Paper · May 26, 2020

How to see the eight Thurston geometries

Tiago Novello, Vinícius da Silva, Luiz Velho, Mikhail Belolipetsky

math.GTarXiv:2005.12772v4
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Abstract

In this expository paper, we present a survey about the history of the geometrization conjecture and the background material on the classification of Thurston's eight geometries. We also discuss recent techniques for immersive visualization of relevant three-dimensional manifolds in the context of the Geometrization Conjecture.

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Submitted
May 26, 2020
Last revised
Sep 14, 2021 (v4)
Subjects
math.GT, cs.GR, math.HO
Geometrization3-manifoldsNumerics & visualization

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper An Introduction to Geometric Topology (Bruno Martelli, 2016)7Martelli 2016 A Realization of Thurstons Geometrization: Discrete Ricci Flow with Surgery (Paul M. Alsing, Warner A. Miller and Shing-Tung Yau, 2017)8Alsing 2017 Canonical geometrization of orientable 3-manifolds defined by vector-colourings of 3-polytopes (Nikolai Erokhovets, 2020)3Erokhovets 2020 Geometrization in Geometry (Izabella Muraro de Freitas and Álvaro Krüger Ramos, 2022)5de Freitas 2022 Large-scale geometry of the Universe (Yassir Awwad and Tomislav Prokopec, 2022)1Awwad 2022 On Ricci Solitons and Harmonic Vector Fields in the Thurston Geometry F⁴ (Halima Boukhari, Hadjer Okbani and Ahmed Mohammed Cherif, 2026)2Boukhari 2026 Modifications of CMB Temperature and Polarization Quadrupole Signals in Thurston Spacetimes (Tanay Gupta, Sukanta Panda and Rajib Saha, 2026)6Gupta 2026 A Lean Formalization of the Hamilton–Perelman Proof of Thurston's Geometrization Conjecture (Yuan Liao, Ziyang Qin, Jui-Hui Chung et al., 2026)4Liao 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Large-scale geometry of the Universe

    Yassir Awwad and Tomislav Prokopec

    Topics Geometrization, 3-manifolds · Shared terms “Thurston geometries”, “eight geometries”, “geometrization conjecture”

  2. On Ricci Solitons and Harmonic Vector Fields in the Thurston Geometry

    Halima Boukhari, Hadjer Okbani and Ahmed Mohammed Cherif

    Topics Geometrization · Shared terms “Thurston geometries”

  3. Canonical geometrization of orientable -manifolds defined by vector-colourings of -polytopes

    Nikolai Erokhovets

    Topics Geometrization · Shared terms “geometrization conjecture”, “Thurston”, “dimensional manifolds”

  4. A Lean Formalization of the Hamilton–Perelman Proof of Thurston's Geometrization Conjecture

    Yuan Liao, Ziyang Qin, Jui-Hui Chung et al.

    Topics Geometrization · Shared terms “geometrization conjecture”, “Thurston”

  5. Geometrization in Geometry

    Izabella Muraro de Freitas and Álvaro Krüger Ramos

    Topics Geometrization · Shared terms “Thurston”

  6. Modifications of CMB Temperature and Polarization Quadrupole Signals in Thurston Spacetimes

    Tanay Gupta, Sukanta Panda and Rajib Saha

    Topics Geometrization · Shared terms “Thurston geometries”, “geometrization conjecture”, “background”

  7. An Introduction to Geometric Topology

    Bruno Martelli

    Topics Geometrization, 3-manifolds · Shared terms “Thurston”

  8. A Realization of Thurstons Geometrization: Discrete Ricci Flow with Surgery

    Paul M. Alsing, Warner A. Miller and Shing-Tung Yau

    Topics Geometrization, Numerics & visualization, 3-manifolds · Shared terms “Thurston”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Geometrization Thurston's conjecture: every closed 3-manifold can be cut along spheres and tori into pieces that each carry one of eight model geometries. Start with: Three dimensional manifolds, Kleinian groups and hyperbolic geometry (Thurston, 1982) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Numerics & visualization Computing and visualising Ricci flow: simulations of neckpinches and critical behaviour, pictures of rotationally symmetric flows, and… Start with: Deforming metrics in the direction of their Ricci tensors (DeTurck, 1983)

In the guidebook

  1. 10A.2
    Finite Rotation Groups and Spherical Space FormsBook 10A · Three-Manifolds and Geometrization

    Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.

  2. 10A.8
    Thurston’s Eight GeometriesBook 10A · Three-Manifolds and Geometrization

    The model geometries, the geometrization conjecture, and why it implies Poincaré.

  3. 10A.9
    Hyperbolic Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

    Ideal tetrahedra, Mostow rigidity, and the thick–thin decomposition.