Retrieving "Local Section" from the archives

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1. Principal Bundle

   Linked via "local sections"

   Since a principal bundle is generally not globally trivial (i.e., $\mathcal{P}$ is not necessarily diffeomorphic to $M \times G$), local trivializations are required.
   Over an open cover $\{U\alpha\}$ of $M$, there exist local sections $s\alpha: U\alpha \to \mathcal{P}$ such that $\pi \circ s\alpha = \text{id}{U\alpha}$. This means that for any $x \in U\alpha$, $s\alpha(x)$ is a unique point in the fiber over $x$.
   The overlap between two charts, $U\alpha \cap U\bet…