Riemannian holonomy group (${\mathrm{Hol}}_p(g)\subset O(T_{p}M,g_p)$)

For a connected Riemannian manifold, the full holonomy group consists of parallel transport maps around all piecewise smooth loops based at $p$. Metric compatibility makes them orthogonal. Basepoint changes conjugate this group by parallel transport. This is full holonomy, not just the subgroup using contractible loops.

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.