Holonomy representation

The holonomy representation is the natural action of the Riemannian holonomy group on $T_{p}M$. It induces representations on cotangent spaces and tensor powers. Irreducibility means that no nonzero proper real linear subspace of $T_{p}M$ is invariant.

A nonzero parallel one-form determines a nonzero parallel vector by metric duality. Its value is fixed by the full Riemannian holonomy group, so its span is an invariant line. This contradicts irreducibility when $\dim M\ge 2$. The dimension condition matters: the standard circle has trivial but irreducible one-dimensional holonomy representation and a nonzero parallel one-form.

Evaluation at a point identifies parallel tensor fields with tensors fixed by the full Riemannian holonomy group. A parallel field is fixed after transport around every loop. Conversely transport a fixed tensor along paths from the basepoint; loop invariance makes the result independent of the path, and local parallel transport shows smoothness and parallelness. Connectedness gives uniqueness.