3-Dimensional Geometries| 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]b − gb[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];
Kruglikov & Tomoda CQG(18)-a1804 [explicit algorithm].
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].
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
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