Editing Connection (vector bundle) (section)

==Local expression==
Let <math>E\to M</math> be a vector bundle of rank <math>k</math>, and let <math>U</math> be an open subset of <math>M</math> over which <math>E</math> trivialises. Therefore over the set <math>U</math>, <math>E</math> admits a local [[smooth frame]] of sections

:<math>\mathbf{e} = (e_1, \dots, e_k);\quad e_i: U \to \left.E\right|_U.</math>
Since the frame <math>\mathbf{e}</math> defines a basis of the fibre <math>E_x</math> for any <math>x\in U</math>, one can expand any local section <math>s:U\to \left.E\right|_U</math> in the frame as
:<math>s = \sum_{i=1}^k s^i e_i</math>
for a collection of smooth functions <math>s^1, \dots, s^k: U \to \mathbb{R}</math>.

Given a connection <math>\nabla</math> on <math>E</math>, it is possible to express <math>\nabla</math> over <math>U</math> in terms of the local frame of sections, by using the characteristic product rule for the connection. For any basis section <math>e_i</math>, the quantity <math>\nabla(e_i)\in \Omega^1(U) \otimes \Gamma(U,E)</math> may be expanded in the local frame <math>\mathbf{e}</math> as

:<math> \nabla (e_i) = \sum_{j=1}^k A_i^{\ j} \otimes e_j,</math>

where <math>A_i^{\ j}\in \Omega^1(U);\, j=1,\dots,k</math> are a collection of local one-forms. These forms can be put into a matrix of one-forms defined by

:<math> A = \begin{pmatrix} A_1^{\ 1} & \cdots & A_k^{\ 1} \\ \vdots & \ddots & \vdots \\ A_1^{\ k} & \cdots & A_k^{\ k} \end{pmatrix}\in \Omega^1(U, \operatorname{End}(\left.E\right|_U))</math>

called the ''local connection form of <math>\nabla</math> over <math>U</math>''. The action of <math>\nabla</math> on any section <math>s: U \to \left.E\right|_U</math> can be computed in terms of <math>A</math> using the product rule as

:<math>\nabla(s) = \sum_{j=1}^k \left(ds^j + \sum_{i=1}^k A_i^{\ j} s^i\right) \otimes e_j.</math>
If the local section <math>s</math> is also written in matrix notation as a column vector using the local frame <math>\mathbf{e}</math> as a basis,

:<math> s = \begin{pmatrix} s^1 \\ \vdots \\ s^k\end{pmatrix},</math>
then using regular matrix multiplication one can write
:<math>\nabla(s) = ds + As</math>
where <math>ds</math> is shorthand for applying the [[exterior derivative]] <math>d</math> to each component of <math>s</math> as a column vector. In this notation, one often writes locally that <math>\left.\nabla\right|_U = d+A</math>. In this sense a connection is locally completely specified by its connection one-form in some trivialisation.

As explained in [[#Affine properties of the set of connections]], any connection differs from another by an endomorphism-valued one-form. From this perspective, the connection one-form <math>A</math> is precisely the endomorphism-valued one-form such that the connection <math>\left.\nabla\right|_U</math> on <math>\left.E\right|_U</math> differs from the trivial connection <math>d</math> on <math>\left.E\right|_U</math>, which exists because <math>U</math> is a trivialising set for <math>E</math>.

=== Relationship to Christoffel symbols ===

In [[pseudo-Riemannian geometry]], the [[Levi-Civita connection]] is often written in terms of the [[Christoffel symbols]] <math>\Gamma_{ij}^{\ \ k}</math> instead of the connection one-form <math>A</math>. It is possible to define Christoffel symbols for a connection on any vector bundle, and not just the tangent bundle of a pseudo-Riemannian manifold. To do this, suppose that in addition to <math>U</math> being a trivialising open subset for the vector bundle <math>E\to M</math>, that <math>U</math> is also a [[Topological_manifold#Coordinate_charts|local chart]] for the manifold <math>M</math>, admitting local coordinates <math>\mathbf{x} = (x^1,\dots,x^n);\quad x^i: U \to \mathbb{R}</math>.

In such a local chart, there is a distinguished local frame for the differential one-forms given by <math>(dx^1,\dots,dx^n)</math>, and the local connection one-forms <math>A_i^{ j}</math> can be expanded in this basis as

:<math>A_i^{\ j} = \sum_{\ell=1}^n \Gamma_{\ell i}^{\ \ j} dx^\ell</math>

for a collection of local smooth functions <math>\Gamma_{\ell i}^{\ \ j} : U \to \mathbb{R}</math>, called the ''Christoffel symbols'' of <math>\nabla</math> over <math>U</math>. In the case where <math>E=TM</math> and <math>\nabla</math> is the Levi-Civita connection, these symbols agree precisely with the Christoffel symbols from pseudo-Riemannian geometry.

The expression for how <math>\nabla</math> acts in local coordinates can be further expanded in terms of the local chart <math>U</math> and the Christoffel symbols, to be given by

:<math> \nabla(s) = \sum_{i,j=1}^k \sum_{\ell=1}^n \left(\frac{\partial s^j}{\partial x^\ell} + \Gamma_{\ell i}^{\ \ j} s^i\right) dx^\ell \otimes e_j.</math>
Contracting this expression with the local coordinate tangent vector <math>\frac{\partial}{\partial x^\ell}</math> leads to
:<math> \nabla_{\frac{\partial}{\partial x^\ell}} (s) = \sum_{i,j=1}^k \left(\frac{\partial s^j}{\partial x^\ell} + \Gamma_{\ell i}^{\ \ j} s^i\right) e_j.</math>

This defines a collection of <math>n</math> locally defined operators

:<math>\nabla_\ell: \Gamma(U,E) \to \Gamma(U,E);\quad \nabla_\ell(s) := \sum_{i,j=1}^k \left(\frac{\partial s^j}{\partial x^\ell} + \Gamma_{\ell i}^{\ \ j} s^i\right)e_j,</math>

with the property that

:<math>\nabla(s) = \sum_{\ell=1}^n dx^\ell \otimes \nabla_\ell(s).</math>

=== Change of local trivialisation ===

Suppose <math>\mathbf{e'}</math> is another choice of local frame over the same trivialising set <math>U</math>, so that there is a matrix <math>g=(g_i^{\ j})</math> of smooth functions relating <math>\mathbf{e}</math> and <math>\mathbf{e'}</math>, defined by
:<math>e_i = \sum_{j=1}^k g_i^{\ j} e'_j.</math>
Tracing through the construction of the local connection form <math>A</math> for the frame <math>\mathbf{e}</math>, one finds that the connection one-form <math>A'</math> for <math>\mathbf{e'}</math> is given by
:<math>{A'}_i^{\ j} = \sum_{p,q=1}^k g_p^{\ j} A_q^{\ p} {(g^{-1})}_i^{\ q} - \sum_{p=1}^k (dg)_p^{\ j} {(g^{-1})}_i^{\ p}</math>
where <math>g^{-1} = \left({(g^{-1})}_i^{\ j}\right)</math> denotes the inverse matrix to <math>g</math>. In matrix notation this may be written
:<math>A' = g A g^{-1} - (dg)g^{-1}</math>

where <math>dg</math> is the matrix of one-forms given by taking the exterior derivative of the matrix <math>g</math> component-by-component.

In the case where <math>E=TM</math> is the tangent bundle and <math>g</math> is the Jacobian of a coordinate transformation of <math>M</math>, the lengthy formulae for the transformation of the Christoffel symbols of the Levi-Civita connection can be recovered from the more succinct transformation laws of the connection form above.