The domestication of hyperspace: Bianchi and Thurston

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The domestication of hyperspace (Bianchi and
Thurston)

• Jeremy Gray
• Open University
• University of Warwick

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Riemannian metrics

• In a system of local coordinates

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Gaussian curvature

• In polar coordinates

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Avoiding

• Differential invariants Lipschitz,
• Christoffel,
• Ricci Curbastro

• n-dimensional linear algebra
• Algebraic surfaces

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But noting

• The Clifford-Klein space problem (Epple 2002).

• Connections from Sophus Lie and Wilhelm Killing
  to Bianchi

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Up to 1890

• Bolyai and Lobachevskii
• Riemann,
• Beltrami,
• Klein,
• Poincaré

• Riemann
• Curvature of higher-dimensional manifolds as
  sectional curvature

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Surfaces of constant curvature

• Constant positive curvature the sphere
• Constant zero curvature the plane and the
  cylinder
• Constant negative curvature non-Euclidean space

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Three-dimensional spaces of constant curvature

• Constant positive curvature the 3-sphere
• Constant zero curvature Euclidean space
• Constant negative curvature non-Euclidean
  3-space

• And ?

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Rudolf Lipschitz
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Minimal surfaces in Euclidean space
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Minimal surfaces in other spaces

• Lipschitz
• an l-dimensional sub-manifold of an n-dimensional
  manifold is a minimal sub-manifold iff its mean
  curvature in every normal direction vanishes.
• A domestication theorem

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Curved ? curved in something

• Blumenthal
• Die Menschen fassen kaum es
• Der Krümmungsmass
• des Raumes

• Translated
• People cannot keep ideas in place
• About the curvature of space

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Friedrich Schur 1885

• Beltrami
• if a space can be given coordinates such that
  the geodesics in the space have equations that
  are linear in the coordinates, then the space is
  of constant curvature.

• Lipschitz
• if a space has constant curvature then it can be
  given a coordinate system such that the geodesics
  have linear equations.

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Wilhelm Killing
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Wilhelm Killing Grundlagen der Geometrie, 1892

• 7 basic judgements (Urtheile) or theorems
  (Sätze) the basic facts (Grundsätze) of
  generalised geometry

• Reduce geometry to the theory of
  finite-dimensional transitive transformation
  groups

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The fundamental ideas of geometry

• rigid body,
• part of a body,
• space,
• part of a space,
• to occupy or cover a space,
• rest,
• motion.

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Grundsätze

• two bodies cannot occupy the same space
• 7) if a body moves and part of it returns
  exactly to where it was before then all of it
  does.

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Grundsatz 8 proper space-forms

• If a point of a solid body in n-dimensional
  space remains at rest while the body moves
• then
• no point of the body can sweep out an
  n-dimensional region of space.

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Proper space forms

• Have constant sectional curvature
• (zero, positive, or negative).
• The proper two-dimensional space forms, are
  precisely the usual cases.

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Frederick Woods MIT

• Space of constant curvature
• Annals of Mathematics, 1902
• Forms of non-Euclidean Space
• Colloquium Address to the American Mathematical
  Society, Boston, 1903

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A three-dimensional geometry

• Should
• accord with the facts of experience within the
  limits of observation (Colloquium, p. 31).
• A Riemannian manifold admitting mobility of
  bodies, i.e. transitivity of motions preserving
  geodesics.
• The appropriate spaces were those of constant
  curvature following Killing, need to analyse
  the possible discrete subgroups of the full group
  of motions. There is a flat three-dimensional
  torus.

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Luigi Bianchi (1856-1928)
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Bianchi 1898

• Sugli spazi a tre dimensioni che ammettono un
  gruppo continuo di movimenti,
• Memorie della Societ\a Italiana delle Scienze,
  (3) 11, 267-352,

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Isometry groups acting on a 3-manifold

• May be of dimensions 1, 2, 3, 4, or 6 (but not 5)
• A 6-dimensional group leads to one of the
  familiar three geometries

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The Bianchi classification

• Nine transitive cases
• Each with a metric that may depend on some
  parameters (and reduce to simpler cases if the
  parameters 0 or 1).

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The possibilities

• The three constant curvature geometries

The geometry of a left-invariant metric on a
compact Lie group
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Lorias Jahrbuch review

• The importance of this result is known to every
  reader who knows the prize problem set by the
  Fürstlich Jablonowski'schen Gesellschaft for the
  year 1901
• A completion of the theory of quadratic
  differential forms in some essential respect
• The prize was not awarded for that topic.

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Bianchi n 3
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Bianchi redivivus

• Taubs paper Empty Space-Times Admitting a Three
  Parameter Group of Motions Annals of
  Mathematics, 53, 472-490
• Ryan and Shepley, Homogeneous Relativistic
  Cosmologies, (1975)

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Thurston in the late 1970s

• Eight model geometries
• 8 versus Bianchis 9?
• Two Bianchi geometries have not Thurston model
  and one does not admit a transitive isometry
  group