Let be a differential manifold, a vector bundle over and a connection for . For a point the holonomy group at is the subgroup of comprising those linear transformations that arise as the holonomy of a loop at .
If the differential manifold is path-connected, the holonomy groups at distinct points are conjugate as subgroups of so we can talk of the holonomy group.
Reduction of structure group to holonomy group
- Pick (arbitrarily) a basis at a particular point
- Now, for each point , the fibre at that point is the set of all bases at that can arise from the basis at by means of transport along a curve using the connection .
- In particular, any two bases at differ by the holonomy of a loop, which lies in . Thus acts freely and transitively on the fibre at each point.
Moreover the holonomy group is the smallest group to which the structure group can be reduced (for the same reasons).