A principal bundle is a mathematical structure that formalizes the concept of a space that has a group acting freely and transitively on its fibers. In the context of differential geometry, principal bundles are essential for understanding how the geometry of a manifold can be described with additional symmetries through the use of a Lie group. This structure is pivotal in many areas such as gauge theory and the study of connections, which are vital for expressing O'Neill's formulas and their applications.
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