Riemannian Geometry| 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 x ∈ X.
$ 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:
One 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. 2D, 3D
and 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
* 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 18 [tactile introduction].
@ Positive scalar curvature: Lesourd et al a2009 [on non-compact manifolds, and the Liouville theorem].
@ 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') := supp ∈ M 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; Lines 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].
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
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