Given a manifold $M$ with its tangent space $TM$ and frame vector field $e \in TM$. However, the transition functions in this tangent bundle are non-smooth. Therefore, the Lie derivative of $e$ with respect to a direction $X$ is given by the equation ($x \in M$):
$L_Xe(x)+e(x+X_+)-e(x-X_-) = L_Xe(x) + J_Xe(x) = 0$.
Choosing $X := X_-+X_+$ sufficiently small it follows that $J_X$ is a jump operator, a derivative which measures the change of $e$ after parallel transporting this field along a singularity. $X_+$ is the amount of $X$ lying at the singularity surface with positive normal vector and $X_-$ the corresponding amount on the surface with negative normal vector.
By dividing above equation by the length of $X$ (which is an infinitesimal number) it follows:
$\nabla_Xe + |X|^{-1}J_Xe= 0$.
The term $|X|^{-1}J_X$ is not a connection (because it tends to infinity); it is a general manifold operator! However I can obtain the holonomy that reads:
$Hol_M(x) = Pexp(- \oint_C (J_X)dt_X)$.
Here, $t_X$ is the unit tangent vector along $X$ and $P$ is the path-ordering operator that orders products by the path $C$. For obtaining the curvature tensor from holonomy it is usually considered that $C$ is an infinitesimal parallelogram. If I do this and order products by path I get:
$Hol_M(x) = exp(- J_{X_4}(x_4)\Delta t_{X_4})exp(- J_{X_3}(x_3)\Delta t_{X_3})$
$\times exp(- J_{X_2}(x_2)\Delta t_{X_2})exp(- J_{X_1}(x_1)\Delta t_{X_1}) (*)$.
The points $x_1,\dots,x_4$ and vector fields $X_1, \dots, X_4$ are ordered along the edges on which the parallelogram is traversed. Now I have to complete the whole exponential product because the operators $\Delta t_{X_i}J_{X_i}$ can have finite values (no infinitesimal values that allow much simplifications).
Is the holonomy operator $(*)$ really the "curvature tensor" for this manifold (after subtracting the 1 which is the trivial element of the holonomy group and eventually dividing by the area of the parallelogram?). Or can I formulate the generalized curvature tensor something mor simple?
If I am assume $J_XJ_X=0$ (i.e. there are no jumps of second order in the same direction) the holonomy would simplify enormously. Is it plausible to use such an assumption?
