive.
rho = arbitrary nonzero real
Einstein constant in Theorem 1.2. It creates the two distinct Ricci eigenvalues 0 and rho; the first family requires rho not equal to 0.
r_j = positive, with (n_j-1)/r_j^2 pairwise distinct
Radii of the sphere factors in Theorem 1.3. They determine the k+1 distinct Ricci eigenvalues and enter the codimension expressions.

assumptions (5)

domain assumption Existence, Einstein property, and codim-two immersion of the warped products in [3, Example 1(a), Proposition 4].
Subsection 2.1: the manifold N^n and its local isometric immersion f are imported from Dajczer-Onti-Vlachos without re-derivation.
domain assumption Corollary 1.1 of [5]: a curvature inhomogeneous constant-Ricci manifold of dimension at least four cannot be immersed as a hypersurface in a real space form.
Used in Theorem 1.2(iii) and Proposition 2.3 to prove that codimension two is minimal. It is a cited same-author preprint result, not reproduced here.
standard math Cartan-Janet theorem supplies a local isometric immersion h of the 2D metric in (4) into R^3.
Subsection 2.1: the profile h is needed to realize the warped product as a rotational submanifold of codimension two.
standard math Gauss and Ricci equations for Euclidean submanifolds.
Used throughout the proof of Theorem 1.4 to relate intrinsic curvature to the second fundamental form and normal curvature.
standard math Hopf-Rinow theorem, Petersen's smoothness criterion for rotationally invariant metrics, and Whitney's theorem on smooth even functions.
Subsection 2.2 and Proposition 2.3: these justify completeness, smooth extension across the origin, injectivity, and properness of the Schwarzschild-Tangherlini embedding.

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Cite this review

Pith. "Pith review of Examples of Curvature Inhomogeneous Submanifolds with Constant Ricci Eigenvalues." pith.science (2026). https://pith.science/paper/S2S6Q5CC

@misc{pith2026260803519,
  author       = {Pith},
  title        = {Pith review of: Examples of Curvature Inhomogeneous Submanifolds with Constant Ricci Eigenvalues},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S2S6Q5CC}},
  note         = {Machine review of arXiv:2608.03519}
}

abstract

We construct two families of curvature inhomogeneous Riemannian manifolds with constant Ricci eigenvalues. The first, derived from the Einstein warped products, has two distinct Ricci eigenvalues and admits a local isometric immersion of minimum codimension two. The second, arising from the Riemannian Schwarzschild--Tangherlini manifold, has $k+1$ distinct Ricci eigenvalues and admits an isometric embedding of codimension $k+2$, which is the smallest within the adapted product class.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

16 extracted references · 15 canonical work pages

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [5]

J. Q. Ge and Y. Y. Zhao,Hypersurfaces with constant Ricci eigenvalues in real space forms, preprint, arXiv:2606.13455 (2026)

work page Pith review arXiv 2026

[2] [3]

Dajczer, C.-R

M. Dajczer, C.-R. Onti, and T. Vlachos,A class of Einstein submanifolds of Euclidean space, J. Geom. Anal.,32(2022), 64, 20

work page 2022

[3] [1]

Bryant, L

R. Bryant, L. Florit, and W. Ziller,Curvature homogeneous hypersurfaces in space forms, Adv. Math.,475(2025), 110338, 21

work page 2025

[4] [2]

Chen,Differential Geometry of Warped Product Manifolds and Submanifolds, World Scien- tific, 2017

B.-Y. Chen,Differential Geometry of Warped Product Manifolds and Submanifolds, World Scien- tific, 2017

work page 2017

[5] [4]

Fronsdal,Completion and embedding of the Schwarzschild solution, Phys

C. Fronsdal,Completion and embedding of the Schwarzschild solution, Phys. Rev. (2),116(1959), 778–781

work page 1959

[6] [6]

Han and J.-X

Q. Han and J.-X. Hong,Isometric Embedding of Riemannian Manifolds in Euclidean Spaces, Mathematical Surveys and Monographs, vol. 130, American Mathematical Society, 2006

work page 2006

[7] [7]

S. W. Hawking,Gravitational instantons, Phys. Lett. A,60(1977), 81–83

work page 1977

[8] [8]

Kasner,Finite representation of the solar gravitational field in flat space of six dimensions, Amer

E. Kasner,Finite representation of the solar gravitational field in flat space of six dimensions, Amer. J. Math.,43(1921), 130–133

work page 1921