Associated vector bundles and frame bundles

Define $P\times_\rho V$ by the relation $(pg,v)\sim(p,\rho(g)v)$. Equivalently, take the quotient of the diagonal right action $(p,v)\cdot g=(pg,\rho(g)^{-1}v)$. These are the same relation; quoting both prevents an inverse from disappearing between sources. Local principal sections identify the quotient with $U_i\times V$, so it is a vector bundle over the base. Addition and scalar multiplication are performed in one fiber and are independent of the chosen representative.

Write an associated section locally as $[s_i(x),v_i(x)]$. On an overlap, $s_j=s_i g_{ij}$ makes $[s_j,v_j]=[s_i,\rho(g_{ij})v_j]$. Consequently the component column in chart $i$ equals the transition matrix times the column in chart $j$. This is a relation between two coordinate descriptions of one vector, not an active rotation of two different vectors. The distinction later determines both the covariant derivative and the gauge-transformation law.

For a rank-$r$ vector bundle $E$, a frame at $x$ is a linear isomorphism $u:\mathbb K^r\to E_x$. The frame bundle has right action $u\cdot g=u\circ g$. Its standard associated bundle maps to $E$ by $[u,v]\mapsto u(v)$. The quotient relation makes this well-defined because $(u\circ g)(v)=u(gv)$. Local frames show that it is a smooth vector-bundle isomorphism. No global frame is presumed in this reconstruction.

A fiber metric reduces real frames to orthonormal frames and complex frames to unitary frames. Such metrics exist over our paracompact manifold bases by partitions of unity, but choosing one is additional geometric data. An orientation reduces a real structure group from $GL(r,\mathbb R)$ to its positive-determinant subgroup. It does not generally reduce it to the identity group or produce a global frame. The tangent bundle of the sphere is an immediate warning that oriented bundles can remain nontrivial.

A section can also be represented by a smooth function $f:P\to V$ satisfying $f(pg)=\rho(g)^{-1}f(p)$. The point $[p,f(p)]$ is then independent of the frame $p$ in a given fiber. Conversely, represent a section uniquely in each frame to obtain such a function. This formulation lets differentiation on the principal bundle produce tensorial operations downstairs. An arbitrary function lacking this equivariance does not define a section, even if its values look plausible in a chosen trivialization.

Dual representations act by inverse transpose, and determinant representations record volume scaling. Tensor products of representations produce tensor-product bundles. For a complex line bundle, tensor power multiplies clutching winding, while taking the dual reverses it. These constructions need not change the base space or the underlying principal bundle, but they change the representation and therefore the resulting associated object. A representation with a nontrivial kernel can even forget part of the principal topology, so reconstructing the principal bundle from one associated bundle is not automatic.