3-Dimensional
Geometries |

**In General** > s.a. riemann
tensor / Geometric Topology; types
of metrics.

* __Types of geometry__:
(Thurston) There are eight, spherical (elliptic), Euclidean, hyperbolic (the
least understood), mixed spherical-Euclidean, mixed hyperbolic-Euclidean,
and three exceptional ones.

* __Geometrization conjecture__:
(Thurston) Every closed, oriented 3-manifold has a natural decomposition into
geometrical pieces (which have one of the eight well-defined types of geometric
structure); This is known to be true for Haken manifolds; If true in general,
Poincaré's
conjecture
would follow.

* __Curvature__: The Weyl tensor *C*_{abcd} vanishes, so the Riemann tensor
depends on the Ricci tensor only,

*R*_{abcd} =
2 (*g*_{a[c }*R*_{d]b}
– *g*_{b[c }*R*_{d]a}) – *R* *g*_{a[c} *g*_{d]b}
.

* __Diagonalization__: Every
3D Riemannian manifold has a diagonalization (Darboux, Cotton, 1800s).

@ __General references__: Thurston 78; Hamilton JDG(82);
Gabai JDG(83), JDG(87), JDG(87);
Thurston AM(86).

@ __And physics__: Gegenberg et al CQG(02)ht [and
M-theory]; > s.a. geometry and topology in cosmology.

@ __Related topics__: Ó Murchadha pr(91)
[Yamabe constant]; Gegenberg & Kunstatter
gq/93 [parametrization];
Birmingham et al PRL(99)
[and boundary structure]; Gegenberg & Kunstatter
CQG(04)ht/03 [Ricci
flow analysis]; Reiris a1002 [relations between Ricci curvature, scalar curvature
and volume radius]; Kowalski & Sekizawa JGP(13) [diagonalization]; Pugliese & Stornaiolo GRG(15)-a1410 [deformations].

@ __As deformations of constant curvature__: Gegenberg & Kunstatter gq/93;
Coll
et
al GRG(02)gq/01.

**Special Cases** > s.a. types
of metrics.

* __Homogeneous__: Characterized by the three eigenvalues of the Ricci
tensor.

* __Spherically symmetric__: They are all conformally flat.

* __With positive R__:
They are all connected sums of elliptic
spaces and
copies of S

@

@

**With Lorentzian Metric** > s.a. 3D
general relativity / lorentzian
geometry; spherical
symmetry.

@ __Types__: Auslander & Markus 59 [flat]; Bona & Coll JMP(94)
[isometry groups];
Gilkey & Nikčević IJGMP(05)
[affine curvature homogeneous]; Calvaruso JMP(07)
[with prescribed Ricci tensor]; Calvaruso & De Leo IJGMP(09)
[pseudo-symmetric].

@ __With constant curvature invariants__: Coley et al CQG(08)-a0710 [all
invariants constant]; Calvaruso JGP(07)
[homogeneous], DG&A(08)
[with distinct constant Ricci eigenvalues].

@ __Classification__: Torres del Castillo & Gómez-Ceballos JMP(03); Milson & Wylleman CQG(13)-a1210 [it requires at most fifth covariant derivatives of the curvature tensor].

**With Riemannian Metric** > s.a. riemannian
geometry.

@ __References__: Carfora & Marzuoli gq/93 [and
simplicial quantum gravity, critical phenomena].

**Related Topics** > s.a. extrinsic
curvature [minimal surfaces]; knots
and physics.

@ __Surfaces__: Montaldo & Onnis JGP(05)
[constant Gauss curvature].

@ __Random models__: Fukuma et al JHEP(15)-a1503 [triangles glued together along multiple hinges].

main page – abbreviations – journals – comments – other
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send feedback and suggestions to bombelli at olemiss.edu – modified 3
apr
2016