holonomy
Walk a vector around a closed loop, parallel-transporting it the whole way so you 'never turn it' — and find it comes back rotated from where it started. That net transformation, picked up purely by going around the loop, is the holonomy of the loop. On a flat plane every loop's holonomy is the identity; on a curved space it is not, and the size of the rotation measures how curved the enclosed region is. Holonomy is the global fingerprint of a connection.
Precisely, fix a connection on a bundle over M and a basepoint p. Each loop gamma based at p gives a parallel-transport map of the fiber E_p to itself, an invertible linear map (for a principal G-bundle, an element of the structure group G). The set of all such maps, over all loops at p, is a subgroup of G called the holonomy group Hol_p; restricting to contractible loops gives the restricted holonomy group, which is connected. The Ambrose-Singer theorem ties this to curvature: the Lie algebra of the holonomy group is spanned by all curvature transformations R(X,Y) parallel-transported back to p. For a small loop bounding a surface element, the holonomy is approximately identity plus the curvature integrated over the enclosed area — curvature is infinitesimal holonomy.
Holonomy organizes a huge amount of geometry. Berger's classification lists the possible holonomy groups of irreducible Riemannian manifolds, and special (reduced) holonomy pins down distinguished geometries: U(n) holonomy means Kahler, SU(n) means Calabi-Yau, Sp(n) means hyperkahler, and G_2 or Spin(7) give the exceptional cases central to string theory. A flat connection is exactly one whose restricted holonomy is trivial, equivalent to local parallel frames existing. One caveat to keep straight: the full holonomy group can be disconnected, and its component group reflects the bundle's topology (which loops are contractible), so holonomy mixes local curvature information with global topological information.
On the unit sphere, parallel-transport a tangent vector around a geodesic triangle (say a sector bounded by two meridians and a piece of the equator). It returns rotated by an angle equal to the area enclosed — the spherical excess. For a triangle covering one-eighth of the sphere (area pi/2) the vector returns turned by pi/2, a quarter turn, with no twisting ever applied by hand. That rotation is the holonomy of the loop, and equals the integral of the Gaussian curvature over the enclosed cap.
Holonomy around a loop equals the curvature integrated over a surface it bounds — the Gauss-Bonnet idea in miniature.
Holonomy depends on the connection, not on the bundle alone: the same bundle with two different connections has different holonomy. And the full holonomy group may be disconnected — its identity component (the restricted holonomy) sees only curvature, while the component group encodes global topology (non-contractible loops), so the two carry genuinely different information.