What follows is a key example to understand gauge theory, principal bundles, connections... and so on.
Suppose we live in a one dimensional world $M=\mathbb{R}$, and that we have a clock in every point of this space. We want to keep track of the movement of an object moving through this world, and we will use the clocks to annotate at what moment the object passes through each point. For intuition, you can think of our object as the shadow of an airplane cast onto a piece of the Earth's surface small enough to look flat to us. It may seem like an overly idealized example, but something similar appears in reality in the case of the description of a solar eclipse, propagating through different time zones of Earth. In our model, however, we are going further and we assume "infinitesimal time zones".
Each clock will be of an special type, idealized, consisting of an infinite length stick with small evenly spaced lines, a little marker moving uniformly along it meaning the passage of time. We are not concerned with how this watch works, we assume its existence.
First we are going to analyse the situation on a single point, with one of these stick-clocks, which we will call $E$. In order to take advantage of this device we need draw evenly spaced lines, and highlight one of them as the origin, making the stroke thicker. To measure time, we count the number of lines between the moving marker and the origin. This way, we have established a bijection from $\mathbb{R}$ to
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Author of the notes: Antonio J. Pan-Collantes
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