Fundamental principle of Riemannian holonomy
= Fundamental principle of Riemannian holonomy
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.