Diffeomorphisms| Diffeomorphisms |
In General > s.a. Hole
Argument; Homeotopy
Group.
* Effect of geometric quantities: In general, 
T = 
v T;
For a metric and a connection
gab =
2 
(a vb)
, 
mab
= a b vm – Rm(ab) cvc
.
* Descriptors: Given
an infinitesimal diffeomorphism of a manifold,
xm 
xm + 
m(x),
the descriptors are the functions m(x),
i.e., the vector field generating the diffeomorphism [@ in Anderson 67].
@ Diffeomorphism groups: in Marsden 74; Michor 80; > s.a. types
of groups [mapping class].
@ Representations: Goldin et al JPA(83) [of R3]; Maxera ht/98 [non-linear
realizations]; Larsson gq/99 [of
extended algebra].
@ Geometry: Khesin & Misiolek JMFM(05)m.DG
[asymptotic directions]; Gordina & Lescot mp/05 [Ricci
curvature of Diff(S1)/S1].
@ Related topics: Dolgopyat CMP(00)
[mostly contracting].
Witt Algebra > s.a. Virasoro
Algebra.
$ Def: The infinite-dimensional Lie algebra of Diff(S1).
* Generators: Given by Ln =
–i exp{in
}
/,
with
commutation relations [Lm, Ln]
= (n–m) Lm+n.
Types of Diffeomorphisms and Manifolds > s.a. sphere [volume-preserving].
* Large diffeomorphims:
The ones not in the component connected to the identity, which form the modular
group.
* 3D: The Smale
conjecture states that the diffeomorphism group Diff(S3)
has the same homotopy type as SO(4); In proving the Smale conjecture, A Hatcher
proposed the Generalized Smale conjecture that for all closed 3-manifolds
Diff(M 3)
is homotopic to Isom(M 3), which
has been proven for many special cases; Don Witt has proposed a correspondence
between the Generalized Smale conjecture and gauge fixing in gauge/diffeomorphism
invariant theories as a procedure for proving it in such situations – the
idea is to use the fact that the diffeo group is the gauge group by which
one divides to obtain the space of gauge-invariant configurations from a
space of gauge-dependent ones, and if both of the latter are known well enough
one can set up an exact sequence with which one calculates all homotopy groups
of the diffeomorphism group.
@ 2D: Thurston BAMS(88) [geometry and dynamics].
@ 3D: Bonatti et al Top(04) [gradient-like]; Benatti & Wilkinson Top(05)
[transitive partially hyperbolic].
@ Compact
Riemannian manifolds: Delanoë DG&A(04) [gradient rearrangement].
@ Generalized: Kunzinger & Steinbauer CQG(99)
[distributional,
pp-wave
example]; Dimitrijevic & Wess ht/04-in
[deformed bialgebra on non-commutative space]; Bahr & Thiemann a0711 [distributional,
and combinatorial lqg].
@ Related topics: Giacomini PRD(04)ht [Poisson
algebra with spacetime bifurcations].
And Physical Theories > s.a. canonical
general relativity; Covariance; symmetries; symmetry
breaking.
* Quantum gravity:
Large diffeomorphisms give rise to theta
sectors.
@ And gauge symmetries: Kuchar & Stone CQG(87)
[parametrised Maxwell field];
Pons
et
al PRD(00)gq/99 [in
Ashtekar variables], JMP(00)gq/99 [in
Einstein-Yang-Mills].
@ And general relativity / quantum gravity:
Chamblin & Gibbons gq/95 [spacetime
topology and diffeomorphism that reverse the time-orientation]; Aldaya & Jaramillo IJMPA(03)gq/02 [representations
of diffeomophism groups]; Larsson a0709-in
[anomalies]; Samanta a0708 [Lagrangian
gravity].
@ Large diffeomorphisms: Giulini BCP(97)gq/95 [and
geons], NPPS(97)gq-in.
@ Related topics: Sorkin in(93) [and particle symmetries]; Pons CQG(03)gq [and
phase space for generally covariant theories].
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
27 jun 2008