Holonomy of a connection

For a supplied point $p$ and connection on $E$, define ${\mathrm{Hol}}_p(\mathrm{\nabla })\subseteq \mathrm{GL}(E_p)$ to be the set of transports around finite piecewise smooth loops based at $p$. Transports are invertible by Parallel transport is a linear isomorphism. The constant loop gives the identity; if $A=P_{\alpha }$ and $B=P_{\beta }$, traversing $\beta$ then $\alpha$ gives $AB$; reversal of $\alpha$ gives $A^{-1}$. These assertions follow from Parallel transport under reparametrization reversal and concatenation, and prove that this set is a subgroup, with associativity inherited from composition of linear maps.

A singleton constant loop suffices for nonemptiness; in rank zero this group consists of the unique automorphism of the zero vector space. There is no such pointed definition on an empty base without a point $p$. No simultaneous choice of loops representing all elements is required: closure checks two supplied representatives. The definition asserts neither homotopy invariance nor path independence. Curvature descriptions and classification of holonomy are deferred.