Metric Differential Geometry
A principal bundle is a mathematical structure that formalizes the idea of having a space (the total space) that locally looks like a product of a base space and a group manifold. It consists of a total space, a base space, and a structure group, where the fibers over each point in the base space are homeomorphic to the group. This concept is essential for understanding various geometric structures, including Riemannian submersions and gauge theories, as it helps in organizing how different spaces relate to each other through symmetry transformations.
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