In the context of differential geometry and algebraic topology, a **connection** on a principal bundle is a mathematical structure that allows one to define and work with notions of parallel transport and differentiability on the bundle. A principal bundle is a mathematical object that consists of a total space \( P \), a base space \( M \), and a group \( G \) (the structure group) acting freely and transitively on the fibers of the bundle.
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A principal connection can be specified by a -equivariant horizontal distribution complementary to the vertical bundle, or equivalently by a Lie-algebra-valued connection form that reproduces infinitesimal generators and has .