 Types of Metrics

Metrics with Symmetries > s.a. axisymmetry; general relativity solutions with symmetries; killing fields; spherical symmetry.

Degenerate Metrics > s.a. extended signatures; geometrodynamics; models in canonical general relativity; quantum gravity and geometry; spin structure.
* Example: [@ Yoneda et al PRD(97)gq] A metric which is flat everywhere but degenerate at x = t = 0 is

ds2 = −[1 − (f'(t) h(x))2] dt2 + [−2 f'(t) h(x) (1−f(t) h'(x))] dt dx + [1 − f(t) h'(x)]2 dx2

((t, x) $$\mapsto$$ (t, xfh)); E.g., f = exp{−t2}, h = x exp{−x2}, −1 < f'(t), h(x) < 1 & −1 < f(t), h'(x) ≤ 1 (= 1 at 0).
@ General references: Kreisel et al AdP(63); Crampin PCPS(68); D'Auria & Regge NPB(82) [instanton]; Koshti & Dadhich CQG(89); Bengtsson CQG(91); Varadarajan CQG(91); Percacci in(92) [quantum field theory approach]; Gratus & Tucker JMP(96)gq [2D]; Dray IJMPD(97)gq [tensor distributions]; Baez CMP(98) [from 2D BF-theory]; Deser CQG(06) [reason for invertibility].
@ Geometry: Borde et al CQG(99)gq [causal continuity]; Stoica IJGMP(11)-a1105; > s.a. causal structure; newton-cartan theory [connections]; singularities.
@ And matter: Cabral & Rivelles CQG(00)ht/99 [particle dynamics]; > s.a. lagrangian systems; quantum fields in curved backgrounds; unified theories.

Constant-Curvature Manifolds > s.a. riemann tensor; sphere.
* Metric: For any signature and dimension d > 2, in stereographic coordinates it is

ds2 = (1 + $$1\over4$$kr2)−2 ηij dxi dxj ,   with   r2:= ηij xi xj ,  ηij = diag(±1, ±1, ..., ±1) .

* Curvature: The Riemann tensor is of the form Rabcd = $$1\over n(n-1)$$R (gac gdbgad gcb), so Rab = $$1\over n$$Rgab; in 4D Cabcd = 0, and Gab = −$$1\over4$$Rgab.
* Examples: The n-sphere (R > 0); For the Lorentzian case, Minkowski space (R = 0), de Sitter (R > 0), anti-de Sitter (R < 0).
@ References: Wolf 87; Dryuma TMP(06)m.DG/05 [3D].

Other Special Types > s.a. 2D manifolds; 3D manifolds; lie groups; types of lorentzian geometries; weyl tensor.
* Zoll metric: A Riemannian gab on a compact M, all of whose geodesics are simply periodic, with period 2π; For example, the standard metric on the 2-sphere.
* Zollfrei metric: A Lorentzian gab on a compact M, all of whose null geodesics are periodic (a conformally invariant property).
* Neutral metric: A pseudo-Riemannian metric of signature (n, n).
@ Vanishing curvature invariants: Pravda et al CQG(02)gq; Coley PRL(02)ht [and string theory].
@ Finiteness results: Grove et al BAMS(89).
@ Bounded curvature: Cheeger & Colding JDG(97) [R bounded below]; > s.a. riemannian geometry; types of lorentzian geometries.
@ Zoll, zollfrei: in Guillemin 89; Nakata JGP(07) [singular self-dual].
@ Other types: Win gq/96 [diagonal, efficient calculation]; Shi & Tam CMP(04) [quasispherical]; Cuccu & Loi JGP(07) [balanced metrics on $$\mathbb C$$n]; Anderson & Herzlich JGP(08) [with prescribed Ricci curvature]; Kalafat et al JGP(13) [self-dual, on non-simply-connected 4-manifolds]; Santalla et al NJP(15)-a1407 [random geometry]; Georgiou & Guilfoyle a1605 [neutral 4-manifolds with null boundary]; > s.a. riemann tensor [in terms of curvature invariants].
@ Singular metrics, distributional curvature: Kunzinger & Steinbauer AAM(02)m.FA/01 [Colombeau]; in Dray in(17)-a1701; > s.a. coordinates [discontinuous transformations]; gravitational-wave solutions [impulsive]; hamiltonian and lagrangian systems.

Information Geometry > s.a. entropy; information; solutions of gauge theories.
* Idea: The introduction of a metric on the space of parameters for models, e.g., in statistical mechanics.
* For probability distributions: If pi is the probability for the i-th event, a natural choice is ds2 = ∑i dpi2 / pi .
@ General references: Streater in(97); Amari & Nagaoka 00; Naudts & Anthonis LNCS-a1506 [extension to non-statistical systems].
@ For probability distributions: Bengtsson AIP(06)qp/05 [Fisher-Rao metric]; > s.a. probabilities in physics; riemannian geometry / actions for gravity.
@ And phase transitions: Janke et al PhyA(04)cm; Kumar et al PRE(12)-a1210 [geodesics, classical and quantum second-order phase transitions]; Maity et al PRE(15)-a1503 [and the renormalization group].
@ Quantum: Álvarez-Jiménez & Vergara IJQI(19)-a1904 [from generating functions]; > s.a. coherent states; origin of quantum theory.

For States of a Physical System > s.a. mixed states; riemannian geometry; thermodynamics [geometry of state space]; types of distances.
* Fubini-Study metric: A complex tensor whose real part is the Riemannian metric that measures the 'quantum distance', and whose imaginary part is the Berry curvature; > s.a. Encyclopedia of Mathematics page; Wikipedia page.
* Cayley-Fubini-Study metric: For a small change dψ from a pure quantum state ψ,

ds2 = $$[\langle{\rm d}\psi \mid {\rm d}\psi\rangle - \langle{\rm d}\psi \mid \psi\rangle\,\langle\psi \mid {\rm d}\psi\rangle] \,/\, \langle\psi \mid \psi\rangle$$ ;

It can be considered as the infinitesimal version of the distinguishability distance d(ψ1,ψ2) := |1 − $$\langle\psi_1|\psi_2\rangle$$ |2.
@ Fubini-Study metric: Anandan PLA(90) [physical meaning]; Cheng a1012 [pedagogical]; Álvarez-Jiménez & Vergara a1605 [and gauge invariance]; > s.a. types of distances; phase space.
@ Other metrics on the space of quantum states: Man'ko et al JPA(17)-a1612 [from relative entropy].