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 T_{x }*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 *T*_{abc}
= 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 d*u* d*v* → d*x*^{2}
+ d*y*^{2}.

@ __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].

**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*) d*x* .

* __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*') := sup_{p
∈ M} sup_{v ∈ 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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jul 2016