Parallel transport compares fibers along a path. Transport around loops need not return a vector unchanged; the resulting holonomy group records the global and infinitesimal twisting of a connection.
Parallel transport
For a connection \(\nabla\) on a vector bundle \(E\to M\), a section \(V(t)\in E_{\gamma(t)}\) is parallel when \[ \frac{DV}{\mathrm{d}t}=0. \] Linear ODE theory yields an isomorphism \[ P_\gamma:E_{\gamma(0)}\to E_{\gamma(1)}. \] It satisfies \[ P_{\gamma_2*\gamma_1}=P_{\gamma_2}P_{\gamma_1},\qquad P_{\gamma^{-1}}=P_\gamma^{-1}, \] and is invariant under orientation-preserving reparametrization.
Holonomy groups
Definition. The holonomy group at \(p\) is \[ \operatorname{Hol}_p(\nabla) =\{P_\gamma:\gamma\text{ is a piecewise smooth loop based at }p\} \subset\operatorname{GL}(E_p). \] The restricted holonomy group uses loops homotopic to the constant loop.
Changing the base point along a path conjugates holonomy groups. For a metric connection, holonomy lies in \(O(n)\); on an oriented Riemannian manifold it lies in \(SO(n)\).
Curvature as infinitesimal holonomy
Proposition. Parallel transport around a small coordinate parallelogram spanned by \(X,Y\) satisfies \[ P_{\partial\Box_\varepsilon} =I-\varepsilon^2R(X,Y)+o(\varepsilon^2) \] up to the stated curvature-sign convention.
Curvature is therefore the second-order failure of transport around an infinitesimal loop to close. Flatness \(R=0\) implies path independence only locally; global holonomy can remain on a non-simply-connected base.
Theorem (Ambrose-Singer). The Lie algebra of the restricted holonomy group is generated by curvature endomorphisms transported back to the base point: \[ P_\gamma^{-1}R_q(X,Y)P_\gamma, \] where \(\gamma\) runs from \(p\) to \(q\).
Flat connections and monodromy
Proposition. A flat connection on a simply connected manifold admits a global parallel frame. On a general connected manifold, a flat bundle determines a representation \[ \rho:\pi_1(M,p)\to\operatorname{GL}(E_p) \] by parallel transport, well-defined up to conjugacy.
Proof. Flatness makes parallel transport invariant under endpoint-preserving homotopy. On a simply connected base, transport a basis at \(p\) to every point; path independence makes the result well-defined and parallel. In general, transport along loops factors through their homotopy classes and composition of loops gives the representation.
Reduced holonomy and geometry
A tensor is parallel if and only if its value at one point is fixed by holonomy. Thus reductions of holonomy produce global geometric structures: \[ \begin{array}{c|c} \operatorname{Hol}(g)\subset U(n)&\text{parallel complex structure; K?hler geometry},\\ \operatorname{Hol}(g)\subset SU(n)&\text{parallel complex volume; Calabi-Yau geometry},\\ \operatorname{Hol}(g)\subset G_2&\text{parallel stable 3-form in dimension 7},\\ \operatorname{Hol}(g)\subset \operatorname{Spin}(7)&\text{parallel 4-form in dimension 8}. \end{array} \] Berger's classification shows that irreducible nonsymmetric Riemannian holonomy groups are severely restricted. Holonomy therefore turns a differential equation \(\nabla T=0\) into a representation-theoretic constraint.
Gauss-Bonnet through holonomy
Example. On an oriented surface, parallel transport around a small region rotates a tangent vector by approximately \(\int K\,\mathrm{d}A\). Boundary turning contributes the geodesic curvature. Their global balance is \[ \int_MK\,\mathrm{d}A+\int_{\partial M}k_g\,\mathrm{d}s =2\pi\chi(M), \] linking infinitesimal holonomy, boundary geometry, and topology.