{"id":"a9a94ef8-44b8-4f00-acb3-3aec4c60f75f","arxiv_id":"2412.16697","paper_version":4,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Sasakian structures are defined for arbitrary contact manifolds using Kähler structures on symplectic R^x-bundles, recovering the classical case and giving a canonical Sasakian product.","lead":"This paper extends Sasakian geometry from contact manifolds that admit a global contact form to all contact manifolds, including non-coorientable ones such as jet bundles over the Möbius band. It achieves this by translating the geometry into homogeneous symplectic and Kähler structures on a principal bundle over the manifold, which also yields a natural product of Sasakian manifolds.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central construction depends on Theorem 3.6 (contact manifolds correspond to symplectic R^x-bundles), which is cited to [20] and not reproved; if this equivalence failed for non-trivializable contact structures, Definition 9.2 would have no objects.","rationale":"The reader's ACCEPT verdict is sound. The local classification of homogeneous Kählerian structures (Theorems 8.8 and 8.9) is proved in detail, and the global statement Theorem 8.11 follows by the described gluing of the local models. The Möbius-band jet bundle example provides a concrete realization of a non-coorientable contact manifold with a Sasakian structure, which supports the central claim. The only genuinely load-bearing step that is not demonstrated in the text is the foundational equivalence between contact manifolds and symplectic R^\\times-bundles (Theorem 3.6). This is a standard result in the authors' program and is consistent with the worked example, so it does not warrant changing the verdict. The proposed check would settle whether this cited equivalence holds in the non-trivializable case that the new definition requires. No other significant technical objection was identified: the canonical lift in Theorem 9.1 is well-defined, the coorientable case is recovered, and the product construction is internally consistent.","tokens_in":30900,"tokens_out":32562,"duration_ms":285703,"concrete_test":"Verify Theorem 3.6 directly for a non-trivializable example: take M=J^1B^* from Example 4.3, form L=TM/C and P=(L^*)^\\times, restrict the canonical symplectic form of T^*M to P, and check explicitly that (i) the restricted 2-form is closed and non-degenerate, (ii) it is 1-homogeneous with respect to the R^\\times action, and (iii) the projection of ker(i_\\nabla\\omega) equals the original contact distribution C. If all three hold for this non-coorientable case, the cited equivalence is confirmed in exactly the regime the definition needs; any failure would invalidate Definition 9.2.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Definition 9.2 and Theorem 9.1 operate on the symplectic cover P of a contact manifold (M,C). The existence, uniqueness, and canonical form P=(L^*)^\\times with L=TM/C are taken from Theorem 3.6, attributed to [20] and not proved in this paper. This is the load-bearing input: the new notion of an (almost) Sasakian structure is defined as compatibility of (P,ω) with the lifted metric fg_M. If the equivalence in Theorem 3.6 were valid only for coorientable contact structures, or if the canonical representative failed to be symplectic for a non-trivializable line bundle L, then the construction would not apply to the non-coorientable manifolds the paper most wants to cover. The local analysis in Theorems 8.8 and 8.9 appears internally consistent, and the gluing discussion around Theorem 8.11 is plausible, but the foundational equivalence is cited rather than demonstrated. The Möbius-band example gives strong evidence that the equivalence holds in the critical non-trivializable case, so this is a dependence to verify rather than a demonstrated error.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes an extension of Sasakian geometry from cooriented contact manifolds to arbitrary contact manifolds understood as contact distributions. The key idea is to work on the symplectic R^×-bundle cover (P,ω) associated with a contact manifold (M,C). The authors classify positively homogeneous Riemannian metrics on such covers that are compatible with ω (Theorems 8.8 and 8.11), characterize the integrable case (Theorem 8.9), exhibit a canonical lift of any Riemannian metric g_M on M to a positively homogeneous metric fg_M on P (Theorem 9.1), and then define (almost) Sasakian structures by requiring (P,ω,fg_M) to be (almost) Kähler (Definition 9.2). They also define a Sasakian product of Sasakian manifolds using products of Kählerian R^×-bundles (Theorem 11.3), and illustrate the theory on the Möbius band and on a non-coorientable jet bundle.","tokens_in":31137,"tokens_out":24453,"duration_ms":199466,"significance":"If the main results hold, this gives a natural, choice-free extension of Sasakian geometry to non-coorientable contact manifolds, recovering the classical cooriented definition as a special case. The local classification in Theorems 8.8 and 8.9 is proved in detail, and the canonical lift in Theorem 9.1 is parameter-free, which are genuine strengths. The global paired formulation of Theorem 8.11 and the Möbius-band example provide concrete evidence that the construction applies precisely in the non-trivializable cases the paper targets. The product construction is also conceptually appealing and appears to be correct in its theorem form, although one of its explicit examples contains a computational error (see minor comments). The main external input, Theorem 3.6 on the equivalence between contact manifolds and symplectic R^×-bundles, is cited from [20]; the paper gives enough of the construction around it to make the dependence transparent, and the Möbius example supports the claimed equivalence, so I do not see a circularity or a demonstrated failure there.","major_comments":[],"minor_comments":[{"comment":"With the parametrization (s1,s2)=(ts/(t+1), s/(t+1)), substitution into equation (49) gives g_M = dt^2/[t(t+1)^2] + t/(t+1) g_M1 + 1/(t+1) g_M2, not dt^2/(t+1)^2 + t/(t+1) g_M1 + 1/(t+1) g_M2 as displayed. Consequently the subsequent formulas for g_C and φC are written for a different metric; the example should be corrected and the paired CR structure recomputed. This is a local error in an illustration and does not affect the proof of Theorem 11.3.","section":"Section 11, Example 11.4"},{"comment":"The canonical equivalence between contact manifolds and symplectic R^×-bundles is quoted from [20] rather than proved. The surrounding text already sketches the construction via the Liouville 1-form and the embedding into T*M, so the dependence is not circular; nevertheless, a short self-contained proof or a more precise statement of the functorial equivalence would make Definition 9.2 easier to verify by the reader.","section":"Section 3, Theorem 3.6"},{"comment":"In the statement of Theorem 11.3, the notation 'ω1 ⊗ ω2' should be 'ω1 ⊕ ω2', since the product symplectic form on P1×!P2 is defined as the direct sum, not a tensor product.","section":"Section 11, Theorem 11.3"},{"comment":"The sentence 'The smooth manifold M1×!M2 is actually an R^×-principal bundle over M1×M2' is confusing because M1×!M2 has already been introduced as the base of the diagonal R^×-bundle P1×!P2. Please clarify that M1×!M2 carries a residual R^×-bundle structure over M1×M2, while the diagonal action makes P1×P2 a principal R^×-bundle over M1×!M2.","section":"Section 11, first paragraph"},{"comment":"In the statement of Theorem 8.11, the phrase 'dµ vanishes on |ξ|' is terse; since µ is a 1-form on M, the condition means i_{|ξ|} dµ = 0. Stating this explicitly would improve readability, especially because the local model has µ = a η with constant a in the integrable case.","section":"Section 8, Theorem 8.11"}],"recommendation":"minor_revision","confidential_remarks":"The central construction appears sound and the paper is a serious contribution. The stress-test concern about Theorem 3.6 is a self-containedness issue rather than a demonstrated error, given the sketch in Section 3 and the Möbius-band evidence. The computational error in Example 11.4 should be fixed before publication, but it is local and does not undermine Theorem 11.3. I recommend minor revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear [Colleague],\n\nThis paper does what the title says: it extends Sasakian geometry from cooriented contact manifolds to general contact distributions. The route is conceptual—use the symplectic R^x-bundle (the symplectic cover) instead of a global contact form, define homogeneous Kähler structures on that bundle, and then read off a generalized Sasakian metric on the base. The canonical lift in Theorem 9.1 is the key move: every Riemannian metric on M determines a unique positively homogeneous metric on P, so \"Sasaki\" becomes \"the Kählerianization is Kähler.\" In the cooriented case this recovers the classical definition. That is a genuine gap-filling result, and the paper earns its place in the subfield.\n\nWhat I found most convincing: the local classification theorems are proved in detail, and the Möbius band example (Section 8.6, Example 9.3) shows the machinery actually works on a non-trivializable contact manifold. The Sasakian product in Section 11 also falls out naturally from products of R^x-bundles, not from an ad hoc formula. That is a real conceptual payoff.\n\nSoft spots: the whole framework leans on Theorem 3.6, the correspondence between contact manifolds and symplectic R^x-bundles, which is cited to [20] and not reproved. If that equivalence failed for non-trivializable structures, Definition 9.2 would have no objects. I don't think it fails—[20] is the second author's own published work, and the Möbius example gives direct evidence in the critical case—but a referee should ask for a precise statement or a proof sketch of the equivalence in the non-trivializable case. Also, Theorem 8.11's global paired-structure formulation is intricate and could use a bit more exposition; the local-to-global gluing is plausible but not fully formalized. These are verification requests, not demonstrated errors.\n\nThe citation pattern is heavy on the authors' own program, but the cited results are the actual tools used, so I don't see that as a flaw.\n\nWho is this for? Anyone working in Sasakian or contact geometry, especially on non-coorientable examples like jet bundles of line bundles. It deserves a serious referee—the central claim is new, the proofs are mostly detailed, and the gap it fills is real. I'd accept it for review and likely accept after minor revision.","headline":"Grabowska–Grabowski–Mohseni give the first coherent Sasakian definition for non-coorientable contact manifolds, and the construction holds up; the only real dependency is a cited symplectic R^x-bundle equivalence that the Möbius example supports.","tokens_in":31688,"tokens_out":2593,"would_cite":true,"duration_ms":20303,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C25","53D10","53D05","32V05","53D35"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every Riemannian metric on a contact manifold determines a canonical lift to the symplectic cover, and the contact manifold is Sasakian exactly when that lift is Kählerian.","keywords":["contact structure","Sasakian manifold","symplectic R^x-bundle","Kählerian R^x-bundle","CR structure","principal bundle","contact product","non-coorientable contact manifold"],"falsifier":"On the non-trivializable contact manifold $J^{1}$B^* over the Möbius band described in Example 9.3, compute the Nijenhuis torsion of the canonical almost complex structure J on its symplectic cover T^*B^×; the paper asserts it vanishes, so a nonzero value would refute the integrability criterion. More generally, take any contact manifold (M,C) with a metric g_M for which the canonical lift is compatible, and check whether failure of the paired CR integrability condition (N_{Φ_C}=0) is accompanied by nonzero N_J; if they ever disagree, Theorem 10.2's equivalence is false.","tokens_in":30694,"feed_emoji":"🌀","tokens_out":6463,"duration_ms":53162,"temperature":0.7,"pith_summary":"This paper argues that Sasakian geometry need not be tied to a global contact form. It shows that any contact manifold (M,C), coorientable or not, has a canonical symplectic cover P, a principal R^x-bundle equipped with a homogeneous symplectic form ω. Given any Riemannian metric g_M on M, the paper constructs a uniquely determined positively homogeneous metric \\tilde g_M on P, called its Kählerianization. The contact manifold is called almost Sasakian exactly when (P,ω,\\tilde g_M) is almost Kähler, and Sasakian when it is Kähler; for cooriented manifolds this recovers the classical definition. If correct, this extends Sasakian geometry to non-coorientable contact manifolds and gives a canonical product of Sasakian manifolds.","feed_headline":"Every contact metric gets a unique Kählerianization","feed_subtitle":"A metric on any contact manifold, even non-coorientable, determines one homogeneous Kähler structure on its symplectic cover.","key_machinery":"The symplectic cover: for a contact manifold (M,C), the annihilator C^o ⊂ T^*M, minus the zero section, is a principal R^x-bundle P over M whose canonical symplectic form is 1-homogeneous, and the contact distribution is the projection of the kernel of the Liouville form θ = i_∇ ω. The second ingredient is positive homogeneity: a tensor is positively homogeneous of degree k when pullback by the R^x action scales it by |s|^k, which allows genuine Riemannian metrics on the R^x-bundle even though the symplectic form is odd under s→−s. The calibration s = g(∇,∇) (or, canonically, the norm on L^*) turns any metric g_M on the base into \\tilde g_M = s((ds/s)^2 + g_M), and compatibility is checked through the almost complex structure J = (ω^♭)^{-1}∘g^♭.","core_discovery":"The central claim is Theorem 9.1 and Definition 9.2: every Riemannian metric g_M on a contact manifold (M,C) determines a unique calibration s on the symplectic cover P, defined by the norm induced on L^* = (TM/C)^*, and hence a canonical positively homogeneous metric \\tilde g_M = s((ds/s)^2 + g_M). Then (C,g_M) is an almost Sasakian structure if and only if (ω,\\tilde g_M) makes P an almost Kählerian R^x-bundle, and a Sasakian structure if and only if the induced almost complex structure is integrable. The authors further characterize all homogeneous almost Kähler structures on symplectic covers: locally they are of the form g = s((ds/s + μ)^2 + |η|^2 + g_C), with μ vanishing on C and a paired almost CR structure on C, and integrability forces μ to be closed (zero when P is non-trivializable) and the paired CR structure to be integrable with the paired Reeb field Killing. In the cooriented case these formulas reduce to the classical cone characterization of Sasakian structures. The framework also produces a canonical Sasakian product, since products of Kählerian R^x-bundles are Kählerian.","pith_inferences":["One implicit consequence is that the category of Sasakian manifolds is closed under products once morphisms are taken to respect the symplectic-cover structure; classical cooriented-only formulations did not have a natural product.","The paired CR structure language suggests an extension of CR geometry to non-orientable distributions, where a complex structure on the contact subbundle is defined only up to an overall sign.","The local freedom in μ (a 1-form vanishing on C) could be tested as a source of genuinely new almost Sasakian metrics on a fixed contact manifold, with integrability selecting the closed ones.","A natural next test is whether the canonical Sasakian product preserves curvature features, for instance whether the product of two Einstein-Sasakian metrics is again Einstein-Sasakian; the paper does not address this."],"forward_implications":["Non-coorientable contact manifolds, such as first jet bundles of non-trivializable line bundles, now have a well-defined Sasakian geometry; the paper works out the Möbius-band example explicitly.","For cooriented manifolds with a chosen contact form, the new definition is equivalent to the classical one, so existing Sasakian geometry is preserved rather than replaced.","A Riemannian metric on a contact manifold is almost Sasakian exactly when it is a Levi metric built from a paired almost CR structure on the contact distribution.","The Sasakian product of two Sasakian manifolds is canonically Sasakian, removing the ad hoc choice of contact form in earlier contact-product constructions.","The Kählerianization of (M,C,g_M) is unique, so checking whether a metric is Sasakian reduces to checking integrability of one canonical almost complex structure on a fixed bundle."],"supporting_citations":[{"why":"Establishes the canonical one-to-one correspondence between contact manifolds and symplectic R^x-bundles that underlies the whole construction.","marker":"[20]"},{"why":"Supplies the classical symplectization picture of a contact form as a homogeneous symplectic form on a cone.","marker":"[1]"},{"why":"Develops the R^x-bundle and Jacobi-bundle viewpoint that the paper relies on for symplectic covers and contact products.","marker":"[7]"},{"why":"Gives the cone-Kähler characterization of Sasakian structures in the cooriented case, the model that Definition 9.2 generalizes.","marker":"[5]"},{"why":"Provides the contact-metric and CR-structure background, including Levi forms and Nijenhuis torsion criteria, used in the integrability results.","marker":"[3]"},{"why":"Introduces the original Sasakian structure whose definition is being extended to general contact manifolds.","marker":"[31]"},{"why":"Supplies the cooriented contact-product formula that the new Sasakian product generalizes.","marker":"[24]"},{"why":"The authors' companion work on regularity and products in contact geometry, cited as the source of the contact-product concept used in Section 11.","marker":"[13]"}],"fun_headline_variants":["Every contact metric defines a unique Kähler cone","Sasakian geometry extended to all contact manifolds","Contact metrics yield canonical Kählerian symplectic covers","General contact manifolds gain Sasakian structures"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction rests on the cited theorem that every contact manifold corresponds, up to isomorphism, to a symplectic R^x-bundle; if that correspondence fails for non-coorientable contact structures, the proposed definition would not apply to the manifolds it is meant to cover.","fun_headline_variants_meta":{"raw":{"variants":["Every contact metric defines a unique Kähler cone","Sasakian geometry extended to all contact manifolds","Contact metrics yield canonical Kählerian symplectic covers","General contact manifolds gain Sasakian structures"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001342,"raw_usage":{"total_tokens":5547,"prompt_tokens":1135,"completion_tokens":4412,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":751,"completion_tokens_details":{"reasoning_tokens":4347}},"tokens_in":751,"tokens_out":4412,"duration_ms":24110,"temperature":1.0,"reasoning_tokens":4347,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T10:21:38.980556+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On the non-trivializable contact manifold $J^{1}$B^* over the Möbius band described in Example 9.3, compute the Nijenhuis torsion of the canonical almost complex structure J on its symplectic cover T^*B^×; the paper asserts it vanishes, so a nonzero value would refute the integrability criterion. More generally, take any contact manifold (M,C) with a metric g_M for which the canonical lift is compatible, and check whether failure of the paired CR integrability condition (N_{Φ_C}=0) is accompanied by nonzero N_J; if they ever disagree, Theorem 10.2's equivalence is false.","supporting_citations":[{"cited_title":"Grabowski, Graded contact manifolds and contact Courant algebroids,J","cited_arxiv_id":null,"evidence_quote":"Establishes the canonical one-to-one correspondence between contact manifolds and symplectic R^x-bundles that underlies the whole construction."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the classical symplectization picture of a contact form as a homogeneous symplectic form on a cone."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Develops the R^x-bundle and Jacobi-bundle viewpoint that the paper relies on for symplectic covers and contact products."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the cone-Kähler characterization of Sasakian structures in the cooriented case, the model that Definition 9.2 generalizes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the contact-metric and CR-structure background, including Levi forms and Nijenhuis torsion criteria, used in the integrability results."},{"cited_title":"Sasaki, Y","cited_arxiv_id":null,"evidence_quote":"Introduces the original Sasakian structure whose definition is being extended to general contact manifolds."},{"cited_title":"Ib´ a˜ nez, M","cited_arxiv_id":null,"evidence_quote":"Supplies the cooriented contact-product formula that the new Sasakian product generalizes."},{"cited_title":"The regularity and products in contact geometry","cited_arxiv_id":"2412.20491","evidence_quote":"The authors' companion work on regularity and products in contact geometry, cited as the source of the contact-product concept used in Section 11."}],"review_version":1}