The DOI matches its Crossref record; 51 of the 51 references it cites were found in a registry or an index; none of the 13 sources its claims cite is checked for support yet.
The Cotton tensor in Riemannian spacetimes
Open accessAbstract
Recently, the study of three-dimensional spaces is becoming of great interest. In these dimensions the Cotton tensor is prominent as the substitute for the Weyl tensor. It is conformally invariant and its vanishing is equivalent to conformal flatness. However, the Cotton tensor arises in the context of the Bianchi identities and is present in any dimension n . We present a systematic derivation of the Cotton tensor. We perform its irreducible decomposition and determine its number of independent components as n ( n 2 − 4)/3 for the first time. Subsequently, we show its characteristic properties and perform a classification of the Cotton tensor in three dimensions. We investigate some solutions of Einstein's field equations in three dimensions and of the topologically massive gravity model of Deser, Jackiw and Templeton. For each class examples are given. Finally, we investigate the relation between the Cotton tensor and the energy–momentum in Einstein's theory and derive a conformally flat perfect fluid solution of Einstein's field equations in three dimensions.
Overview
CoChat Summary
TL;DR
The paper gives a systematic exterior-form derivation and irreducible decomposition of the Cotton tensor in Riemannian spacetimes, showing that in Riemannian space only its tensor piece survives with \(n(n^2-4)/3\) independent components. It then classifies the Cotton tensor in three dimensions, relates it to Einstein and topologically massive gravity field equations, and derives a conformally flat perfect-fluid solution.
Key findings
- The Cotton 2-form is derived from the Bianchi identities as \(C_\alpha := DL_\alpha\), with \(L_\alpha\) the Schouten 1-form; its trace and axial pieces vanish in Riemannian space, leaving only TENCOT with \(n(n^2-4)/3\) independent components — 5 in \(n=3\) and 16 in \(n=4\).
- In three dimensions the Cotton tensor is conformally invariant, and its vanishing is necessary and sufficient for conformal flatness; in more than three dimensions the vanishing of the Weyl 2-form is the relevant conformal-flatness condition.
- The 3D Cotton tensor is classified by its eigenvalues/Jordan form into classes I, I', D, II, III, N, and O, paralleling the Petrov classification; examples are given for all classes, including a 3D Reissner-Nordström analog, a Gödel solution, and DJT solutions.
- The Cotton tensor is related to matter via the differentiated Einstein equation, \(C_\alpha = (-1)^{n-1+\mathrm{ind}}\ell D\star[\Sigma_\alpha - \frac{1}{n-1}e_\alpha\rfloor(\Sigma_\beta\wedge\vartheta^\beta)]\); all vacuum Einstein solutions have vanishing Cotton 2-form, and in 4D the Bach tensor is recovered as a conserved 3-form.
Methods
- Theoretical derivation using exterior differential forms, Bianchi identities, irreducible decomposition with respect to the Lorentz group, and variational derivation from the 3D Chern-Simons action with Lagrange multipliers enforcing vanishing torsion and nonmetricity.
- Classification of the 3D Cotton tensor via the generalized eigenvalue problem \(C_{\alpha\beta}-\lambda g_{\alpha\beta}=0\), with Jordan normal forms and Segre notation; examples are checked by computer algebra using Reduce-Excalc.
- No empirical sample; the analysis is mathematical, with illustrative metric examples and a conformally flat static spherically symmetric perfect-fluid solution of Einstein’s field equations in 3D.
Limitations
- The paper states that it found no “sensible” solutions to the Einstein or DJT field equations of Class II or III; the examples for those classes are metrics whose Cotton tensor degenerates on surfaces/hypersurfaces, not field-equation solutions.
- Evident from the text: the work is purely theoretical/mathematical, with no observational data or empirical validation.
- Evident from the text: the Lorentzian classification can involve complex eigenvalues, and the paper notes this point “seems to have been overlooked” by some earlier authors.
Model-written summary. Check it against the paper before relying on it.
Integrity signals
Registry
Cites
Statistics
Verification
Claims the paper makes
0 Valid · 5 Not valid · 11 Not checked · 63 Own results and interpretations
A result the paper states — it cites no source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“From the construction it is clear that the Weyl 2-form is totally traceless, i. e., e α ⌋Weyl αβ = -e β ⌋Weyl αβ = 0 , e α ⌋e β ⌋Weyl αβ = 0 .(25)”
§ The Cotton tensor in Riemannian spacetimes > II. BIANCHI IDENTITIES AND THE IRREDUCIBLE DECOMPOSITION OF THE CURVATURE
boosted- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
A result the paper states — it cites no source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The Bianchi identities imply full integrability of this system.”
§ The Cotton tensor in Riemannian spacetimes > C α as a variational derivative
plainstates a cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: unclear — It is unclear why the paper says it holds.
A result the paper states — it cites no source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“for these spaces the Weyl tensor is always zero”
§ The Cotton tensor in Riemannian spacetimes > I. INTRODUCTION
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
A result the paper states — it cites no source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Hence, the vanishing of the Weyl 2-form is also the sufficient condition for conformal flatness.”
§ The Cotton tensor in Riemannian spacetimes > n = 2
plainstates a cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
A result the paper states — it cites no source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“However, the Cotton tensor is only conformally invariant in three dimensions.”
- Also stated in: § IV. CONFORMAL CORRESPONDENCE · plain
“Thus, in n = 3, where the Weyl 2-form vanishes, the Cotton 2-form becomes conformally invariant.”
§ The Cotton tensor in Riemannian spacetimes > I. INTRODUCTION
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- Also stated in: § IV. CONFORMAL CORRESPONDENCE · plain
48 more claims
Create a free account to see all 79 claims and to verify the 13 references still open.
- Probably not a substantial claim. 10
- It may state two separate claims as one. 16
- The quote only partly states this claim. 3
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
Each contradiction between two passages listed here was confirmed by an independent check.
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ The Cotton tensor in Riemannian spacetimes > B. Lorentzian signature
“types D and II with only one independent eigenvalue λ 1 = λ 2 = -2λ 3 are always real.”
§ The Cotton tensor in Riemannian spacetimes > B. Lorentzian signature
“λ 1 = λ 2 = - 1 2 λ 3 = µ 3 3 . (122”
Citations
Cites
The large $N$ limit of superconformal field theories and supergravity
Maldacena, J. (1998) · Advances in Theoretical and Mathematical Physics · doi:10.4310/atmp.1998.v2.n2.a1 · Cited by 14,524
Anti de Sitter space and holography
Witten, E. (1998) · Advances in Theoretical and Mathematical Physics · doi:10.4310/atmp.1998.v2.n2.a2 · Cited by 13,070
The Large N Limit of Superconformal Field Theories and SupergravityPreprint
Maldacena, J. (1998) · Cited by 11,886
Gauge theory correlators from non-critical string theory
Gubser, S. S., Klebanov, I. R., & Polyakov, A. M. (1998) · Physics Letters B · doi:10.1016/s0370-2693(98)00377-3 · Cited by 9,307
Gauge Theory Correlators from Non-Critical String Theory
Gubser, S. S., Klebanov, I. R., & Polyakov, A. M. (1998) · arXiv · doi:10.48550/arxiv.hep-th/9802109 · Cited by 9,307
The Large-N Limit of Superconformal Field Theories and Supergravity
Maldacena, J. (1999) · International Journal of Theoretical Physics · doi:10.1023/a:1026654312961 · Cited by 6,704
Black hole in three-dimensional spacetime
Bañados, M., Teitelboim, C., & Zanelli, J. (1992) · Physical Review Letters · doi:10.1103/physrevlett.69.1849 · Cited by 3,831
Topologically massive gauge theories
Deser, S., Jackiw, R., & Templeton, S. (1982) · Annals of Physics · doi:10.1016/0003-4916(82)90164-6 · Cited by 2,740
Eisenhart, L. P. (1950) · Princeton University Press eBooks · doi:10.1515/9781400884216 · Cited by 2,606
Republication of: The dynamics of general relativity
Arnowitt, R., Deser, S., & Misner, C. W. (2008) · General Relativity and Gravitation · doi:10.1007/s10714-008-0661-1 · Cited by 2,284
Research works printed but not linked: 3
- Polchinski J, 1998, String Theory, 2 vol
- Tsantilis E, 1996, Relativity and Scientific Computing
- Misner C W, 1973, Gravitation
Cited by
Items 101–152 of 152
Local gravity theories in conformal superspace
Gomes, H. D. A. (2018) · Annals of Physics · doi:10.1016/j.aop.2018.05.014 · Cited by 3
Cotton Tensor in Riemannian Spacetimes
García-Díaz, A. A. (2017) · Cambridge University Press eBooks · doi:10.1017/9781316556566.021 · Cited by 3
Quadratic Curvature Gravity with Second Order Trace and Massive Gravity Models in Three DimensionsPreprint
Baykal, A. (n.d.) · Cited by 3
Gauge symmetry and partially Lagrangian systems
Lyakhovich, S. L., & Sinelnikov, N. A. (2025) · Annals of Physics · doi:10.1016/j.aop.2025.170236 · Cited by 2
Towards a general solution of the Hamiltonian constraints of General Relativity
Tiemblo, A., & Tresguerres, R. (2006) · General Relativity and Gravitation · doi:10.1007/s10714-006-0361-7 · Cited by 2
A derivation of Weyl-Lanczos equations
Baykal, A., & Ünal, B. (2018) · The European Physical Journal Plus · doi:10.1140/epjp/i2018-11995-x · Cited by 2
Exact anyon black hole solutions
Xu, W. (2018) · The European Physical Journal C · doi:10.1140/epjc/s10052-018-6348-1 · Cited by 2
Carroll-Cotton Tensors and Gravitational Radiation at Null InfinityPreprint
Fiorucci, A., Pekar, S., Petropoulos, P. M., & Vilatte, M. (2026) · arXiv (Cornell University) · doi:10.48550/arxiv.2608.28741 · Cited by 2
Cotton Solitons on Three Dimensional Almost $\alpha$-paracosymplectic Manifolds
Erken, İ. K., Savur, B., & Özkan, M. (2023) · International Electronic Journal of Geometry · doi:10.36890/iejg.1316716 · Cited by 1
Poddar, R. (2024) · Letters in Mathematical Physics · doi:10.1007/s11005-024-01840-w · Cited by 1
The Anderson-Friedman Absolute Objects Program: Several Successes, One Difficulty
Pitts, J. B. (2006) · PhilSci-Archive (University of Pittsburgh) · Cited by 1
Twisted entropy functional for three-manifolds and flows induced on the boundary
Cartas-Fuentevilla, R. (2014) · International Journal of Geometric Methods in Modern Physics · doi:10.1142/s0219887814500339 · Cited by 1
Brane structure from a scalar field in general covariant Horava-Lifshitz gravity
Bazeia, D., Brito, F. A., & Costa, F. G. (2015) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.91.046001 · Cited by 1
Metamaterial acoustics on the Poincaré disk
Tung, M. M. (2020) · Mathematical Methods in the Applied Sciences · doi:10.1002/mma.6912 · Cited by 1
Dilaton–Inflaton Friedmann–Robertson–Walker Cosmologies
García-Díaz, A. A. (2017) · Cambridge University Press eBooks · doi:10.1017/9781316556566.011 · Cited by 1
Cosmic Time and the Initial State of the Universe
Soo, C. (2023) · Universe · doi:10.3390/universe9120489 · Cited by 1
TOPOLOGICAL GRAVITY, KALUZA-KLEIN REDUCTION, AND THE KINK
Iorio, A. (2006) · The Ninth Marcel Grossmann Meeting · doi:10.1142/9789812704030_0287 · Cited by 1
Supersymmetry, black holes and holography in three dimensions
Alkaç, G. (2017) · University of Groningen research database (University of Groningen / Centre for Information Technology) · Cited by 1
Absolute Objects, Counterexamples and General Covariance
Pitts, J. B. (2007) · PhilSci-Archive (University of Pittsburgh)
Schwarzschild and Kerr Solutions
(2009) · Birkhäuser Basel eBooks · doi:10.1007/978-3-7643-9971-9_14
Mielke, E. W. (2017) · Mathematical physics studies · doi:10.1007/978-3-319-29734-7_10
Gravitational systems in asymptotically anti-de Sitter space-times
Martinon, G. (2017) · theses.fr (ABES) · doi:10.70675/14963eedz09fdz4ffbz9d78z5411bd436c57
García-Díaz, A. A. (2017) · Cambridge University Press eBooks · doi:10.1017/9781316556566.017
Static Perfect Fluid Stars with Λ
García-Díaz, A. A. (2017) · Cambridge University Press eBooks · doi:10.1017/9781316556566.007
García-Díaz, A. A. (2017) · Cambridge University Press eBooks · doi:10.1017/9781316556566.019
A Shortcut to (2+1) Cyclic Symmetric Stationary Solutions
García-Díaz, A. A. (2017) · Cambridge University Press eBooks · doi:10.1017/9781316556566.005
Black Holes Coupled To Nonlinear Electrodynamics
García-Díaz, A. A. (2017) · Cambridge University Press eBooks · doi:10.1017/9781316556566.013
García-Díaz, A. A. (2017) · Cambridge University Press eBooks · doi:10.1017/9781316556566.022
García-Díaz, A. A. (2017) · Cambridge University Press eBooks · doi:10.1017/9781316556566.004
Friedmann–Robertson–Walker Cosmologies
García-Díaz, A. A. (2017) · Cambridge University Press eBooks · doi:10.1017/9781316556566.010
Stationary Circularly Symmetric Perfect Fluids with Λ
García-Díaz, A. A. (2017) · Cambridge University Press eBooks · doi:10.1017/9781316556566.009
Low-Energy (2+1) String Gravity
García-Díaz, A. A. (2017) · Cambridge University Press eBooks · doi:10.1017/9781316556566.016
García-Díaz, A. A. (2017) · Cambridge University Press eBooks · doi:10.1017/9781316556566.002
García-Díaz, A. A. (2017) · Cambridge University Press eBooks · doi:10.1017/9781316556566.001
Scalar Field Non-Minimally Coupled to (2+1) Gravity
García-Díaz, A. A. (2017) · Cambridge University Press eBooks · doi:10.1017/9781316556566.015
Bianchi-Type (BT) Spacetimes in TMG; Petrov Type D
García-Díaz, A. A. (2017) · Cambridge University Press eBooks · doi:10.1017/9781316556566.018
Dilaton Field Minimally Coupled to (2 + 1) Gravity
García-Díaz, A. A. (2017) · Cambridge University Press eBooks · doi:10.1017/9781316556566.014
García-Díaz, A. A. (2017) · Cambridge University Press eBooks · doi:10.1017/9781316556566.003
García-Díaz, A. A. (2017) · Cambridge University Press eBooks · doi:10.1017/9781316556566.020
García-Díaz, A. A. (2017) · Cambridge University Press eBooks · doi:10.1017/9781316556566.008
Perfect Fluid Static Stars; Cosmological Solutions
García-Díaz, A. A. (2017) · Cambridge University Press eBooks · doi:10.1017/9781316556566.006
García-Díaz, A. A. (2017) · Cambridge University Press eBooks · doi:10.1017/9781316556566.012
Kroon, J. A. V. (2022) · Cambridge University Press eBooks · doi:10.1017/9781009291347.027
Extremal Rotating BTZ Black Holes Cannot Be Dressed in (anti-)Self-Dual Maxwell Field
Maeda, H., & Podolský, J. (2024) · Progress of Theoretical and Experimental Physics · doi:10.1093/ptep/ptae160
The exterior calculus of quadratic gravity
Arık, M., Baykal, A., Dereli, T., & Tanrıverdi, T. (2025) · International Journal of Modern Physics D · doi:10.1142/s0218271825500282
Cosmic transit in phantom massive Chern–Simons models: confrontations with astronomical observations
El‐Nabulsi, R. A., & Anukool, W. (2025) · Modern Physics Letters A · doi:10.1142/s0217732325500853
Interpreting gravitational fields of topologically massive gravity using geodesic deviation
Papajčík, M., & Podolský, J. (2025) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.111.124003
Physical constraints from holographic duality
Cheamsawat, K. (2020) · Spiral (Imperial College London) · doi:10.25560/88226
Beckering Vinckers, U. K. (2021) · Open University of Cape Town (University of Cape Town)
Algebraic classification of 2+1 geometries
Papajčík, M., & Podolský, J. (2026) · Journal of Physics Conference Series · doi:10.1088/1742-6596/3177/1/012021
Gauge Theory of Gravity and the AdS/CFT Correspondence
Fukuyama, T. (2026) · Progress of Theoretical and Experimental Physics · doi:10.1093/ptep/ptag182
On AdS4 Holography - Towards applications to 2+1 dimensional graphene-like systems
MATRECANO, R. (2022) · PORTO@Iris
Metadata
- Authors
- García, A. A., Hehl, F. W., Heinicke, C., & Macías, A.
- Affiliations
- Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional (MX)
- University of Cologne (DE)
- Universidad Autónoma Metropolitana (MX)
- Venue
- Classical and Quantum Gravity
- Publisher
- IOP Publishing
- Published
- 2004-01-22
- Volume
- 21
- Issue
- 4
- Pages
- 1099-1118
- ISSN
- 0264-9381 (ISSN-L)
- 1361-6382
- OpenAlex
- W2109289785
- Cited by
- 148
- References
- 50
- Citations by year
- 2026: 4 · 2025: 5 · 2024: 8 · 2023: 4 · 2022: 4 · 2021: 10 · 2020: 5 · 2019: 2 · 2018: 4 · 2017: 29 · 2016: 5 · 2015: 3 · 2014: 10 · 2013: 7 · 2012: 5
- FWCI
- 2.67
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Classical and Quantum Gravity (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Funders
- Deutsche Forschungsgemeinschaft (528/20-1)
- Consejo Nacional de Ciencia y Tecnología
- Topics
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Physical Sciences › Mathematics › Geometry and Topology › Advanced Differential Geometry Research
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Cosmology and Gravitation Theories
- Keywords
- Cotton tensor, 2+1 dimensions, conformal flatness, conformal invariance, irreducible decomposition, Bianchi identities, Einstein field equations, topologically massive gravity, energy-momentum tensor, perfect fluid solutions
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
A language model drew the claims from the paper on 2026-11-14, under version 3 of the claim definition (model deepseek-v4.1-flash, prompt extract-v3).
A second model labelled each claim on 2026-11-14 (model jev-1.13-20260917, definition revision gold-guideline-v0.2).
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
Data sources
Bibliographic data: Semantic Scholar, OpenAlex, Crossref.
Contains information from the Semantic Scholar Academic Graph Datasets, made available under the ODC Attribution License (ODC-BY 1.0).
Semantic Scholar’s data is described in: Kinney, R., et al. (2023). The Semantic Scholar Open Data Platform. arXiv:2301.10140.
