Riemannian Geometry  

In General > s.a. connections; riemann tensor / 2D manifolds and 3D manifolds; differential geometry; metric tensors.
$ Weak Riemannian manifold / structure: A manifold X with a smooth assignment of a weakly non-degenerate inner product (not necessarily complete) on Tx X, for all xX.
$ Riemannian manifold / structure: A weak one with non-degenerate inner product (the model space is isomorphic to a Hilbert space); This means a Euclidean metric on the tangent bundle; Alternatively, a Riemann-Cartan manifold with vanishing torsion, i.e., with Tabc = 0.
* Conditions: Any (paracompact) manifold can be given one, and any one can be deformed into any other, since at each point the set of possible metrics is a convex set (not true in the Lorentzian case).
* Moduli space: Gromov's topological moduli spaces \(\cal M\)(n, ρ).
* Invariants: In a purely metric geometry, the Euler class and the Pontryagin class are useful invariants for characterizing the topological properties of the manifolds; In a metric-affine geometry, where torsion comes into play, one can define a torsional invariant.
* Relationship Euclidean-Minkowskian: Can make a Wick rotation; Or use u = x + i y, v = x − i y to map du dv → dx2 + dy2.
@ General references: in Geroch CMP(69).
@ Texts, overviews: Bonola 55, Coxeter 57 ["non-Euclidean"]; Yano & Kon 84; Willmore 93 [IIb, including complex]; Lee 97; Petersen 97 [III]; Berger 00, 03 [intro]; Godinho & Natário 14 [III, with applications]; Marsh a1412.
@ Global: Petersen BAMS(99) [curvature and topology]; Markvorsen & Min-Oo 03.
@ Invariants: Connes IJGMP(08)-a0810 [unitary invariant]; Nieh a1309 [torsional topological invariant].
@ Related topics: Coleman & Korté JMP(94) [G-structures]; Ferry Top(98) [Gromov-Hausdorff limits of polyhedra]; Rylov m.MG/99, m.MG/00 [defining topology from metric]; Papadopoulos JMP(06) [essential constants]; Calderón a0905 [Ricardo's formula].
> Online resources: see Wikipedia page.

Types and Examples > s.a. 3D geometries; 4D geometries; euclidean geometry; riemann tensor; metrics [characterization] and types of metrics.
* Example: An example of a weak Riemannian structure is the space X:= C([0,1], \(\mathbb R\)), with

\(\langle\)f | g\(\rangle\):= \(\int_0^1\)f(x) g(x) dx .

* Hyperbolic: Founded by Lobachevsky in 1829, normally means the geometry of constant negative curvature spaces.
@ Hyperbolic: Milnor BAMS(82) [rev]; Anderson 05 [II]; Vermeer T&A(05) [plane, Ungar's addition and gyration].
@ Non-positive curvature: Ballmann 95; Eberlein 96; Bridson & Häfliger 99; Taimina 09 [tactile introduction].
@ Constant scalar curvature: Mach & Ó Murchadha CQG(14) [spherically symmetric, any dimension].
@ With curvature bounds: Cheeger & Colding JDG(97) + MR, JDG(00) + MR [lower bounds].
@ On quantum states: Petz & Sudár JMP(96) [density matrices]; Dittmann JGP(99)qp/98, Slater JGP(01)qp/00 [Bures metric]; Petz JPA(02)qp/01 [Fisher metric]; Pandya & Nagawat PE(06)qp/02; Pandya qp/03 [Lorentzian]; Andai JMP(03)mp; > s.a. mixed states.
@ On spaces of connections: Gibbons & Manton PLB(95)ht [Yang-Mills monopoles]; Orland ht/96 [Yang-Mills configurations].
@ Information metric: Groisser & Murray dg/96 [instantons]; Parvizi MPLA(02)ht [non-commutative instantons]; > s.a. types of metrics.
@ Singular manifolds: Botvinnik G&T(01)m.DG/99 [Sullivan-Baas singularities]; > s.a. distributions.
@ Other topics: Atzmon MPLA(97)qp [on \(\cal C\)]; Mendoza et al JMP(97) [1D, fluctuating]; Anastopoulos & Savvidou AP(03)qp [on phase space]; Deng & Hou JPA(04) [Randers metrics]; Hiai & Petz a0809 [on positive-definite matrices]; Berestovskii & Nikonorov DG&A(08) [δ-homogeneous]; in Abramowicz a1212 [circles, geodesic, circumferential and curvature radii]; > s.a. instantons [moduli space]; manifolds [G-manifolds]; Osserman Manifold.

Space of Riemannian Metrics on a Manifold > s.a. Cheeger-Gromov Theory; distance; foliations; geomeotrodynamics; metrics.
* Topology: For a 3-manifold M it is contractible but with non-trivial global differentiable structure, a cone on the vector space of symmetric bilinear forms on M.
* Metric structure: A distance on the space of Riemannian metrics on a manifold M is

d(h, h') := suppM  supv ∈ TpM, v ≠ 0 | ln [h(v,v) / h'(v,v)] | .

@ General references: Fischer JMP(86) [structure of superspace]; Gao JDG(90) [convergence]; Schmidt gq/01-proc [and general relativity], gq/01-proc [3D homogeneous]; Gomes a0909 [structure of subspace of metrics with no Killing vector fields]; Sormani a1006-fs, a1606-conf [on the convergence of sequences of Riemannian manifolds]; Clarke JDG(13) [completion of the manifold of Riemannian metrics].
@ Cotton flow: Kisisel et al CQG(08)-a0803 [3D]; Kilicarslan et al JHEP(15)-a1502 [3D].
@ Other flows: Letelier IJTP(08) [Riemann-Christoffel flow]; > s.a. Ricci Flow; Yamabe Flow.
@ Metric structure: in Eder GRG(80); Peters pr(87); Gromov 81, 98 [for geometries]; Seriu PRD(96)gq, CMP(00)gq/99 [based on spectra].

Concepts and Results > s.a. Covariant Derivative; Curves and geodesics; diffeomorphisms; Hopf-Rinow Theorem.
@ Generalizations: de Beauce & Sen ht/04 [discretizations]; > s.a. non-commutative geometry; phase space [quantum]; Sub-Riemannian Manifolds.
@ Related topics: in Molzahn et al AP(92) [length scales]; Cabrerizo et al JGP(12) [isotropic submanifolds].

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