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 Cabcd vanishes, so the Riemann tensor depends on the Ricci tensor only,

Rabcd = 2 (ga[c Rd]bgb[c Rd]a) – R ga[c gd]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 S2 × S1; The moduli space for the orientable compact case is path-connected.
@ General references: Kowalski & Prüfer MA(94) [with distinct constant Ricci eigenvalues]; Beig gq/96 [conformally flat 3-manifolds, transverse-traceless tensors]; Doyle & Rossetti G&T(04) [isospectral but non-isometric compact manifolds]; Grosche PAN(07)qp/05 [Darboux spaces, path integrals]; Dryuma TMP(06) [constant curvature]; Coda AM-a0907 [with positive scalar curvature].
@ With Killing vector felds: Gürses CQG(10)-a1007 [Ricci tensor in terms of the Killing vector].

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].

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