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arxiv: 2606.02611 · v1 · pith:4IZV3Z4Inew · submitted 2026-05-25 · 🌀 gr-qc · math.DG

Spacetime triple wormhole

Vincent Herr , Aim\'e Fournier , Andrew J.S Hamilton This is my paper

Pith reviewed 2026-06-29 21:14 UTC · model grok-4.3

classification 🌀 gr-qc math.DG
keywords spacetime wormholetriple wormholeDupin hypercyclideEinstein field equationsmulti-neck solutionsynchronous coordinatesnegative energy densityasymptotically flat manifold
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The pith

A Dupin hypercyclide metric tensor provides an exact solution to Einstein's field equations for a triple wormhole spacetime.

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

The paper constructs a spacetime containing three wormhole necks by spherically inverting a 3-torus to form a Dupin hypercyclide and embedding it in synchronous coordinates. This yields a simple metric tensor that the authors assert satisfies Einstein's field equations exactly, producing diagonal Ricci and stress-energy tensors with a Riemann tensor having only six nonzero entries. The construction answers the question from Einstein and Rosen in 1935 about the existence of multi-neck wormhole solutions by providing a non-vacuum, intra-universe example without thin shells or spherical symmetry. The solution features negative energy density to keep the necks open, although some geodesics pass through regions of only positive energy density. The manifold is unbounded, asymptotically flat, and admits global isothermal coordinates.

Core claim

Asserting this metric tensor as an exact solution of Einstein's field equations in global coordinates generates diagonal Ricci and stress-energy tensors, and a Riemann curvature tensor with only six nonzero entries. This non-vacuum solution answers affirmatively the question posed by Einstein and Rosen (1935) of whether or not multi-neck solutions exist.

What carries the argument

The Dupin hypercyclide metric tensor obtained by spherical inversion of an equal-radii 3-torus and placed in a synchronous reference frame.

Load-bearing premise

The constructed Dupin hypercyclide metric, when placed in synchronous coordinates, satisfies Einstein's field equations exactly without additional assumptions, coordinate patching, or post-hoc adjustments.

What would settle it

Substituting the given metric tensor components into the Einstein tensor calculation and checking whether it equals the claimed diagonal stress-energy tensor with the specified nonzero Riemann components.

Figures

Figures reproduced from arXiv: 2606.02611 by Aim\'e Fournier, Andrew J.S Hamilton, Vincent Herr.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: shows steps in constructing a PDC, an inverted 2-torus that meets the spatial requirements of Definition 1. The orange coordinate circles cannot be continuously shrunk to a point; the gray area is non-simply connected. Identifying corresponding points of parallel 6 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
read the original abstract

We describe a multi-neck spacetime wormhole with a simple metric tensor and a simple injective map without coordinate patching. An intra-universe, non-thin-shell, non-spherically-symmetric 3-neck spacetime wormhole is geometrically constructed by spherically inverting a 3-torus. We place the resulting Dupin hypercyclide in a synchronous reference frame. The three necks are arranged around a central point and satisfy topological and geometric spacetime wormhole definitions. Asserting this metric tensor as an exact solution of Einstein's field equations in global coordinates generates diagonal Ricci and stress-energy tensors, and a Riemann curvature tensor with only six nonzero entries. The local inertial frame at every point of the coordinate system is comoving with the triple wormhole. This non-vacuum solution answers affirmatively the question posed by Einstein and Rosen (1935) of whether or not multi-neck solutions exist. The wormhole solution contains negative energy density as is expected to hold the necks open; however, geodesic paths through each neck exist which encounter only positive energy density. The spatial manifold is a trivariate Dupin hypercyclide. The spherically inverted equal-radii 3-torus is unbounded, asymptotically flat and admits a global isothermal coordinate system that further simplifies the curvature tensors.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript constructs an intra-universe triple wormhole by spherically inverting an equal-radii 3-torus to produce a Dupin hypercyclide, places the resulting spatial manifold in synchronous coordinates, and asserts that the resulting metric is an exact non-vacuum solution of Einstein's equations. This solution is claimed to yield diagonal Ricci and stress-energy tensors together with a Riemann tensor possessing only six nonzero components, to be asymptotically flat, and to admit geodesic paths through each neck that encounter only positive energy density. The construction is presented as answering the 1935 Einstein-Rosen question on the existence of multi-neck solutions.

Significance. If the central assertion were verified by explicit curvature calculations, the result would supply a concrete, globally coordinated example of a non-spherically-symmetric, non-thin-shell multi-neck wormhole with controlled energy-density sign changes. Such an explicit metric could serve as a test case for traversability studies and for examining the relationship between topology and curvature in general relativity.

major comments (2)
  1. [Abstract] Abstract (paragraph beginning 'Asserting this metric tensor...'): the central claim that the Dupin hypercyclide metric in synchronous coordinates satisfies Einstein's equations exactly, producing diagonal Ricci and stress-energy tensors and a Riemann tensor with only six nonzero entries, is stated without any explicit metric components, Christoffel symbols, Riemann components, or contractions. This assertion is load-bearing for the entire result; its verification is required before the solution can be accepted as exact.
  2. [Abstract] Abstract and subsequent text: the manuscript asserts that the local inertial frame is comoving with the triple wormhole and that the spatial manifold is asymptotically flat, yet supplies no coordinate expressions or asymptotic analysis that would confirm these properties or the claimed global isothermal coordinate system.
minor comments (1)
  1. The reference to Einstein and Rosen (1935) is appropriate but should be supplemented by a brief statement of the precise question those authors posed, to clarify how the present construction addresses it.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed review and for identifying the need for explicit verification of the central claims. We agree that the assertions regarding the exact solution status require supporting calculations and will revise the manuscript accordingly to include them.

read point-by-point responses

  1. Referee: [Abstract] Abstract (paragraph beginning 'Asserting this metric tensor...'): the central claim that the Dupin hypercyclide metric in synchronous coordinates satisfies Einstein's equations exactly, producing diagonal Ricci and stress-energy tensors and a Riemann tensor with only six nonzero entries, is stated without any explicit metric components, Christoffel symbols, Riemann components, or contractions. This assertion is load-bearing for the entire result; its verification is required before the solution can be accepted as exact.

    Authors: We agree that the manuscript as submitted does not provide the explicit curvature calculations needed to verify the claim. In the revised version we will insert the metric components in synchronous coordinates, the full set of nonzero Christoffel symbols, the six nonzero Riemann components, and the resulting diagonal Ricci and stress-energy tensors obtained by contraction. These additions will allow direct confirmation that the metric satisfies Einstein's equations exactly. revision: yes

  2. Referee: [Abstract] Abstract and subsequent text: the manuscript asserts that the local inertial frame is comoving with the triple wormhole and that the spatial manifold is asymptotically flat, yet supplies no coordinate expressions or asymptotic analysis that would confirm these properties or the claimed global isothermal coordinate system.

    Authors: We acknowledge that coordinate expressions and asymptotic analysis are absent. The revision will add the explicit coordinate transformation to the comoving synchronous frame, the asymptotic expansion of the metric at large distances demonstrating flatness, and the explicit form of the global isothermal coordinates together with the simplified curvature tensors that result. revision: yes

Circularity Check

0 steps flagged

No circularity: geometric construction asserted as exact solution without reduction to inputs.

full rationale

The paper constructs the metric via spherical inversion of a 3-torus to obtain a Dupin hypercyclide, places it in synchronous coordinates, and asserts that this yields an exact solution of Einstein's equations with diagonal Ricci and stress-energy tensors plus a Riemann tensor having only six nonzero entries. No quoted equations, self-citations, fitted parameters, or ansatzes reduce this assertion to the construction by definition or force the outcome statistically. The central claim is presented as an independent geometric result rather than a renaming, self-definition, or load-bearing self-citation, making the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the construction is described at the level of geometric operations and an assertion of exact solvability.

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Works this paper leans on

126 extracted references · 48 canonical work pages · 3 internal anchors

  1. [1]

    Richard Arnowitt and Stanley Deser and CHarles W. Misner. Republication of: The dynamics of general relativity. General relativity and gravitation. 2008. doi:10.1007/s10714-008-0661-1

    work page doi:10.1007/s10714-008-0661-1 2008

  2. [2]

    doi:https://doi-org.colorado.idm.oclc.org/10.1007/b138899 , isbn=

    Computer graphics and geometric modeling , author=. doi:https://doi-org.colorado.idm.oclc.org/10.1007/b138899 , isbn=

    work page doi:10.1007/b138899

  3. [3]

    Anderson and Dieter R

    Paul R. Anderson and Dieter R. Brill. Gravitational geons revisited. Physical Review. 1997

    1997

  4. [4]

    1962 , edition=

    Vectors, tensors, and the basic equations of fluid mechanics , author=. 1962 , edition=

    1962

  5. [5]

    Unit-lapse versions of the Kerr spacetime , journal =

    Joshua Baines and Thomas Berry and Alex Simpson and Matt Visser. Unit-lapse versions of the Kerr spacetime , journal =. 2020. doi:10.1088/1361-6382/abd071

    work page doi:10.1088/1361-6382/abd071 2020

  6. [6]

    Banchoff

    Thomas F. Banchoff. The spherical two-piece property and tight surfaces in spheres. Journal of Differential Geometry. 1970

    1970

  7. [7]

    Banchoff

    Thomas F. Banchoff. The flat torus in the 3-sphere. 1995 remake of 1968 original

    1995

  8. [8]

    Banchoff

    Thomas F. Banchoff. Geometry of the Hopf mapping and Pinkall's tori of given conformal type. Computers in Algebra. 1988

    1988

  9. [10]

    Scalar fields, energy conditions and traversable wormholes

    Carlos Barceló and Matt Visser. Scalar fields, energy conditions and traversable wormholes. Classical and Quantum Gravity. 2000. doi:10.1088/0264-9381/17/18/318

    work page doi:10.1088/0264-9381/17/18/318 2000

  10. [11]

    Twilight for the energy conditions?

    Carlos Barceló and Matt Visser. Twilight for the energy conditions?. Physics Letters B. 1999. doi:10.1016/S0370-2693(99)01117-X

    work page doi:10.1016/s0370-2693(99)01117-x 1999

  11. [12]

    On the Parameterization of Rational Ringed Surfaces and Rational Canal Surfaces

    Bohumír Bastl and Bert Jüttler and Miroslav Lávička and Tino Schulz and Zbyněk Šír. On the Parameterization of Rational Ringed Surfaces and Rational Canal Surfaces. Mathematics in Computer Science. 2014

    2014

  12. [13]

    2010 , isbn=

    Numerical relativity; Solving Einstein's equations on the computer , author=. 2010 , isbn=

    2010

  13. [14]

    Exact arrangements on tori and Dupin cyclides

    Eric Berberich and Michael Kerber. Exact arrangements on tori and Dupin cyclides. Mathematics in Computer Science. 2014

    2014

  14. [15]

    Crittenden

    Richard Lawrence Bishop and Richard J. Crittenden. Geometry of manifolds. 1968

    1968

  15. [16]

    Dieter R. Brill. On the positive definite mass of the Bondi-Weber-Wheeler time-symmetric gravitational waves. Annals of Physics. 1959. doi:https://doi.org/10.1016/0003-4916(59)90055-7

    work page doi:10.1016/0003-4916(59)90055-7 1959

  16. [17]

    2012 , publisher=

    Charles Dupin (1784-1873) and his influence on France; The contributions of a mathematician, educator, engineer, and statesman , author=. 2012 , publisher=

    2012

  17. [18]

    Burstall and Neil M

    Francis E. Burstall and Neil M. Donaldson and Franz Pedit and Ulrich Pinkall. Isothermic submanifolds of symmetric R-spaces. Mathematics and Statistics Department Faculty Publication Series. 2011. doi:http://dx.doi.org/10.1515/crelle.2011.075

    work page doi:10.1515/crelle.2011.075 2011

  18. [19]

    Annales scientifiques de l’É.N.S

    Sur la structure des groupes infinis de transformation , author=. Annales scientifiques de l’É.N.S. 3e série. doi:http://dx.doi.org/10.24033/asens.538

    work page doi:10.24033/asens.538

  19. [20]

    Riemann Geometry in an Orthogonal Frame, From lectures delivered by \'E lie Cartan at the Sorbonne in 1926-27

    Cartan, \'E lie. Riemann Geometry in an Orthogonal Frame, From lectures delivered by \'E lie Cartan at the Sorbonne in 1926-27. 2001. doi:10.1142/4808

    work page doi:10.1142/4808 1926

  20. [21]

    H. B. G. Casimir. On the attraction between two perfectly conducting plates. Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen. 1948

    1948

  21. [22]

    On the cyclide

    Arthur Cayley. On the cyclide. Quarterly journal of pure and applied mathematics. 1873

  22. [23]

    Cecil and Shiing-Shen Chern

    Thomas E. Cecil and Shiing-Shen Chern. Dupin submanifolds in Lie sphere geometry (updated version). Mathematics Department Faculty Scholarship. 2020

    2020

  23. [24]

    and Ryan, Patrick J

    Cecil, Thomas E. and Ryan, Patrick J. Tight spherical embeddings. Global Differential Geometry and Global Analysis. 1979

    1979

  24. [25]

    Cecil and Patrick J

    Thomas E. Cecil and Patrick J. Ryan. Conformal geometry and the cyclides of Dupin. Canadian Journal of Mathematics. 1980

    1980

  25. [26]

    2015 , publisher=

    Geometry of hypersurfaces , author=. 2015 , publisher=. doi:10.1007/978-1-4939-3246-7 , isbn=

    work page doi:10.1007/978-1-4939-3246-7 2015

  26. [27]

    Chandru and D

    V. Chandru and D. Dutta and C.M. Hoffman. On the geometry of Dupin cyclides. The Visual Computer. 1989

    1989

  27. [28]

    Shape from metric

    Albert Chern and Felix Knoppel and Ulrich Pinkall and Peter Shroder. Shape from metric. ACM Trans. Graph. 2018. doi:https://doi.org/10.1145/3197517.3201276

    work page doi:10.1145/3197517.3201276 2018

  28. [29]

    Finding Conformal and Isometric Immersions of Surfaces

    Albert Chern and Felix Knoppel and Franz Pedit and Ulrich Pinkall and Peter Shroder. Finding conformal and isometric immersions of surfaces. arXiv:1901.09432v1 [math.DG]. 2019. doi:https://arxiv.org/abs/1901.09432

    work page internal anchor Pith review Pith/arXiv arXiv 1901

  29. [30]

    2003 , publisher=

    General relativity and the Einstein equations , author=. 2003 , publisher=

    2003

  30. [31]

    Journal de Mathématiques Pures et Appliquées , volume =

    On the recovery of a surface with prescribed first and second fundamental forms , author =. Journal de Mathématiques Pures et Appliquées , volume =. 2002 , issn =. doi:https://doi.org/10.1016/S0021-7824(01)01236-3 , url =

    work page doi:10.1016/s0021-7824(01)01236-3 2002

  31. [32]

    Ciarlet and Cristinel Mardare

    Philippe G. Ciarlet and Cristinel Mardare. Recovery of a manifold with boundary and its continuity as a function of its metric tensor. Journal de Mathématiques Pures et Appliquées. 2004. doi:http://dx.doi.org/10.1016/j.matpur.2004.01.004

    work page doi:10.1016/j.matpur.2004.01.004 2004

  32. [33]

    2005 , publisher=

    An introduction to differential geometry with applications to elasticity , author=. 2005 , publisher=

    2005

  33. [34]

    Preliminary Sketch of Biquaternions

    William Clifford. Preliminary Sketch of Biquaternions. Proceedings of the London Mathematical Society , volume =. 1873 , month =. doi:10.1112/plms/s1-4.1.381 , url =

    work page doi:10.1112/plms/s1-4.1.381

  34. [35]

    Ron Cowen. Space. Time. Entanglement. Nature. 2015

    2015

  35. [36]

    Values of Tvµ and the Christoffel symbols for a line element of considerable generality

    Herbert Dingle. Values of Tvµ and the Christoffel symbols for a line element of considerable generality. Proceedings of the National Academy of Sciences of the USA. 1933

    1933

  36. [37]

    1976 , edition =

    Differential geometry of curves & surfaces , author =. 1976 , edition =

    1976

  37. [38]

    Drost (communicated by Prof

    J. Drost (communicated by Prof. H. A. Lortentz. The field of a single centre in Einstein's theory of gravitation and the motion of a particle in that field. Proceedings of the Royal Netherlands Academy of Arts and Sciences. 1917

    1917

  38. [39]

    1822 , publisher=

    Applications de Geometrie et de Mechanique , author=. 1822 , publisher=

  39. [40]

    On the relativity principle and the conclusions drawn from it

    Albert Einstein. On the relativity principle and the conclusions drawn from it. Jahrbuch der Radioaktivität und Elektronik. 1907

    1907

  40. [41]

    On the Influence of Gravitation on the Propagation of Light

    Albert Einstein. On the Influence of Gravitation on the Propagation of Light. Annalen der Physik. 1911

    1911

  41. [42]

    The field equations of gravitation

    Albert Einstein. The field equations of gravitation. Annalen der Physik. 1915

    1915

  42. [43]

    Die Grundlage der allgemeinen Relativitätstheorie

    Albert Einstein. Die Grundlage der allgemeinen Relativitätstheorie. Annalen der Physik. 1911. doi:10.1002/andp.19163540702

    work page doi:10.1002/andp.19163540702 1911

  43. [44]

    Can quantum-mechanical description of physical reality be considered complete?

    Albert Einstein and Boris Podolsky and Nathan Rosen. Can quantum-mechanical description of physical reality be considered complete?. Physical Review. 1935. doi:http://dx.doi.org/10.1103/PhysRev.47.777

    work page doi:10.1103/physrev.47.777 1935

  44. [45]

    The particle problem in the general theory of relativity

    Albert Einstein and Nathan Rosen. The particle problem in the general theory of relativity. Physical Review. 1935. doi:http://dx.doi.org/10.1103/PhysRev.48.73

    work page doi:10.1103/physrev.48.73 1935

  45. [46]

    1909 , publisher=

    A treatise on the differential geometry of curves and surfaces , author=. 1909 , publisher=

    1909

  46. [47]

    Homer G. Ellis. Ether flow through a drainhole: A particle model in general relativity. Journal of Mathematical Physics. 1973

    1973

  47. [48]

    Lectures on minimal surfaces in R3

    Yu Fang. Lectures on minimal surfaces in R3. Proceedings of the Center for Mathematics and its Applications. 1996

    1996

  48. [49]

    CYCLIDE DE DUPIN, Dupin’s Cyclide, dupinsche Zyklide

    Robert Ferréol and Jacques Mandonnet. CYCLIDE DE DUPIN, Dupin’s Cyclide, dupinsche Zyklide. 2017

    2017

  49. [50]

    Fischer and Jerrold E

    Arthur E. Fischer and Jerrold E. Marsden. Linearization stability of the Einstein equations. Bulletin of the American Mathematical Society. 1973

    1973

  50. [51]

    Fischer and Jerrold E

    Arthur E. Fischer and Jerrold E. Marsden. Beiträge zur Einsteinschen Gravitationstheorie (Comments on Einstein's Theory of Gravity). Physikalische Zeitschrift. 1916

    1916

  51. [52]

    Danial Forghani and S

    S. Danial Forghani and S. Habib Mazharimousavi and Mustafa Halilsoy. Asymmetric thin-shell wormholes. The European Physical Journal C. 2018. doi:https://doi.org/10.1140/epjc/s10052-018-5776-2

    work page doi:10.1140/epjc/s10052-018-5776-2 2018

  52. [53]

    1912 , publisher=

    Lectures on the differential geometry of curves and surfaces , author=. 1912 , publisher=

    1912

  53. [54]

    personal correspondence and DupinParabolic3Cyclide.nb 12/30/2020

    Aimé Fournier. personal correspondence and DupinParabolic3Cyclide.nb 12/30/2020. 2020

    2020

  54. [55]

    personal correspondence 1/1/2021

    Aimé Fournier. personal correspondence 1/1/2021. 2021

    2021

  55. [56]

    personal correspondence and expanded DupinParabolic3Cyclide.nb 1/5/2021

    Aimé Fournier. personal correspondence and expanded DupinParabolic3Cyclide.nb 1/5/2021. 2021

    2021

  56. [57]

    personal correspondence and minor revisions of DupinParabolic3Cyclide.nb 1/9/2021

    Aimé Fournier. personal correspondence and minor revisions of DupinParabolic3Cyclide.nb 1/9/2021. 2021

    2021

  57. [58]

    personal correspondence 2/5/2021 with comments on my Whiteboard-20210202-PDE-embedding-shrinking.jpg

    Aimé Fournier. personal correspondence 2/5/2021 with comments on my Whiteboard-20210202-PDE-embedding-shrinking.jpg. 2021

    2021

  58. [59]

    Fuller and John A

    Robert W. Fuller and John A. Wheeler. Causality and multiply connected space-time. Physical Review. 1962

    1962

  59. [60]

    On Conformal Representation, translated by Herbert P

    Carl Friedrich Gauss. On Conformal Representation, translated by Herbert P. Evans. A source book in mathematics. 1822

  60. [61]

    2021 , publisher=

    General relativity, black holes, and cosmology , author=. 2021 , publisher=

    2021

  61. [62]

    personal correspondence and tetrad.nb 12/26/2020

    Andrew Hamilton. personal correspondence and tetrad.nb 12/26/2020. 2020

    2020

  62. [63]

    personal correspondence 12/31/2020

    Andrew Hamilton. personal correspondence 12/31/2020. 2020

    2020

  63. [64]

    personal correspondence and Dupin4D.nb 1/30/2021

    Andrew Hamilton. personal correspondence and Dupin4D.nb 1/30/2021. 2021

    2021

  64. [65]

    personal correspondence 2/7/2021

    Andrew Hamilton. personal correspondence 2/7/2021. 2021

    2021

  65. [66]

    Stephen Hawking and W. Israel. General relativity: An Einstein centenary survey. 2010

    2010

  66. [67]

    300 Years of gravitation

    Stephen Hawking and Israel. 300 Years of gravitation. ?. 19__

  67. [68]

    A triple-micro-wormhole particle model , year =

    Vincent Herr. A triple-micro-wormhole particle model , year =

  68. [69]

    Physics Review D , volume =

    Geometric structure of the generic static traversable wormhole throat , author =. Physics Review D , volume =. 1997 , month =. doi:10.1103/PhysRevD.56.4745 , url =

    work page doi:10.1103/physrevd.56.4745 1997

  69. [70]

    Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche

    Hopf. Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche. Mathematische Annalen. 1931

    1931

  70. [71]

    W. Israel. Singular hypersurfaces and thin shells in general relativity. Il Nuovo Cimento. 1966

    1966

  71. [72]

    Surya Kiran Kanumilli and C. N. Pope. Super-geometrodynamics in higher dimensions. Classical and Quantum Gravity. 2018. doi:https://doi.org/10.1088/1361-6382/aae32b

    work page doi:10.1088/1361-6382/aae32b 2018

  72. [73]

    1885 , publisher =

    Die nicht-euklidischen Raumformen in analytischer Behandlung , author =. 1885 , publisher =

  73. [74]

    Ueber die Clifford-Klein'schen Raumformen

    Wilhelm Killing. Ueber die Clifford-Klein'schen Raumformen. Mathematische Annelen. 1891. doi:http://doi.org/10.1007/BF01206655

    work page doi:10.1007/bf01206655

  74. [75]

    The Casimir force: background, experiments, and applications

    Steven K Lamoreaux. The Casimir force: background, experiments, and applications. Reports on Progress in Physics. 2005. doi:http://dx.doi.org/10.1088/0034-4885/68/1/R04

    work page doi:10.1088/0034-4885/68/1/r04 2005

  75. [76]

    L. D. Landau and E. M. Lifshitz. 99 The synchronous reference system. The Classical Theory of Fields, Translated from the Russion by Morton Hamermesh, 99 The synchronous reference system. 1971

    1971

  76. [78]

    E. M. Lifshitz and I. M. Khalatnikov. On the singularities of cosmological solutions of the gravitational equations. I. Soviet Physics JETP. 1961. doi:http://dx.doi.org/10.1007/s40314-014-0116-0

    work page doi:10.1007/s40314-014-0116-0 1961

  77. [79]

    Generation of orthogonal curvilinear grids on the sphere surface based on Laplace-Beltrami equations

    Fuqiang Lu and Zhiyao Song and Xin Fang. Generation of orthogonal curvilinear grids on the sphere surface based on Laplace-Beltrami equations. IOP Conference Series: Earth and Environmental Science. 2013. doi:http://dx.doi.org/10.1088/1755-1315/19/1/012012

    work page doi:10.1088/1755-1315/19/1/012012 2013

  78. [80]

    Cool horizons for entangled black holes

    Juan Maldacena and Leonard Susskind. Cool horizons for entangled black holes. Fortschritte der Physik. 2013. doi:http://dx.doi.org/10.1002/prop.201300020

    work page doi:10.1002/prop.201300020 2013

  79. [81]

    Weisstein

    Eric W. Weisstein. Cyclide. 2021

    2021

  80. [82]

    On the cyclide

    James Clerk Maxwell. On the cyclide. The quarterly journal of pure and applied mathematics. 1868

Showing first 80 references.