{"id":"465643f1-b015-497a-a760-560e67e2f733","arxiv_id":"2608.03519","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"New examples show constant Ricci eigenvalues do not imply curvature homogeneity, with sharp low-codimension Euclidean immersions.","lead":"This paper constructs two families of manifolds whose Ricci eigenvalues are constant while the full curvature changes from point to point. They can be placed inside flat Euclidean space with the smallest possible number of extra dimensions, two or k+2.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Minimum-codimension claims for the first family and for N_0^n rest on unverified same-author preprint [5]; the adapted-minimality claim for W_m^k is self-contained.","rationale":"The reader's weakest assumption correctly identifies the reliance on [5] as a substantive dependency for unqualified minimality claims. However, the reader overstates the scope by listing Theorem 1.3(iii): that theorem's adapted-minimality statement is proved in Theorem 1.4(a) using only Gauss equation and the Ricci-flatness of N_0^n, independent of [5]. The external dependency is real for Theorem 1.2(iii) and Proposition 2.3, where [5, Cor. 1.1] is the only obstruction to codimension-one immersions. The paper's internal differential-geometric arguments--Proposition 2.1's Kretschmann computation, the explicit embedding (13), and Theorem 1.4's algebraic proof--are explicit and checkable. I verified the induced-metric computation for (13) and the eigenvalue argument in Theorem 1.4(a); both are sound. Thus the central adapted-minimality claim stands, but the paper's broader 'minimum codimension' claims for the first family remain conditional on an unverified same-author preprint. The CONDITIONAL verdict is therefore appropriate; no adjustment is needed.","tokens_in":10335,"tokens_out":23840,"duration_ms":237817,"concrete_test":"Independently re-derive [5, Cor. 1.2] in the special case of the product N^n×R^ℓ: assume a codimension-one isometric immersion into Euclidean space exists, then use the Gauss and Codazzi equations to prove the Ricci eigenvalues are constant and the Kretschmann scalar is constant, contradicting (5). If this derivation succeeds, Theorem 1.2(iii) holds without relying on [5]. If a counterexample is found instead, the minimality claim is false.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's unqualified minimality claims--Theorem 1.2(iii) (codimension two is smallest for M=N^n×R^ℓ) and Proposition 2.3 (codimension two is smallest for N_0^n)--depend on Corollary 1.1 of the same-authors' preprint [5], which is cited but not proved or formalized. Section 3.1 explicitly invokes it to exclude codimension-one immersions. If [5, Cor. 1.2] were false, these claims would fail, although the immersions themselves would remain valid. The second family's sharpness claim, Theorem 1.4(a), is independent: it proves adapted codimension ≥ k+2 via the Gauss equation and Ricci-flatness, with no reliance on [5]. Thus the strongest_claim as stated is secure, but the first family's headline 'minimum codimension two' is conditional on an external unpublished result.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":10570,"tokens_out":8744,"duration_ms":102088,"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":[{"comment":"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}.","section":"§3.1, Theorem 1.2(iii) and Proposition 2.3"},{"comment":"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.","section":"Abstract and Introduction"}],"minor_comments":[{"comment":"The title in the arXiv rendering contains broken spacing: 'CUR V ATURE' and 'CONST ANT'. This should be fixed in the final version.","section":"Title/header"},{"comment":"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.","section":"Proposition 2.3, proof of injectivity"},{"comment":"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.","section":"Theorem 1.4(a)"},{"comment":"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.","section":"Remark 2.4 and Appendix A"}],"recommendation":"major_revision","confidential_remarks":"The manuscript's most valuable contribution, Theorem 1.4 and the second family, is self-contained and sound. The first family's minimality claims, however, rest on the authors' own unpublished preprint [5]. If the editorial policy is that such dependencies are acceptable when clearly flagged, the current paper can be revised to state the conditional status. If the policy requires all load-bearing results to be proved in the submitted manuscript, then the paper is not yet in publishable form. I recommend major revision rather than rejection, because the core constructions and the sharpness result for the second family are independent and convincing."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nI 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.\n\nBottom 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].","headline":"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.","tokens_in":11037,"tokens_out":3472,"would_cite":true,"duration_ms":34457,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C42","53C40","53C25","53B25"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["constant Ricci eigenvalues","curvature inhomogeneous","isometric immersions","Einstein warped product","Schwarzschild–Tangherlini metric","Kretschmann scalar"],"falsifier":"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.","tokens_in":10234,"feed_emoji":"📐","tokens_out":14098,"duration_ms":145915,"temperature":0.7,"pith_summary":"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.","feed_headline":"Constant Ricci eigenvalues do not force curvature homogeneity","feed_subtitle":"Warped products and sphere-crossed Schwarzschild–Tangherlini spaces embed at sharp codimension.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"supplies the Einstein warped-product metrics N^n and their codimension-two rotational isometric immersions that the first construction uses directly.","marker":"[3]"},{"why":"states Corollary 1.1, the cited obstruction to hypersurface immersions of curvature-inhomogeneous constant-Ricci-eigenvalue manifolds; minimality rests on it.","marker":"[5]"},{"why":"provides the warped-product curvature formulas used to compute the Kretschmann scalar and prove curvature inhomogeneity.","marker":"[2]"},{"why":"is cited for the Cartan–Janet theorem that gives the local immersion h of the profile into R^3, completing the codimension-two immersion.","marker":"[6]"},{"why":"shows the polar metric on L_0^2 extends smoothly across the origin, making the Schwarzschild–Tangherlini base complete.","marker":"[10]"},{"why":"justifies that smooth even functions of t are smooth functions of t^2, used to extend the rotational embedding across the bolt.","marker":"[16]"},{"why":"gives the alternative classification of Einstein hypersurfaces that rules out codimension-one immersions of the first family.","marker":"[11]"},{"why":"defines the classical Schwarzschild–Tangherlini metric of which N_0^n is the Riemannian counterpart.","marker":"[12, 14]"}],"fun_headline_variants":["Constant Ricci eigenvalues ≠ curvature homogeneity","Sharp codimension two for two-eigenvalue warped product embeddings","Schwarzschild–Tangherlini products embed at minimum codimension","k+2 codimensions minimal for Ricci-constant inhomogeneous spaces"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Constant Ricci eigenvalues ≠ curvature homogeneity","Sharp codimension two for two-eigenvalue warped product embeddings","Schwarzschild–Tangherlini products embed at minimum codimension","k+2 codimensions minimal for Ricci-constant inhomogeneous spaces"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00067,"raw_usage":{"total_tokens":2812,"prompt_tokens":585,"completion_tokens":2227,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":329,"completion_tokens_details":{"reasoning_tokens":2157}},"tokens_in":329,"tokens_out":2227,"duration_ms":19102,"temperature":1.0,"reasoning_tokens":2157,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T17:27:00.480406+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Dajczer, C.-R","cited_arxiv_id":null,"evidence_quote":"supplies the Einstein warped-product metrics N^n and their codimension-two rotational isometric immersions that the first construction uses directly."},{"cited_title":"Hypersurfaces with Constant Ricci Eigenvalues in Real Space Forms","cited_arxiv_id":"2606.13455","evidence_quote":"states Corollary 1.1, the cited obstruction to hypersurface immersions of curvature-inhomogeneous constant-Ricci-eigenvalue manifolds; minimality rests on it."},{"cited_title":"Chen,Differential Geometry of Warped Product Manifolds and Submanifolds, World Scien- tific, 2017","cited_arxiv_id":null,"evidence_quote":"provides the warped-product curvature formulas used to compute the Kretschmann scalar and prove curvature inhomogeneity."},{"cited_title":"Han and J.-X","cited_arxiv_id":null,"evidence_quote":"is cited for the Cartan–Janet theorem that gives the local immersion h of the profile into R^3, completing the codimension-two immersion."},{"cited_title":"Petersen,Riemannian Geometry, 3rd ed., Graduate Texts in Mathematics, vol","cited_arxiv_id":null,"evidence_quote":"shows the polar metric on L_0^2 extends smoothly across the origin, making the Schwarzschild–Tangherlini base complete."},{"cited_title":"Whitney,Differentiable even functions, Duke Math","cited_arxiv_id":null,"evidence_quote":"justifies that smooth even functions of t are smooth functions of t^2, used to extend the rotational embedding across the bolt."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"gives the alternative classification of Einstein hypersurfaces that rules out codimension-one immersions of the first family."}],"review_version":1}