Connections as horizontal distributions

The vertical space at a point of $P$ is the kernel of the differential of the projection. A principal connection chooses a smooth complement $H_p$ that is carried to $H_{pg}$ by right translation. Equivalently, it is a Lie-algebra-valued one-form $\omega$ that returns $\xi$ on the fundamental vector generated by $p\exp(t\xi)$ and transforms equivariantly under right multiplication. Its kernel is precisely the horizontal distribution. The normalization identifies vertical motion with the correct Lie-algebra element.

Pulling $\omega$ back along a section gives a local potential $A_i=s_i^*\omega$. Moving from $s_i$ to $s_j=s_i g_{ij}$ introduces both conjugation and the derivative of the changing frame. Thus a connection form does not transform like a tensor. Dropping the derivative term would make ordinary derivatives of sections disagree on overlaps. A global principal form always exists for a connection, but a single base-valued matrix potential requires a global trivialization.

For the standard matrix representation, define the local covariant derivative of a component column by $\nabla v_i=dv_i+A_i v_i$. Since $v_i=g_{ij}v_j$, the connection overlap law makes $\nabla v_i=g_{ij}\nabla v_j$. For a general representation, replace $A_i$ by its image under the differential representation. This compatibility, together with the Leibniz rule, makes the derivative intrinsic even though its separate terms depend on the local frame. It is a differential operator, not ordinary fiberwise multiplication.

Two connections differ by an adjoint-bundle-valued one-form: their inhomogeneous derivative terms cancel. Consequently the space of connections on a fixed bundle is affine. It has no preferred zero unless extra data select one. Convex combinations with coefficients summing to one remain connections, and a subordinate partition of unity glues local choices. These existence statements use paracompactness. They should not be confused with a claim that a globally flat connection exists on every bundle; flatness is a further nonlinear condition.

Along a path in one trivializing chart, write a principal lift as $p(t)=s(\gamma(t))g(t)$. Horizontality is equivalent to the matrix equation below. The same equation transports associated columns through $v(t)=U(t)v(0)$. Principal connections admit transport over each finite piecewise smooth path; local ordinary differential equations patch uniquely, and the group structure prevents a finite-path escape of the principal lift. This conclusion is special to the compatible principal connection, not an unrestricted completeness assertion for every Ehresmann connection.

For a constant commuting matrix potential, transport is an ordinary exponential. For a varying noncommuting potential, writing the exponential of the integrated matrix generally fails; time ordering records the successive multiplication. A change of frame changes endpoint matrices, while the underlying map between fibers remains the same. This is a useful practical test of a formula: a purported transport law must both solve its differential equation and transform correctly at the two endpoints.