Vector bundle connections. Connection form.

 Let \(E \rightarrow M\) a smooth vector bundle on a manifold \(M\). Let \(\Gamma(E)\) be the space of sections. A connection on \(E\) is an \(\mathbb{R}\)-linear map
\[
\nabla : \Gamma(E) \rightarrow \Gamma\left(E \otimes T^{*} M\right)
\]
satisfyng Leibniz rule
\[
\nabla(\sigma f)=(\nabla \sigma) f+\sigma \otimes d f
\]

This way, if \(X\) is a vector field one can define a notion of derivative of a section
\[
\nabla_{X} : \Gamma(E) \rightarrow \Gamma(E)
\]
by means of \(\nabla_{X} \sigma=(\nabla \sigma)(X)\).


If we think of sections like a generalization of functions or maps (I personally call them twisted functions) the connection is a device that let us derive this functions.
 

Connection form

Suppose we choose a particular local frame \(e=\{e_i\}\) (that is, we restrict to an open subset \(U\subseteq M\)). If we apply the connection to every \(e_i\) we obtain
\[
\nabla(e_i)=\sum_j e_i \otimes \omega_i^j
\]

The square matrix \(\omega=(\omega_i^j)\) of 1-forms is called the connection form. If our vector bundle is the tangent bundle, the connection form is not other thing that the usual Christoffel symbols!
 

Given a particular local section of \(E\), say
\[
\sigma= \sum_i \sigma_i e_i
\]
we have
\[
\nabla(\sigma)=\nabla(\sum_i \sigma_i e_i)=\sum_i e_i\otimes d\sigma_i+\sum_{ij}\sigma_i e_j\otimes \omega_i^j
\]
and renaming \(i\) and \(j\) in the second sum
\[
\nabla(\sigma)=\sum_i e_i\otimes\left( d \sigma_i+\sum_j \sigma_j \omega_j^i\right)
\]

So if we call \(\widetilde{\sigma}\) and \(\widetilde{\nabla(\sigma)}\) to the column vector of components of \(\sigma\) and \(\nabla(\sigma)\) respectively we have
\[
\widetilde{\nabla(\sigma)}=(d+\omega)\widetilde{\sigma}
\]
where \(d\) is the differential and \(\omega\) is a matrix of 1-forms.

Observe that an arbitrary matrix of 1-forms is a priori enough to define locally a connection.


If we choose a different frame \(e'=\{e'_i\}\) in \(U\), being
\[
e'=e g
\]
with \(g\) a matrix of smooth functions, it can be show that the new connection form will be
\[
\omega'=g^{-1} dg+g^{-1} \omega g
\]

This let us define a global connection form as a collection of matrices of 1-forms defined in an open covering of \(M\) satisfying certain compatibility condition in the overlaps.


Horizontal subspaces



Consider now the space \(TE\).
For every \(e\in E\), with \(\pi(e)=x\in M\), we have a natural linear space \(T_e E_x\) that is called the vertical space. The union in \(e\in E\) gives us a subbundle \(V\subseteq TE\) called the vertical bundle.

At a first glance, there is no natural choice for an horizontal bundle, but \(\nabla\) determines one. We will define
\[
H_e:=ds_x (T_x M)
\]
where \(s\) is a section such that \(s(x)=e\) and \(\nabla_X s=0\) for every \(X\in T_x M\).

The idea of an horizontal subbundle gives rise to Ehresman connections.


In the next entry we are going to apply these ideas to a very simple model to get a deep understanding of them.


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