Riemannian
Geometry| Riemannian
Geometry |
In General > s.a. [connection;
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 Lorentzian case).
* Moduli space: Gromov's
topological moduli spaces 
(n, 
).
* 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: Bonola 55, Coxeter 57 ["non-Euclidean"]; Yano & Kon
84; Willmore 93 [IIb, including complex]; Petersen 97 [III]; Berger 00, 03
[intro].
@ Global: Petersen BAMS(99) [curvature and topology]; Markvorsen
& Min-oo 03.
@ 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].
Types and Examples > s.a. 3D geometries; 4D
geometries; euclidean
geometry; types
of metrics.
* Example: An example of
a weak Riemannian structure is the space X:= C([0,1], R),
with
f | g 
:= 
01 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 & Haefliger
99.
@ With curvature bounds: Cheeger & Colding JDG(97), JDG(00)
[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 connections: Gibbons & Manton PLB(95)ht [Yang-Mills
monopoles]; Orland
ht/96 [Yang-Mills].
@ Information metric: Groisser & Murray dg/96 [instantons];
Parvizi
MPLA(02)ht [nc
instantons].
@ Singular manifolds: Botvinnik G&T(01)m.DG/99 [Sullivan-Baas
singularities]; > s.a. distributions.
@ Other topics: Atzmon MPLA(97)qp [on 
];
Mendoza et al JMP(97)
[1D, fluctuating]; Anastopoulos & Savvidou
AP(03)qp [on
phase space]; Deng & Hou JPA(04)
[Randers metrics]; > s.a. instanton [moduli
space], manifolds [G-manifolds], Osserman
Manifold.
Space of Riemannian Metrics on a Manifold > s.a. distance;
foliations;
geomeotrodynamics;
Cheeger-Gromov Theory.
* 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.
* Ricci flow: The study of the
initial value problem 
t gab
= –Rab + (2/n) r gab,
with gab(0)
= gab, for an n-dimensional
manifold M (usually closed), with gab Riemannian, and r the
averaged scalar curvature over M; Results: Solutions always
exist
for finite t, and continue while the curvature is bounded; Isometries
and
volume are preserved; Under certain circumstances, the flow converges to
a
sphere.
* 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-in
[and general relativity], gq/01-in
[3D homogeneous].
@ Ricci flow: Hamilton JDG(82), JDG(85),
in(88); Carfora & Marzuoli CQG(88);
Carfora et al JDG(90);
Cao & Chow BAMS(99);
Samuel & Chowdhury CQG(07),
CQG(08)
[gradient formulation ito entropy and general relativity]; Holzegel et al a0706 [with
surgery, for biaxial Bianchi IX metrics]; Woolgar a0708-in
[applications in physics]; Carroll a0710 [and
quantum theory]; > s.a. relativistic cosmology.
@ Other flows: Kisisel et al a0803 [Cotton
flow, 3D]; Letelier IJTP(08)
[Riemann-Christoffel 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; diffeomorphisms; geodesics;
Hopf-Rinow Theorem.
@ Discretizations: de Beauce & Sen ht/04.
@ Related topics: in Molzahn et al AP(92)
[length scales].
Main page – Abbreviations – Journals – Comments – Other
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
25 may 2008