Editing Connection form (section)

==Vector bundles==
{{see also|Connection (vector bundle)}}

===Frames on a vector bundle===
{{main|Frame bundle}}
Let <math>E</math> be a [[vector bundle]] of fibre dimension <math>k</math> over a [[differentiable manifold]] <math>M</math>.  A '''local frame''' for <math>E</math> is an ordered [[basis of a vector space|basis]] of [[section (fiber bundle)|local sections]] of <math>E</math>.  It is always possible to construct a local frame, as vector bundles are always defined in terms of [[local trivialization]]s, in analogy to the [[atlas (topology)|atlas]] of a manifold. That is, given any point <math>x</math> on the base manifold <math>M</math>, there exists an open neighborhood <math>U \subseteq M</math> of <math>x</math> for which the vector bundle over <math>U</math> is locally trivial, that is isomorphic to <math>U \times \mathbb R^k</math> projecting to <math>U</math>. The vector space structure on <math>\mathbb R^k</math> can thereby be extended to the entire local trivialization, and a basis on <math>\mathbb R^k</math> can be extended as well; this defines the local frame. (Here the real numbers are used, although much of the development can be extended to modules over rings in general, and to vector spaces over complex numbers <math>\mathbb C</math> in particular.)

Let <math>\mathbf e = (e_\alpha)_{\alpha = 1, 2, \dots, k}</math> be a local frame on <math>E</math>.  This frame can be used to express locally any section of <math>E</math>.  For example, suppose that <math>\xi</math> is a local section, defined over the same open set as the frame <math>\mathbb e</math>. Then 
:<math>\xi = \sum_{\alpha=1}^k e_\alpha \xi^\alpha(\mathbf e)</math>
where <math>\xi^\alpha(\mathbf e)</math> denotes the ''components'' of <math>\xi</math> in the frame <math>\mathbf e</math>.  As a matrix equation, this reads
:<math>\xi = {\mathbf e}
\begin{bmatrix}
\xi^1(\mathbf e)\\
\xi^2(\mathbf e)\\
\vdots\\
\xi^k(\mathbf e)
\end{bmatrix}=
{\mathbf e}\, \xi(\mathbf e)
</math>

In [[general relativity]], such frame fields are referred to as [[Tetrad formalism|tetrads]]. The tetrad specifically relates the local frame to an explicit coordinate system on the base manifold <math>M</math> (the coordinate system on <math>M</math> being established by the atlas).

===Exterior connections===
{{main|Exterior covariant derivative}}
A [[connection (vector bundle)|connection]] in ''E'' is a type of [[differential operator]]
:<math>D : \Gamma(E) \rightarrow \Gamma(E\otimes T^*M) = \Gamma(E)\otimes\Omega^1M</math>
where Γ denotes the [[sheaf (mathematics)|sheaf]] of local [[section (fibre bundle)|sections]] of a vector bundle, and Ω<sup>1</sup>''M'' is the bundle of differential 1-forms on ''M''.  For ''D'' to be a connection, it must be correctly coupled to the [[exterior derivative]].  Specifically, if ''v'' is a local section of ''E'', and ''f'' is a smooth function, then
:<math>D(fv) = v\otimes (df) + fDv</math>
where ''df'' is the exterior derivative of ''f''.

Sometimes it is convenient to extend the definition of ''D'' to arbitrary [[vector-valued differential form|''E''-valued forms]], thus regarding it as a differential operator on the tensor product of ''E'' with the full [[exterior algebra]] of differential forms.  Given an exterior connection ''D'' satisfying this compatibility property, there exists a unique extension of ''D'':
:<math>D : \Gamma(E\otimes\Lambda^*T^*M) \rightarrow \Gamma(E\otimes\Lambda^*T^*M)</math>
such that
:<math> D(v\wedge\alpha) = (Dv)\wedge\alpha + (-1)^{\text{deg}\, v}v\wedge d\alpha</math>
where ''v'' is homogeneous of degree deg ''v''.  In other words, ''D'' is a [[derivation (abstract algebra)|derivation]] on the sheaf of graded modules Γ(''E'' ⊗ Ω<sup>*</sup>''M'').

===Connection forms===
The '''connection form''' arises when applying the exterior connection to a particular frame '''e'''.  Upon applying the exterior connection to the ''e''<sub>''α''</sub>, it is the unique ''k'' × ''k'' matrix (''ω''<sub>''α''</sub><sup>''β''</sup>) of [[one-form]]s on ''M'' such that
:<math>D e_\alpha = \sum_{\beta=1}^k e_\beta\otimes\omega^\beta_\alpha.</math>
In terms of the connection form, the exterior connection of any section of ''E'' can now be expressed. For example, suppose that ''ξ'' = Σ<sub>''α''</sub> ''e''<sub>''α''</sub>''ξ''<sup>''α''</sup>.  Then
:<math>D\xi = \sum_{\alpha=1}^k D(e_\alpha\xi^\alpha(\mathbf e)) = \sum_{\alpha=1}^k e_\alpha\otimes d\xi^\alpha(\mathbf e) + \sum_{\alpha=1}^k\sum_{\beta=1}^k e_\beta\otimes\omega^\beta_\alpha \xi^\alpha(\mathbf e).</math>

Taking components on both sides,
:<math>D\xi(\mathbf e) = d\xi(\mathbf e)+\omega \xi(\mathbf e) = (d+\omega)\xi(\mathbf e)</math>
where it is understood that ''d'' and ω refer to the component-wise derivative with respect to the frame '''e''', and a matrix of 1-forms, respectively, acting on the components of ''ξ''.  Conversely, a matrix of 1-forms ''ω'' is ''a priori'' sufficient to completely determine the connection locally on the open set over which the basis of sections '''e''' is defined.

====Change of frame====
In order to extend ''ω'' to a suitable global object, it is necessary to examine how it behaves when a different choice of basic sections of ''E'' is chosen.  Write ''ω''<sub>''α''</sub><sup>''β''</sup> = ''ω''<sub>''α''</sub><sup>''β''</sup>('''e''') to indicate the dependence on the choice of '''e'''.

Suppose that '''e'''{{prime}} is a different choice of local basis.  Then there is an invertible ''k'' × ''k'' matrix of functions ''g'' such that
:<math>{\mathbf e}' = {\mathbf e}\, g,\quad \text{i.e., }\,e'_\alpha = \sum_\beta e_\beta g^\beta_\alpha.</math>
Applying the exterior connection to both sides gives the transformation law for ''ω'':
:<math>\omega(\mathbf e\, g) = g^{-1}dg+g^{-1}\omega(\mathbf e)g.</math>
Note in particular that ''ω'' fails to transform in a [[tensor]]ial manner, since the rule for passing from one frame to another involves the derivatives of the transition matrix ''g''.

====Global connection forms====
If {''U''<sub>''p''</sub>} is an open covering of ''M'', and each ''U''<sub>''p''</sub> is equipped with a trivialization '''e'''<sub>''p''</sub> of ''E'', then it is possible to define a global connection form in terms of the patching data between the local connection forms on the overlap regions.  In detail, a '''connection form''' on ''M'' is a system of matrices ''ω''('''e'''<sub>''p''</sub>) of 1-forms defined on each ''U''<sub>''p''</sub> that satisfy the following compatibility condition
:<math>\omega(\mathbf e_q) = (\mathbf e_p^{-1}\mathbf e_q)^{-1}d(\mathbf e_p^{-1}\mathbf e_q)+(\mathbf e_p^{-1}\mathbf e_q)^{-1}\omega(\mathbf e_p)(\mathbf e_p^{-1}\mathbf e_q).</math>
This ''compatibility condition'' ensures in particular that the exterior connection of a section of ''E'', when regarded abstractly as a section of ''E'' ⊗ Ω<sup>1</sup>''M'', does not depend on the choice of basis section used to define the connection.

===Curvature===
{{main|Curvature form}}
The '''curvature two-form''' of a connection form in ''E'' is defined by
:<math>\Omega(\mathbf e) = d\omega(\mathbf e) + \omega(\mathbf e)\wedge\omega(\mathbf e).</math>
Unlike the connection form, the curvature behaves tensorially under a change of frame, which can be checked directly by using the [[Poincaré lemma]].  Specifically, if '''e''' → '''e''' ''g'' is a change of frame, then the curvature two-form transforms by
:<math>\Omega(\mathbf e\, g) = g^{-1}\Omega(\mathbf e)g.</math>
One interpretation of this transformation law is as follows.  Let '''e'''<sup>*</sup> be the [[dual basis]] corresponding to the frame ''e''.  Then the 2-form
:<math>\Omega={\mathbf e}\Omega(\mathbf e){\mathbf e}^*</math>
is independent of the choice of frame.  In particular, Ω is a vector-valued two-form on ''M'' with values in the [[endomorphism ring]] Hom(''E'',''E'').  Symbolically,
:<math>\Omega\in \Gamma(\Lambda^2T^*M\otimes \text{Hom}(E,E)).</math>

In terms of the exterior connection ''D'', the curvature endomorphism is given by
:<math>\Omega(v) = D(D v) = D^2v\, </math>
for ''v'' ∈ ''E'' (we can extend ''v'' to a local section to define this expression).  Thus the curvature measures the failure of the sequence
:<math>\Gamma(E)\ \stackrel{D}{\to}\ \Gamma(E\otimes\Lambda^1T^*M)\ \stackrel{D}{\to}\ \Gamma(E\otimes\Lambda^2T^*M)\ \stackrel{D}{\to}\ \dots\ \stackrel{D}{\to}\ \Gamma(E\otimes\Lambda^nT^*(M))</math>
to be a [[chain complex]] (in the sense of [[de Rham cohomology]]).

===Soldering and torsion===
Suppose that the fibre dimension ''k'' of ''E'' is equal to the dimension of the manifold ''M''.  In this case, the vector bundle ''E'' is sometimes equipped with an additional piece of data besides its connection: a [[solder form]].  A '''solder form''' is a globally defined [[vector-valued form|vector-valued one-form]] θ ∈ Ω<sup>1</sup>(''M'',''E'') such that the mapping
:<math>\theta_x : T_xM \rightarrow E_x</math>
is a linear isomorphism for all ''x'' ∈ ''M''.  If a solder form is given, then it is possible to define the '''[[torsion (differential geometry)|torsion]]''' of the connection (in terms of the exterior connection) as
:<math>\Theta = D\theta.\, </math>
The torsion Θ is an ''E''-valued 2-form on ''M''.

A solder form and the associated torsion may both be described in terms of a local frame '''e''' of ''E''.  If θ is a solder form, then it decomposes into the frame components
:<math>\theta = \sum_i \theta^i(\mathbf e) e_i.</math>
The components of the torsion are then
:<math>\Theta^i(\mathbf e) = d\theta^i(\mathbf e) + \sum_j \omega_j^i(\mathbf e)\wedge \theta^j(\mathbf e).</math>
Much like the curvature, it can be shown that Θ behaves as a [[Covariance and contravariance of vectors|contravariant tensor]] under a change in frame:
:<math>\Theta^i(\mathbf e\, g)=\sum_j g_j^i \Theta^j(\mathbf e).</math>

The frame-independent torsion may also be recovered from the frame components:
:<math>\Theta = \sum_i e_i \Theta^i(\mathbf e).</math>

===Bianchi identities===
The [[Bianchi identities]] relate the torsion to the curvature. The first Bianchi identity states that

:<math>D\Theta=\Omega\wedge\theta</math>

while the second Bianchi identity states that

:<math>\, D \Omega = 0.</math>

===Example: the Levi-Civita connection===
As an example, suppose that ''M'' carries a [[Riemannian metric]]. If one has a [[vector bundle]] ''E'' over ''M'', then the metric can be extended to the entire vector bundle, as the [[bundle metric]]. One may then define a connection that is compatible with this bundle metric, this is the [[metric connection]].  For the special case of ''E'' being the [[tangent bundle]] ''TM'', the metric connection is called the [[Riemannian connection]]. Given a Riemannian connection, one can always find a unique, equivalent connection that is [[torsion tensor|torsion-free]]. This is the [[Levi-Civita connection]] on the tangent bundle ''TM'' of ''M''.<ref>See {{harvtxt|Jost|2011}}, chapter 4, for a complete account of the Levi-Civita connection from this point of view.</ref><ref>See {{harvtxt|Spivak|1999a}}, II.7 for a complete account of the Levi-Civita connection from this point of view.</ref>

A local frame on the tangent bundle is an ordered list of vector fields {{nowrap|1='''e''' = (''e''<sub>''i''</sub> {{!}} ''i'' = 1, 2, ..., ''n'')}}, where {{nowrap|1=''n'' = dim ''M''}}, defined on an open subset of ''M'' that are linearly independent at every point of their domain.  The [[Christoffel symbols]] define the Levi-Civita connection by
:<math>\nabla_{e_i}e_j = \sum_{k=1}^n\Gamma_{ij}^k(\mathbf e)e_k.</math>
If ''θ'' = {{mset|1=''θ''<sup>''i''</sup> {{!}} ''i'' = 1, 2, ..., ''n''}}, denotes the [[dual basis]] of the [[cotangent bundle]], such that ''θ''<sup>''i''</sup>(''e''<sub>''j''</sub>) = ''δ''<sup>''i''</sup><sub>''j''</sub> (the [[Kronecker delta]]), then the connection form is
:<math>\omega_i^j(\mathbf e) = \sum_k \Gamma^j{}_{ki}(\mathbf e)\theta^k.</math>

In terms of the connection form, the exterior connection on a vector field {{nowrap|1=''v'' = Σ<sub>''i''</sub>''e''<sub>''i''</sub>''v''<sup>''i''</sup>}} is given by
:<math> Dv=\sum_k e_k\otimes(dv^k) + \sum_{j,k}e_k\otimes\omega^k_j(\mathbf e)v^j.</math>
One can recover the Levi-Civita connection, in the usual sense, from this by contracting with ''e''<sub>i</sub>:
:<math> \nabla_{e_i} v = \langle Dv, e_i\rangle = \sum_k e_k \left(\nabla_{e_i} v^k + \sum_j\Gamma^k_{ij}(\mathbf e)v^j\right)</math>

====Curvature====
The curvature 2-form of the Levi-Civita connection is the matrix (Ω<sub>''i''</sub><sup>''j''</sup>) given by
:<math>
\Omega_i{}^j(\mathbf e) = d\omega_i{}^j(\mathbf e)+\sum_k\omega_k{}^j(\mathbf e)\wedge\omega_i{}^k(\mathbf e).
</math>
For simplicity, suppose that the frame '''e''' is [[Holonomic basis|holonomic]], so that {{nowrap|1=''dθ''<sup>''i''</sup> = 0}}.<ref>In a non-holonomic frame, the expression of curvature is further complicated by the fact that the derivatives dθ<sup>i</sup> must be taken into account.</ref>  Then, employing now the [[summation convention]] on repeated indices,
:<math>\begin{array}{ll}
\Omega_i{}^j &= d(\Gamma^j{}_{qi}\theta^q) + (\Gamma^j{}_{pk}\theta^p)\wedge(\Gamma^k{}_{qi}\theta^q)\\
&\\
&=\theta^p\wedge\theta^q\left(\partial_p\Gamma^j{}_{qi}+\Gamma^j{}_{pk}\Gamma^k{}_{qi})\right)\\
&\\
&=\tfrac12\theta^p\wedge\theta^q R_{pqi}{}^j
\end{array}
</math>
where ''R'' is the [[Riemann curvature tensor]].

====Torsion====
The Levi-Civita connection is characterized as the unique [[metric connection]] in the tangent bundle with zero torsion.  To describe the torsion, note that the vector bundle ''E'' is the tangent bundle.  This carries a canonical solder form (sometimes called the [[canonical one-form]], especially in the context of [[classical mechanics]]) that is the section ''θ'' of {{nowrap|1=Hom(T''M'', T''M'') = T<sup>∗</sup>''M'' ⊗ T''M''}} corresponding to the identity endomorphism of the tangent spaces.  In the frame '''e''', the solder form is {{nowrap|1=''θ'' = Σ<sub>''i''</sub> ''e''<sub>''i''</sub> ⊗ ''θ''<sup>''i''</sup>}}, where again ''θ''<sup>''i''</sup> is the dual basis.

The torsion of the connection is given by {{nowrap|1=Θ = ''Dθ''}}, or in terms of the frame components of the solder form by
:<math>\Theta^i(\mathbf e) = d\theta^i+\sum_j\omega^i_j(\mathbf e)\wedge\theta^j.</math>
Assuming again for simplicity that '''e''' is holonomic, this expression reduces to
:<math>\Theta^i = \Gamma^i{}_{kj} \theta^k\wedge\theta^j</math>,
which vanishes if and only if Γ<sup>''i''</sup><sub>''kj''</sub> is symmetric on its lower indices.

Given a metric connection with torsion, one can always find a single, unique connection that is torsion-free, this is the Levi-Civita connection. The difference between a Riemannian connection and its associated Levi-Civita connection is the [[contorsion tensor]].