Linear
and Affine Connections| Linear
and Affine Connections |
On the Bundle of (Orthonormal) Frames > s.a. metric [examples];
tetrads [calculation of curvature].
$ Linear connection:
A linear connection on a smooth manifold X is a
connection on the principal fiber bundle F(X) of frames on X.
$ Levi-Civita, metric or
Riemannian connection: Given the reduced bundle O(M) over M (i.e.,
a Riemannian metric g),
it is the only connection such that (i) 
has
no torsion, mab = mba,
and (ii) Parallel transport preserves the metric, gab;m =
0.
$ Riemann-Cartan connection: The 1-form
ij :=
ij – Kij ,
where is the Levi-Civita
connection 1-form, and K the contorsion,
with
T i = 
j
Kji.
* Cartan structure equations: They describe the affine connection on
a
manifold through the torsion and curvature tensors (which in turn can be obtained
from the connection); If T is the torsion 2-form, the
coframe, 
the curvature
2-form, and the connection,
d + = T , =
d + .
Christoffel Symbols
$ First kind: {ab, c}
= 
(
bgac
+ agbc –
cgab).
$ Second kind: {a \atop bc}
= gad {bc, d}.
* And connection: Given a choice
of coordinates, the condition that a
linear connection be the metric connection for a metric gab is
expressed by
abc =
{a \atop bc} = gmn (gnb,a
+ gan,b – gab.n)
.
* Gauge transformation: For
an infinitesimal diffeomorphism generated by va, 
mab = 
abvm – Rm(ab)c vc.
On the Tangent Bundle > s.a. coordinates [Fermi transport]; torsion.
* History: Linear connections were invented by Cartan to geometrize
Newtonian
gravitation.
* Existence: They always exist on a manifold with a countable basis.
* Parallel transport: To
define
one, start with x 
M,
a curve
through x,
and a lift of to
the frame bundle, F(M);
Fix a vector X Tx(M);
Then the parallel transport
of X is
T(M)(t):=
F(M)(t)
X.
* Covariant derivative: Defined in T(M) of Z = T(M)(t)
= bm (/xm)
wrt Y = cm (/xm)
as
Y Z:=
(d/dt)T(M)
= (dxm(t)/dt) DmF(M)(t)
X ,
where Dm is the covariant derivative in F(M); One finds
Y Z =
cn bm;n /xm
, bm;n:=
bm,n – Amns bs, Amns = mns .
* Flat connection: If
the torsion and curvature both vanish, the connection is called flat, and there
exist local
coordinates in which the connection vanishes;
Then, in other coordinates, =
h–1dh.
* On a Riemannian manifold: The connection can be written as
abc =
{a \atop bc} + Kabc ,
where {a \atop bc} are the Christoffel symbols, Qabc = Ka[bc] the
torsion, and Mabc = Ka(bc) the
non-metricity.
* Weyl vector: The trace of
the non-metricity, Ba:= Mabc gbc.
Special Types of Metrics > s.a. FRW
models;
spherical symmetry; schwarzschild
spacetime;
symmetry.
@ Ricci-flat: Armstrong JGP(07) [classification].
References > s.a. finsler
geometry; Lagrange Spaces [non-linear
connection]; Parallel Transport;
torsion.
@ General: Levi-Civita RCMP(17).
@ Connection and metric: Schmidt CMP(73);
Hall GRG(88);
Edgar JMP(92);
Thompson CQG(93), JGP(96);
Hall & Lonie JPA(06)gq/05 [Lorentzian,
4D]; Cocos JGP(06)
[condition for symmetric connection to be Levi-Civita]; Atkins mp/06;
Atkins a0804 [connections that are only locally Levi-Civita].
@ Non-commutative geometry: Dubois-Violette
et al LMP(95)ht/94 [quantum
plane]; Madore et al CQG(95)ht/94 [matrix
geometries]; Madore CQG(96)ht/95;
Mourad CQG(95)ht/94;
Sitarz ht/95;
Masson & Serié JMP(05)
[invariant].
@ Other generalizations: Milani & Shafei Deh Abad LMP(97)ht/96 [quantized
manifolds, geodesics]; Sobreiro & Vasquez Otoya a0711-in
[non-metricity in gravity theories]; Casanova et al MPLA(08)-a0712 [with
torsion and non-metricity, Schouten classification].
@ And field theory: Barut et al HPA(93);
in DeWitt & Molina-París MPLA(98)ht [Vilkovisky
connection on space of histories].
main page – abbreviations – journals – comments – other
sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified
9 aug 2009