an associated bundle

Once you have a principal G-bundle — a bundle of symmetries — you can manufacture a companion bundle out of any space the group acts on. The principal bundle supplies the twisting (how fibers are glued as you move around), and a chosen action of G on a model fiber F supplies the contents. The result is the associated bundle: it twists exactly like P but its fibers are copies of F. From one frame bundle you thereby get the tangent bundle, all the tensor bundles, the density bundle, spinor bundles, and more — all sharing P's gluing.

Precisely: given a principal G-bundle P -> M and a left action of G on a space F, form P times F and quotient by the diagonal G-action (p, f) ~ (p.g, g^{-1}.f). The quotient, written P times_G F, is a fiber bundle over M with fiber F, and its transition functions are the images of P's transition functions g_{ij}: U_i cap U_j -> G under the action on F. When F = R^k and G acts by a linear representation rho: G -> GL(k), the associated bundle is a vector bundle — this is how representations of the structure group turn into vector bundles. The twist 'g^{-1}' in the gluing is what makes sections of the associated bundle correspond to equivariant functions P -> F, the clean way to compute with them.

This construction is the engine that produces every geometric object from one principal bundle: the standard representation of GL(n) gives TM, its dual gives T*M, tensor powers give tensor fields, the determinant representation gives the density/orientation line, and for a spin structure the spin representation gives spinor fields. In gauge theory, matter fields are sections of bundles associated to the gauge group's representations. The slogan: a principal bundle plus a representation equals a vector bundle, and turning the crank of associated bundles is how a single symmetry bundle generates the whole zoo of physical and geometric fields.