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) Γ has no torsion,
Γmab
= Γmba, and
(ii) Parallel transport preserves the metric, gab;m = 0.
$ Riemann-Cartan connection:
The 1-form
ωi j := Γi j − Ki j ,
where Γ is the Levi-Civita connection 1-form, and K the contorsion,
with T i
= θ 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, θ the
coframe, Ω the curvature 2-form, and ω the connection,
dθ + ω ∧ θ = T , Ω = dω + ω ∧ ω .
> Online resources: For affine connections, see Wikipedia page and Encyclopedia of Mathematics page; for Levi-Civita connection, see Wikipedia page.
Christoffel Symbols
$ First kind: The Christoffel
symbols of the first kind are defined as {ab, c}
= \(1\over2\)(∂bgac
+ ∂agbc −
∂cgab).
$ Second kind: The Christoffel
symbols of the second kind are defined as \(\{{a \atop bc}\} = g^{ad} \{bc, d\}\).
* And connection: Given a choice
of coordinates, the components of the linear connection compatible with a metric
gab are expressed by
Γabc = \(\{{a \atop bc}\}\) = \(1\over2\)gad (gbd,c + gdc,b − gbc,d) .
* Gauge transformation: For an infinitesimal diffeomorphism generated by va, δΓmab = ∇a∇b vmd − 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 γ
through x, and a lift of γ to the frame bundle,
γF(M); Fix a vector
X ∈ Tx(M);
Then the parallel transport of X is
γT(M)(t) := γF(M)(t) X.
* Covariant derivative: Defined in T(M) of Z = γT(M)(t) = bm (∂/∂xm) with respect to Y = cm (∂/∂xm) as
∇Y Z:= (d/dt)γT(M) = (dxm(t)/dt) Dm γF(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 = Γmns .
* 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, Γ = h−1dh.
* On a Riemannian manifold: The connection can be written as
Γabc = \(\{{a \atop bc}\}\) + 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. FLRW models; spherical symmetry;
schwarzschild spacetime; symmetry.
@ References: Armstrong JGP(07) [Ricci-flat, classification];
Guadagnini et al NPB(17)-a1710 [3D, flat].
References
> s.a. holonomy; Non-Metricity
[including in gravity]; 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);
Cocos JGP(06) [condition for symmetric connection to be Levi-Civita];
Atkins mp/06;
Atkins a0804 [connections that are only locally Levi-Civita];
> s.a. formulations of general relativity.
@ Lorentz connections: Hall & Lonie JPA(06)gq/05 [4D];
Pereira AIP(12)-a1210 [in general relativity and teleparallel gravity].
@ 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];
> s.a. types of geodesics.
@ And field theory: Barut et al HPA(93);
in DeWitt & Molina-París MPLA(98)ht [Vilkovisky connection on space of histories].
@ On use in gravitational theories:
Kofinas JModP(20)-a1712;
Queiruga a1912 [as spacetime defects and foam].
@ Other generalizations: Milani & Shafei Deh Abad LMP(97)ht/96 [quantized manifolds, geodesics];
Pham MJM(15)-a1408 [higher affine connections].
> Other generalizations: see finsler
geometry; Lagrange Spaces [non-linear connection];
newton-cartan theory [degenerate metrics].
main page
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– other sites – acknowledgements
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