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1. Connection

   Linked via "vector bundle"

   Mathematical Definition and Connection Theory
   Formally, connection theory centers on defining a connection $\nabla$ on a vector bundle $E \to M$ over a smooth manifold $M$. This structure allows for the definition of parallel transport. If a vector $v$ at a point $x \in M$ is parallel-transported along a smooth path $\gamma: [0, 1] \to M$ such that $\gamma(0) = x$ and $\gamma(1) = y$, the resulting vector $T\gamma v$ at $y$ is its parallel transport. The transformation $P…

2. Exterior Derivative

   Linked via "vector bundle"

   Connection to Affine Structures
   The operation $\text{d}$ is closely related to connections in an Affine Connection (or more generally, a principal bundle connection). While the standard exterior derivative operates intrinsically on forms, the exterior covariant derivative $D$ incorporates the geometry of the connection $\nabla$ when considering forms with values in a vector bundle. The f…

3. Gauge Theory

   Linked via "vector bundles"

   Gauge theory is a mathematical framework originating in differential geometry that underpins the description of fundamental physical interactions. At its core, gauge theory formalizes the principle that the physical laws should remain unchanged (invariant) under certain local transformations of the fields describing the system. These transformations are known as gauge transformations, and the associated fields required to maintain this invariance are termed [ga…