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Imagine an arbitrary closed loop in some Riemannian manifold $M$ with the Levi-Civita connection $\nabla$. If this curve $\gamma$ were infinitesimal (parametric distance ~ $\epsilon$) and given by 2 vector fields $U,V$, then starting from a point $A$ and parallel transporting some vector $W$ around the loop, we obtain the vector $$ \tau^{\gamma}_{A,A} W = W - \epsilon^2 R(U,V)W + \ldots$$ where $$ R(U,V) \equiv [\nabla_{U}, \nabla_{V}] - \nabla_{[U,V]} $$ is the curvature tensor and $$ \tau^{\gamma}_{A,A}: T_AM \rightarrow T_AM $$ is the operator of parallel transport around $\gamma$.

My question is, is it possible to find such an operator (explicitly) for a general non-infinitesimal loop? I tried to decompose the loop into infinitesimal loops (as if in Kelvin-Stokes theorem) and to construct the operator from them, but I am stuck at how to traverse the infinitesimal loops. I guess the operator should somehow depend on the curvature tensor, since it is an identity for vanishing curvature. What is the dependence?

N.B.: This is not a homework question. I am just curious.

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Not in general. Such map is actually called holonomy. See holonomy.

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