Curvature, the Bianchi identity, and holonomy

Applying $\nabla=d+A$ twice to a column-valued zero-form cancels the terms containing its first derivative and leaves multiplication by $F=dA+A\wedge A$. This two-form transforms by conjugation under a frame change, so it is an adjoint-bundle-valued object. In coordinates its components contain derivatives of the potential and the commutator of its coefficient matrices. For a line bundle the commutator term vanishes, but that simplification must not be imported into a higher-rank calculation.

The covariant derivative of an endomorphism-valued $k$-form is $D_A\eta=d\eta+A\wedge\eta-(-1)^k\eta\wedge A$. Expanding the curvature formula gives the Bianchi identity. In degree two its final sign is minus. This identity is structural, not an extra field equation imposed on a connection. It explains why invariant polynomials of curvature will become closed forms: the apparently nonzero ordinary derivative terms cancel after the invariant pairing is taken.

For the transport convention $\dot U=-A(\dot\gamma)U$, a small positively oriented coordinate rectangle has transport $I-F(X,Y)\,\text{area}$ to leading order in a fixed frame. The sign is tied to this transport convention and the orientation of the traversal. Holonomy at a chosen principal frame consists of all resulting group elements from based loops. Replacing that frame conjugates the group. Statements about the actual element therefore need a base frame, while conjugacy classes have a more intrinsic meaning.

A flat connection is locally gauge-equivalent to the zero potential on sufficiently small simply connected neighborhoods. Globally, its holonomy can factor through the fundamental group and remain nontrivial. On a connected simply connected base, flatness does force trivial loop transport. The restricted holonomy subgroup uses null-homotopic loops. For smooth principal connections, the Ambrose–Singer theorem identifies its Lie algebra through curvature values transported back from accessible points. This is a statement about an immersed Lie subgroup, not necessarily a closed subgroup of the structure group.

Consider the globally defined one-form $A=i a\,d\theta$ on a trivial circle line bundle, where $d\theta$ denotes the invariant angular one-form. Its curvature is zero, but one positive circuit has transport $e^{-2\pi i a}$. Integer values of $a$ give identity transport; noninteger values generally do not. If $a$ is irrational, the cyclic subgroup generated by this transport is dense in the circle. Its closure has positive-dimensional Lie algebra even though the flat connection has zero restricted holonomy algebra.

This example is a safeguard when using holonomy as a global diagnostic. Taking a closure changes the group under discussion, and curvature alone does not compute every component of the full group. Conversely, finding one loop with nonidentity transport does not by itself establish nonzero curvature, because topology may account for it. A useful analysis specifies the base, the connection regularity, the allowed loops, and whether holonomy means the immersed subgroup or its closure before drawing an infinitesimal conclusion.