Connections in Fiber Bundles

Let us consider consider the fiber bundle \(\pi : Y \to X\) where \(X\) is a line and \(Y\) is a plane. We draw \(X\) curved so that the tangent lines become more apparent:

In the figure above, the fibers \(Y_x\) of \(Y\) are in grey; the tangent space \(T_yY\) is drawn with the canonical vertical subspace \(V_yY\) visible, and the dotted horizontal line is the missing horizontal space.

This is the important part: while there is a well-defined vertical subbundle \(VY \coloneqq \ker T\pi\) inside \(TY\), there is no horizontal subbundle. The reason is that in the holonomic coordinates \((\dot{x}, \dot{y})\) of the fiber coordinates¹ \((x, y)\) the equation \(\dot{x} = 0\) is well-defined (preserved under change of fiber coordinates) but the equation \(\dot{y} = 0\) is not, as I explain later.

Suppose we had such a horizontal subbundle \(HY\), what could we do with it? If we had a vector field \(\tau\) on \(X\), we could lift it to a vector field \(\Gamma\tau\) on \(Y\):

It is not obvious from the figure² above but the lift is performed such that it projects back down to \(\tau\) via \(T\pi\). Since \(T\pi : (\dot{x}, \dot{y}) \mapsto \dot{x}\), the form the vector field must take on \(Y\) is

\begin{align} \Gamma : \partial_x & \mapsto \partial_x + C \partial_y, & C \in \mathbb{R}. \end{align}

This is all there is to the definition of a connection (from the perspective of horizontal bundles) with two minor details:

1. There should be a transformation law for \(\Gamma\) so that \(C\) stays invariant between different fiber coordinates.
2. Typical fiber bundles have larger dimensions with coordinates \((x^\lambda, y^i)\) and thus instead of the term \(C\partial_y\) we have \(\Gamma^i_\lambda \partial_i\) involved.³

Connections are typically introduced as a means to differentiate vector fields. If \(s : X \to Y\) is a section and \(\tau : X \to TX\) a vector field, how do we form \(\nabla_\tau s\)? The formula is:

\begin{align} \nabla_\tau s & = (\partial_\lambda s^i - \Gamma^i_\lambda)\tau^\lambda \partial_i, \end{align}

where \(\Gamma^i_\lambda\) is evaluated at \(s(x)\) (the minus sign is a convention and you may encounter it in either direction.) The result lies in the vertical subbundle \(VY\). When \(Y\) is a vector bundle, there is a canonical identification of \(VY\) with those of \(Y\), and the result can be said to be a section \(\nabla_\tau s : X \to Y\). The term \(\partial_\lambda s^i\) is the only term with first derivatives; other term is a zero-order correction without which the expression would not be coordinate-invariant: if a frame \(e_i\) is given for a vector bundle and a path is given by \(y(t) \coloneqq y^i(t) e_i(t)\), then the derivative \(y'(t)\) contains zero-order terms of the form \(y^i(t)e'_i(t) = y^i(t)\omega_i^j(t)e_j(t)\) which are going to be different for different coordinates; hence the zero-order correction is required.

Using the covariant derivative we can introduce parallel transport as the path of \(Y\) solving the equation \(\nabla_{\frac{d\gamma}{dt}} s = 0\) where \(s(0) = s_0 \in T_{y_0}Y\) and \(\gamma(t)\) is a path on \(X\) with \(\gamma(0) = \pi(y_0)\).

We can also view parallel transport from the perspective of horizontal bundles; \(\frac{d\gamma}{dt}\) lifts to a path \(\Gamma\frac{d\gamma}{dt}\) on \(TY\). The solution \(s(t)\) to the equation \(\frac{ds}{dt} = \Gamma\frac{d\gamma}{dt}\) is the parallel transport of \(s_0\) along \(\gamma\).

With this example we can arrive at a line bundle: if \(Y \to X\) is our fiber bundle and \(\gamma : I \hookrightarrow X\) is our path, then there is a new fiber bundle \(\gamma^*Y \to I\) with connection \(\gamma^*\Gamma\) given by \((\gamma^*\Gamma)\partial_t = \Gamma\frac{d\gamma}{dt}\), or in coordinates \(\partial_t \mapsto \partial_t + \Gamma^i_\lambda\frac{d\gamma^\lambda}{dt}\partial_i\). A common notation for the covariant derivative \(\nabla_{\frac{d\gamma}{dt}}\) of \(\gamma^*\Gamma\) is \(\frac{D}{dt}\).

Before demonstrating the answer, let me explain the reason, which is invariance. What does invariance mean? All objects in Mathematics eventually become numbers, (to which I also offer the following quote):

The older I get, the more I believe that at the bottom of most deep mathematical problems there is a combinatorial problem. —Israel M. Gelfand, Lecture to Courant Institute (1990)

How does this apply for the theory of manifolds? For example, vectors are a set of coordinates, with each coordinate being a number at every point of a coordinate chart. Invariance then becomes the pointwise agreement of these numbers for different charts. This is what the following transition function formula states:

\begin{align} \label{eq:holonomic-transition} \dot{z}'^\lambda & = \frac{\partial z'^\lambda}{\partial z^\mu}\dot{z}^\mu. \end{align}

Invariance can take other more global forms (i.e. not pointwise) such as for example the minimal number of critical points of any function \(f\in C^\infty(M)\) for a given manifold \(M\), or the de Rham cohomology class of some differential form. Those too are just numbers when all abstraction is eliminated, but the process by which they are obtained is different than evaluating functions at a coordinate chart point.

The formula in \eqref{eq:holonomic-transition} is the transition function formula for the holonomic coordinates of \(TZ\) for a manifold \(Z\) (see [1] Ch. 1.) Applied to \(TY\) for a fiber bundle \(Y \to X\) we obtain:

\begin{equation} \label{eq:vert-bundle-coords} \begin{aligned} \dot{x}'^\lambda & = \frac{\partial x'^\lambda}{\partial x^\mu}\dot{x}^\mu + \cancel{\frac{\partial x'^\lambda}{\partial y^i}}\dot{y}^i, \\ \dot{y}'^i & = \frac{\partial y'^i}{\partial x^\mu}\dot{x}^\mu + \frac{\partial y'^i}{\partial y^j}\dot{y}^j. \end{aligned} \end{equation}

Note that the cancelled term is because the \(x\)-coordinates are independent of changes in the fibers in tangent bundles. The definition of the vertical bundle \(VY\) is given by \(\dot{x}^\lambda = 0\) locally. To prove that this is an invariant definition, we must prove that if \(\dot{x}^\lambda = 0\) holds in one chart, then \(\dot{x}'^\lambda = 0\) holds in another (the pointwise equality of numbers I alluded to above). Plugging in \(\dot{x}^\lambda = 0\) in \eqref{eq:vert-bundle-coords}, we obtain the following system:

\begin{equation} \begin{aligned} \dot{x}'^\lambda & = 0, \\ \dot{y}'^i & = \frac{\partial y'^i}{\partial y^j}\dot{y}^j. \end{aligned} \end{equation}

The important point here is that the equality \(\dot{x}'^\lambda = 0\) holds in the new chart. On the other hand, setting for example \(\dot{y}^i = 0\), which would be a guess for the a definition of the horizontal bundle, the following equations are obtained instead:

\begin{equation} \begin{aligned} \dot{x}'^\lambda & = \frac{\partial x'^\lambda}{\partial x^\mu}\dot{x}^\mu, \\ \dot{y}'^i & = \frac{\partial y'^i}{\partial x^\mu}\dot{x}^\mu. \end{aligned} \end{equation}

In these new coordinates, the equality \(\dot{y}'^i = 0\) is not preserved, and this is exactly the reason why a definition of the horizontal bundle by \(\dot{y}^i = 0\) is not well-defined.