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];
Ferrando & Sáez CQG-a2004 [with a transitive group of isometries].

**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].

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