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Holonomy group

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Definition

Let M be a differential manifold, E a vector bundle over M and \nabla a connection for E. For a point m \in M the holonomy group at m is the subgroup of GL(E_p) comprising those linear transformations that arise as the holonomy of a loop at m.

If the differential manifold is path-connected, the holonomy groups at distinct points are conjugate as subgroups of GL(E_p) so we can talk of the holonomy group.

Related notions

Facts

Reduction of structure group to holonomy group

Any path-connected differential manifold can be treated as a principal bundle with structure group being the holonomy group. The description is as follows:

  • Pick (arbitrarily) a basis at a particular point p
  • Now, for each point m, the fibre at that point is the set of all bases at m that can arise from the basis at p by means of transport along a curve using the connection \nabla.
  • In particular, any two bases at m differ by the holonomy of a loop, which lies in H. Thus H 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).