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,
Γ^{m}_{ab}
= Γ^{m}_{ba}, and
(ii) Parallel transport preserves the metric, *g*_{ab;m} = 0.

$ __Riemann-Cartan connection__:
The 1-form

*ω*_{i}^{ j}
:= Γ_{i}^{ j}
− *K*_{i}^{ j} ,

where Γ is the Levi-Civita connection 1-form, and *K* the contorsion,
with *T*^{ i}
= *θ*^{ j}∧
*K*_{j}^{i}.

* __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,

d*θ* + *ω* ∧ *θ* = *T* ,
Ω = d*ω* + *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*}
= \(1\over2\)(∂_{b}*g*_{ac}
+ ∂_{a}*g*_{bc} −
∂_{c}*g*_{ab}).

$ __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
*g*_{ab} are expressed by

Γ^{a}_{bc}
= \(\{{a \atop bc}\}\) = \(1\over2\)*g*^{ad}
(*g*_{bd,c}
+ *g*_{dc,b}
− *g*_{bc,d}) .

* __Gauge transformation__: For
an infinitesimal diffeomorphism generated by *v*^{a},
δΓ^{m}_{ab}
= ∇_{a}∇_{b}
*v*^{m}d −
*R*^{m}_{(ab)c}
*v*^{c}.

**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, *γ*_{F(M)};
Fix a vector *X* ∈ *T*_{x}(*M*);
Then the parallel transport of *X* is

*γ*_{T(M)}(*t*) :=
*γ*_{F(M)}(*t*) *X*.

* __Covariant derivative__:
Defined in T(*M*) of *Z*
= *γ*_{T(M)}(*t*)
= *b*^{m}
(∂/∂*x*^{m})
with respect to *Y* = *c*^{m}
(∂/∂*x*^{m}) as

∇_{Y }*Z*:=
(d/d*t*)*γ*_{T(M)}
= (d*x*^{m}(*t*)/d*t*)
*D*_{m}
*γ*_{F(M)}(*t*)
*X* ,

where *D*_{m} is the
covariant derivative in F(*M*); One finds

∇_{Y} *Z*
= *c*^{n}
*b*^{m}_{;n}
∂/∂*x*^{m} ,
*b*^{m}_{;n}
:= *b*^{m}_{,n}
− *A*^{m}_{ns}
*b*^{s},
*A*^{m}_{ns}
= Γ^{m}_{ns} .

* __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*^{−1}d*h*.

* __On a Riemannian manifold__: The connection can be written as

Γ^{a}_{bc} = \(\{{a \atop bc}\}\)
+ *K*^{a}_{bc} ,

where \(\{{a \atop bc}\}\) are the Christoffel symbols,
*Q*^{a}_{bc}
= *K*^{a}_{[bc]} the
torsion, and *M*^{a}_{bc}
= *K*^{a}_{(bc)} the
non-metricity.

* __Weyl vector__: The trace of the non-metricity,
*B*^{a}:=
*M*^{a}_{bc}
*g*^{bc}.

**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. 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-metricity__: Casanova et al MPLA(08)-a0712 [and torsion, Schouten classification];
Foster et al PRD(17)-a1612 [constraints from Lorentz violation].

@ __Non-metricity and gravity__:
Sobreiro & Vasquez Otoya BJP(10)-a0711;
Mol AACA(17)-a1406 [and general relativity];
Järv et al PRD(18)-a1802 [non-metricity formulation of general relativity and scalar-tensor theory];
Rünkla & Vilson PRD(18)-a1805 [scalar-nonmetricity theories of gravity];
Lazkoz et al a1907 [observational constraints];
> s.a. brans-dicke theory; torsion in physical theories.

@ __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 a1712.

@ __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].

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