{"id":"f8ce0a32-3f79-45a8-9a8f-f06b9c74e430","arxiv_id":"2608.08781","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"The authors characterize co-Kähler-like and flat Sasaki-with-torsion manifolds, classify compact 5-dimensional ∇-Einstein manifolds, and prove gauge equivalence of their new strong flow to generalized Ricci flow.","lead":"Sasaki with torsion manifolds are odd-dimensional spaces with a twisted connection, appearing in string theory. This paper classifies the flat and Einstein-like cases in dimensions 5 and 7 and builds a flow that preserves the twist while matching the generalized Ricci flow.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Classification hinges on the unproved [35] lemma that produces the parallel field V=θ♯−grad f; without verifying its exact hypotheses, the V≠0/V=0 dichotomy in Theorem 5.16 is unsupported.","rationale":"The reader's weakest assumption identifies the same load-bearing concern: the compact classification is imported from the very recent preprint [35], with the splitting argument from [6] as a secondary external input. I focused on the [35] lemma because it controls the entire V≠0/V=0 dichotomy and therefore every branch of Theorem 5.16. The paper is otherwise carefully written, with self-contained proofs for the classification conditional on that lemma; I found no internal inconsistency or sign of circularity. Since the reader's CONDITIONAL verdict already reflects exactly this dependence, no verdict adjustment is needed. If the [35] lemma were verified, the classification would be substantially supported; if not, the main theorem would collapse to the V=0 cases that are proved internally. The flow section also relies on a terse parabolicity argument, but that is not the central claim being stress-tested here.","tokens_in":37357,"tokens_out":8231,"duration_ms":95452,"concrete_test":"Obtain the statement of the result in [35, Kennon–Streets, arXiv:2511.20773] that is cited, and check line-by-line that its hypotheses match Definition 5.5 exactly: compact, dimension 5, ρ∇=0 and dH=0, with no extra condition such as c≠0 or V≠0. Then track the cited derivation through [33] and [26]: confirm that the generalized Ricci soliton equation supplied by [33] is gradient in the sense needed, and that the conclusion V=θ♯−grad f is ∇-parallel, unique up to the stated normalization, and when nonzero lies in F⊥ξ. If any of these checks fails or requires an extra hypothesis, Theorem 5.16 is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central classification (Theorem 5.16) splits on the existence, uniqueness, and horizontality of a ∇-parallel vector field V=θ♯−grad f, stated in Section 5.1 immediately before Proposition 5.9 and attributed to the recent preprint [35]. This statement is not proved in the paper; the surrounding remarks cite [33] and [26] but do not supply the derivation. Every branch of the theorem depends on it: Proposition 5.9 uses \"V∈F⊥ξ and V, φV parallel\" to conclude flatness in case V≠0, and the V=0 cases use θ=df, which forces the transverse geometry to be conformally Kähler and drives the f-constant/f-nonconstant dichotomy. If [35] in fact requires an additional hypothesis (e.g., some normalization of the generalized Ricci soliton, c≠0, or a priori nonvanishing of V), or if the uniqueness of f fails, then cases (1)–(4) need not be exhaustive. This is the load-bearing external input, and it is not machine-checked or reproduced here.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies Sasaki-with-torsion (SWT) structures, the odd-dimensional counterparts of Hermitian manifolds with torsion, with the Friedrich–Ivanov connection ∇. It proves Theorem 3.3: a SWT structure is co-Kähler-like (∇ satisfies the first Bianchi identity) if and only if the torsion H is closed and parallel. Theorem 4.7 characterizes ∇-flat SWT manifolds, up to finite cover, as quotients of simply connected odd-dimensional Lie groups with left-invariant bi-invariant structures. The central geometric object is the ∇-Einstein condition (strong SWT plus ρ∇=0), introduced as an analogue of Bismut Hermite-Einstein; Examples 5.7 and 5.8 give non-compact examples in dimensions 5 and 7. Theorem 5.16 classifies compact 5-dimensional ∇-Einstein manifolds into four cases according to the behaviour of V=θ♯−grad f, where the existence of this ∇-parallel vector field is imported from the preprint [35]. Section 5.2 sketches the 7-dimensional case. Section 6 develops a general U(n)′-flow framework, introduces the strong SWT flow (6.44)–(6.45), proves short-time existence, preservation of the strong condition, and gauge equivalence to generalized Ricci flow.","tokens_in":37637,"tokens_out":12815,"duration_ms":130222,"significance":"The paper is valuable both for the classification and for the flow theory. If Theorem 5.16 is fully supported, it gives a complete structural description of compact 5-dimensional ∇-Einstein SWT manifolds, and the appearance of the Box equation (5.32) links the geometry to the 6-dimensional Bismut Hermite-Einstein theory of [6] and to explicit physics examples such as the orthotoric family L_{a,b,c}. The flow part is novel, and Proposition 6.8 derives the gauge equivalence to generalized Ricci flow rather than assuming it; the preservation of the strong condition via a linear transversely parabolic equation for Ψ=dH is a clean argument. Most of the local computations (Theorem 3.3, Theorem 4.7, the examples, and the algebra in Proposition 6.8) are explicit and checkable. The main weakness is that the compact classification depends on external inputs, primarily the recent preprint [35], whose exact hypotheses are not reproduced in the manuscript.","major_comments":[{"comment":"The dichotomy in Theorem 5.16 is driven entirely by the existence and uniqueness of a ∇-parallel vector field V=θ♯−grad f, attributed to the recent preprint [35] in the unnumbered paragraph preceding Proposition 5.9. The manuscript neither proves this statement nor states its precise hypotheses: the normalization of f, the compactness hypotheses, and whether V∈F⊥ξ is part of the conclusion are not specified. Proposition 5.9 uses V∈F⊥ξ and the parallelism of V and φV to conclude that the only nonzero curvature component is R∇(X,Y)(α,φα), which then vanishes by ρ∇=0, and every case with V=0 uses θ=df. If [35] in fact requires an additional hypothesis, such as a normalization condition on the generalized Ricci soliton or a nonvanishing assumption on V, the four cases of Theorem 5.16 need not be exhaustive. Please include a proof or a precise statement of the imported lemma and verify that it applies verbatim to the ∇-Einstein SWT manifolds defined in Definition 5.5.","section":"§5.1, paragraph before Proposition 5.9"},{"comment":"This branch invokes the splitting argument of Corollary 1.2 of [6] via a \"transverse version of Theorem 1 in [7]\" after asserting that the transverse Ricci tensor of the Kähler metric has two distinct non-negative constant eigenvalues, one of which is zero. The imported result is load-bearing: it is what upgrades the parallelism of K to a global Riemannian splitting and ultimately forces N³≅SU(2). The precise statement of the transverse splitting theorem, its hypotheses (compactness, eigenvalue assumptions, the meaning of \"transverse version\") and the verification of those hypotheses in the present setting are not given. Please state and prove the splitting lemma in the present notation, or give a reference whose hypotheses match this situation exactly.","section":"§5.1, proof of Theorem 5.16, case (b)(i)"},{"comment":"The 7-dimensional analysis is presented as a sequence of claims importing [35, Theorem 5.6] and a \"generalization of Theorem 5.2 to local R^{2k+1}-bundles\", and the footnote to (5.34) acknowledges a sign and normalization discrepancy in the constants c_μ and c_ν. As written, equations (5.33), (5.34), and the generalized Box equation (5.35) cannot be verified from the manuscript, and the discrepancy is left unresolved. Since the section is presented as the paper's treatment of compact ∇-Einstein manifolds in dimension 7, please supply the missing derivation and fix the conventions, or explicitly mark the section as provisional.","section":"§5.2, equations (5.33)–(5.35)"},{"comment":"The short-time existence and uniqueness statement for the strong SWT flow is not proved in detail. After equation (6.46) the text asserts that the gauge-fixed system is transversely parabolic and that the DeTurck trick or the results in [9] apply, but the explicit gauge-fixing vector field and the principal symbol computation are not written down. Given that Theorem 6.5 is a main analytic result of the flow section, please provide the explicit gauge-fixed system or a precise reduction to [9] so that parabolicity and uniqueness are checkable from the paper.","section":"§6.2, Theorem 6.5"}],"minor_comments":[{"comment":"The Box equation in Theorem 1.2 is printed as \"2˜sT = (˜sT)²/2 − |gRic|²\", whereas equation (5.32) gives \"□˜sT = (˜sT)²/2 − |˜Ric|²\"; the missing □ and the spurious factor 2 should be corrected.","section":"Introduction, Theorem 1.2"},{"comment":"The heading \"∇-Hemite Einstein manifolds\" contains a typo; it should read \"∇-Hermite-Einstein manifolds\".","section":"Table of contents and Section 5 heading"},{"comment":"The proof begins with \"By Theorem 5.2, the ∇-Hermite–Einstein condition is equivalent to...\", but the statement used is Proposition 5.2; the cross-reference should be corrected.","section":"Proof of Theorem 5.12, first sentence"},{"comment":"The name \"Friderich-Ivanov connection\" is misspelled; it should be \"Friedrich–Ivanov connection\".","section":"Section 4, paragraph before Theorem 4.7"},{"comment":"The terminology is not uniform: Definition 5.5 uses \"∇-Einstein\", while the introduction and Theorem 5.16 use \"∇-Hermite–Einstein\". A sentence identifying the two terms would avoid confusion.","section":"Throughout, Definition 5.5 vs. Introduction"}],"recommendation":"major_revision","confidential_remarks":"The main obstacle to acceptance is the dependence of the central classification on the very recent preprint [35], whose content is not reproduced in this manuscript. It would be helpful if the authors either proved the V=θ♯−grad f lemma in an appendix or coordinated with [35] so that the two papers can be refereed together. The editor may also wish to confirm that the other recent preprints cited as load-bearing, namely [8], [20], and [50], are available in the form cited here."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nWhat you should know: this paper transfers the Hermitian Bismut program to Sasaki-with-torsion geometry and delivers a real classification theorem in dimension 5, plus a new flow that is gauge-equivalent to generalized Ricci flow. It is mostly solid, but the compact classification currently rests on a lemma imported from a very recent preprint, so treat Theorem 5.16 as conditional until that input is verified.\n\nWhat is actually new: the co-Kähler-like characterization (Theorem 3.3), the ∇-flat classification (Theorem 4.7), the compact 5-dimensional ∇-Einstein classification (Theorem 5.16), and the flow theory in Section 6, especially Proposition 6.8. The link to the La,b,c manifolds from AdS3×Y7 is a nice concrete connection to physics. The paper is carefully written, proves Theorem 3.3 and Proposition 6.8 in detail, and does not oversell.\n\nSoft spots, in proportion. Theorem 5.16 splits on the existence of a ∇-parallel vector field V = θ♯ − grad f, stated in Section 5.1 and attributed to [35]. This lemma is not proved here. If [35] carries extra hypotheses—normalization, c≠0, or nonvanishing—the dichotomy V≠0/V=0 may fail to be exhaustive. That is the main load-bearing external input, and the stress-test note is right to flag it. The f-constant subcase also imports Corollary 1.2 of [6], though that is published. Minor issues: Example 5.8 is sketched, the parabolicity of Theorem 6.5 is terse and leans on [20] and [9], and the 7-dimensional section is explicitly exploratory.\n\nThe parts proved in the text look sound to me on reading. I found no circularity and no fitted data; the authors flag exactly which results are external. I could not machine-check the long computations.\n\nThis deserves a serious referee. The referee should ask for the hypotheses of the [35] lemma to be stated in full and verified, or for the theorem to be presented as conditional. It is a useful paper for anyone in Sasakian geometry, torsion connections, or geometric flows.\n\nMy recommendation: engage, and push for the [35] dependency to be resolved or made explicit before this becomes the canonical citation.","headline":"A solid transfer of the Bismut program to Sasaki-with-torsion geometry, with a genuine 5-dimensional classification that currently rests on an unproved preprint lemma from [35].","tokens_in":38179,"tokens_out":2811,"would_cite":true,"duration_ms":29394,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C15","53C25","53C29","53C55"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper classifies compact five-dimensional ∇-Einstein Sasaki-with-torsion manifolds into four explicit local classes.","keywords":["Sasaki with torsion","∇-Einstein manifolds","Bismut connection","co-Kähler-like manifolds","geometric flows","generalized Ricci flow","almost contact metric structures","string backgrounds"],"falsifier":"Look for a compact five-dimensional $\\nabla$-Einstein Sasaki-with-torsion manifold whose universal cover is not one of the four models in Theorem 5.16—for instance, a compact example with $V\\neq 0$ but non-flat $\\nabla$, or a $V=0$, $c\\neq 0$, $f$ non-constant example whose transverse Kähler scalar curvature is constant or fails $\\square \\tilde{s}^T = (\\tilde{s}^T)^2/2 - |\\widetilde{\\mathrm{Ric}}|^2$. Since the theorem is conditional, the sharpest test is to disprove the imported lemma: exhibit a compact $\\nabla$-Einstein Sasaki-with-torsion manifold on which no normalized $f$ makes $\\theta^\\sharp-\\operatorname{grad} f$ parallel.","tokens_in":37129,"feed_emoji":"🌀","tokens_out":13072,"duration_ms":118932,"temperature":0.7,"pith_summary":"This paper studies Sasaki manifolds with torsion—the odd-dimensional counterpart of Hermitian manifolds with a Bismut connection—and asks what the analogue of the Bismut–Hermite–Einstein condition is. It introduces $\\nabla$-Einstein Sasaki-with-torsion manifolds, defined by vanishing of the $\\varphi$-Ricci form and closed torsion, and proves that in compact dimension five these manifolds fall into exactly four explicit local classes. Two classes are flat, one is a mapping torus over a compact Kähler Ricci-flat 4-manifold, and the remaining class consists of local $S^1$-bundles over Kähler surfaces whose strictly positive non-constant scalar curvature solves the Box equation $\\square \\tilde{s}^T = (\\tilde{s}^T)^2/2 - |\\widetilde{\\mathrm{Ric}}|^2$. Because such structures are exactly the odd-dimensional string backgrounds in dimension five, the classification provides a concrete catalogue of compact five-dimensional string backgrounds. The paper also introduces a geometric flow for Sasaki-with-torsion structures, proves short-time existence and preservation of the strong (closed torsion) condition, and shows the flow is gauge-equivalent to generalized Ricci flow.","feed_headline":"Compact 5-D torsion geometries fall into four explicit classes","feed_subtitle":"Four cases: flat Lie-group quotients, a mapping torus, and S^1-bundles over Kähler surfaces.","key_machinery":"The load-bearing object is the $\\nabla$-parallel vector field $V=\\theta^\\sharp-\\operatorname{grad} f$, where $\\theta$ is the Lee form of the transverse Hermitian structure; on compact $\\nabla$-Einstein manifolds its existence is imported from a recent preprint. Proposition 5.2 turns the $\\nabla$-Einstein condition $\\rho^\\nabla=0$ into the algebraic system $c\\, d\\eta=\\rho^B$ with $c$ constant and $dH=0$, so the transverse geometry is constrained by the Bismut Ricci form of the base. Conformally rescaling the transverse metric by $e^{-f}$ makes the base Kähler and forces its scalar curvature to obey the Box equation. The flat cases are controlled by Theorem 4.7, which identifies $\\nabla$-flat Sasaki-with-torsion manifolds, up to finite cover, with quotients $G/\\mathbb{Z}^k$ of simply connected odd-dimensional Lie groups carrying bi-invariant metrics and left-invariant normal almost contact structures. For the flow, the machinery is the decomposition of infinitesimal deformations of a $U(n)'$-structure into irreducible modules, which selects the evolution equations for $\\eta$ and $g^T$ and yields the gauge equivalence with generalized Ricci flow.","core_discovery":"The paper's central claim is Theorem 5.16: every compact five-dimensional $\\nabla$-Einstein Sasaki-with-torsion manifold belongs to exactly one of four classes. If $V=\\theta^\\sharp-\\operatorname{grad} f\\neq 0$, the connection $\\nabla$ is flat and the universal cover is isometric to $\\mathbb{R}\\times(\\mathbb{R}\\times SU(2))$. If $V=0$ and $c=0$, the manifold is a mapping torus over a compact Kähler Ricci-flat 4-manifold. If $V=0$ and $c\\neq 0$ with $f$ constant, $\\nabla$ is flat and the universal cover is $SU(2)\\times\\mathbb{C}$ with the standard left-invariant Sasaki structure on $SU(2)$. If $V=0$ and $c\\neq 0$ with $f$ non-constant, $\\nabla$ is non-flat and the manifold is locally an $S^1$-bundle over a 4-dimensional Kähler manifold whose strictly positive non-constant scalar curvature satisfies $\\square \\tilde{s}^T = (\\tilde{s}^T)^2/2 - |\\widetilde{\\mathrm{Ric}}|^2$; this transverse geometry is exactly the one found on six-dimensional Bismut–Hermite–Einstein manifolds.","pith_inferences":["The same $V$-dichotomy is likely to organize compact $\\nabla$-Einstein Sasaki-with-torsion manifolds in all odd dimensions: the 7-dimensional analysis already shows $V\\neq 0$ forces an $\\mathbb{R}^3$-bundle over a 4-dimensional Hermitian base, suggesting a hierarchy of Box-type equations indexed by the number of parallel directions.","Because the strong Sasaki-with-torsion flow is gauge-equivalent to generalized Ricci flow, established long-time and singularity results for the latter could transfer to this setting, turning the static classification into a dynamical existence proof for $\\nabla$-Einstein structures.","The non-flat 5-dimensional transverse geometry coincides with the transverse geometry of 6-dimensional Bismut–Hermite–Einstein manifolds, so the classification points to a fibration correspondence: unit $S^1$-bundles over such 6-manifolds should produce 5-dimensional $\\nabla$-Einstein manifolds, and the paper's examples realize the forward direction.","A testable extension is to run the strong Sasaki-with-torsion flow on the explicit non-compact examples (Examples 5.7 and 5.8) and check whether the Box equation emerges as a scalar constraint; convergence of the flow would give a dynamical construction of $\\nabla$-Einstein structures."],"forward_implications":["Every compact $\\nabla$-Einstein Sasaki-with-torsion 5-manifold is now locally known: the four cases of Theorem 5.16 give the complete set of universal covers and fibrations.","The only non-flat compact case has a Kähler surface base with strictly positive non-constant scalar curvature solving the Box equation, and the orthotoric orbifold family $S_{a,b,c}$ yields explicit examples $L_{a,b,c}$ with $c=2$.","$\\nabla$-flat Sasaki-with-torsion manifolds are, up to finite cover, quotients $G/\\mathbb{Z}^k$ of simply connected odd-dimensional Lie groups with bi-invariant metric and left-invariant normal almost contact structure.","In dimension 7, compact $\\nabla$-Einstein manifolds with $V\\neq 0$ have a local $\\mathbb{R}^3$-bundle structure over a 4-dimensional Hermitian base, and when $c=0$ the torsion satisfies a generalized Box equation (5.35); trivial bundle constructions give new examples.","The strong Sasaki-with-torsion flow exists uniquely for short time, preserves the strong condition, is gauge-equivalent to generalized Ricci flow, and has every $\\nabla$-Einstein structure as a stationary point."],"supporting_citations":[{"why":"Supplies the key lemma that on a compact $\\nabla$-Einstein manifold there is a unique normalized $f$ with $V=\\theta^\\sharp-\\operatorname{grad} f$ parallel; the classification's dichotomy rests on it.","marker":"[35]"},{"why":"With [26], shows the data $(\\nabla,g,H,\\theta^\\sharp)$ form a generalized Ricci soliton, which combined with compactness yields a gradient soliton and the parallel vector field.","marker":"[33]"},{"why":"Provides the generalized Ricci flow framework used to identify the soliton structure and to prove gauge equivalence of the strong Sasaki-with-torsion flow.","marker":"[26]"},{"why":"Supplies the splitting argument for the $f$-constant subcase and identifies the transverse 4-dimensional geometry with that of Bismut–Hermite–Einstein 6-manifolds.","marker":"[6]"},{"why":"The Bismut-flat classification whose odd-dimensional analogue, Theorem 4.7, characterizes the flat cases of Theorem 5.16.","marker":"[47]"},{"why":"Constructs the Friedrich–Ivanov connection and derives the curvature and spinor identities underlying the $\\nabla$-Einstein condition.","marker":"[23]"},{"why":"Gives the Bianchi-identity criterion for metric connections with skew torsion used in Theorem 3.3, which feeds into the flat classification.","marker":"[34]"},{"why":"Introduces the Box equation that the transverse Kähler scalar curvature must satisfy in the non-flat compact case.","marker":"[27]"}],"fun_headline_variants":["Compact 5-D torsion Sasaki: only four geometries","All compact 5-D torsion Sasaki fall into four classes","Four classes pin down every compact 5-D torsion Sasaki","Compact 5-D torsion Sasaki: four exclusive types"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The classification rests on an imported lemma, quoted from a recent preprint, that every compact $\\nabla$-Einstein Sasaki-with-torsion manifold admits a unique normalized smooth function $f$ for which $V=\\theta^\\sharp-\\operatorname{grad} f$ is $\\nabla$-parallel; the $V\\neq 0$ versus $V=0$ split and the four-case structure of Theorem 5.16 depend on it, and the $f$-constant subcase further imports a splitting result from another paper.","fun_headline_variants_meta":{"raw":{"variants":["Compact 5-D torsion Sasaki: only four geometries","All compact 5-D torsion Sasaki fall into four classes","Four classes pin down every compact 5-D torsion Sasaki","Compact 5-D torsion Sasaki: four exclusive types"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001618,"raw_usage":{"total_tokens":6440,"prompt_tokens":950,"completion_tokens":5490,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":566,"completion_tokens_details":{"reasoning_tokens":5421}},"tokens_in":566,"tokens_out":5490,"duration_ms":44940,"temperature":1.0,"reasoning_tokens":5421,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T04:25:29.764289+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Look for a compact five-dimensional $\\nabla$-Einstein Sasaki-with-torsion manifold whose universal cover is not one of the four models in Theorem 5.16—for instance, a compact example with $V\\neq 0$ but non-flat $\\nabla$, or a $V=0$, $c\\neq 0$, $f$ non-constant example whose transverse Kähler scalar curvature is constant or fails $\\square \\tilde{s}^T = (\\tilde{s}^T)^2/2 - |\\widetilde{\\mathrm{Ric}}|^2$. Since the theorem is conditional, the sharpest test is to disprove the imported lemma: exhibit a compact $\\nabla$-Einstein Sasaki-with-torsion manifold on which no normalized $f$ makes $\\theta^\\sharp-\\operatorname{grad} f$ parallel.","supporting_citations":[{"cited_title":"The canonical symmetry reduction of string backgrounds","cited_arxiv_id":"2511.20773","evidence_quote":"Supplies the key lemma that on a compact $\\nabla$-Einstein manifold there is a unique normalized $f$ with $V=\\theta^\\sharp-\\operatorname{grad} f$ parallel; the classification's dichotomy rests on it."},{"cited_title":"Ivanov and A","cited_arxiv_id":null,"evidence_quote":"With [26], shows the data $(\\nabla,g,H,\\theta^\\sharp)$ form a generalized Ricci soliton, which combined with compactness yields a gradient soliton and the parallel vector field."},{"cited_title":"Garcia-Fernandez and J","cited_arxiv_id":null,"evidence_quote":"Provides the generalized Ricci flow framework used to identify the soliton structure and to prove gauge equivalence of the strong Sasaki-with-torsion flow."},{"cited_title":"Apostolov, G","cited_arxiv_id":null,"evidence_quote":"Supplies the splitting argument for the $f$-constant subcase and identifies the transverse 4-dimensional geometry with that of Bismut–Hermite–Einstein 6-manifolds."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The Bismut-flat classification whose odd-dimensional analogue, Theorem 4.7, characterizes the flat cases of Theorem 5.16."},{"cited_title":"Friedrich and S","cited_arxiv_id":null,"evidence_quote":"Constructs the Friedrich–Ivanov connection and derives the curvature and spinor identities underlying the $\\nabla$-Einstein condition."},{"cited_title":"Ivanov and N","cited_arxiv_id":null,"evidence_quote":"Gives the Bianchi-identity criterion for metric connections with skew torsion used in Theorem 3.3, which feeds into the flat classification."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the Box equation that the transverse Kähler scalar curvature must satisfy in the non-flat compact case."}],"review_version":1}