In $\S 3$ of this document the following is stated:

Let $M$ be a $2$-dimensional Riemannian manifold and $R$ denote the curvature tensor. Let $p$ be a point in $M$ and $X_0, Y_0\in T_p M$ be linearly independent. Extend $X_0$ and $Y_0$ locally around $p$ to get commuting vector fields $X$ and $Y$. Thus $R(X, Y)= [\nabla_X, \nabla_Y]$, where $\nabla$ is the Levi-Civita connection. For each $t>0$ small enough, let $\gamma_t$ be the loop formed by flowing for $\sqrt{t}$ time along $X$, then for $\sqrt{t}$ time along $Y$, then flowing for $\sqrt{t}$ time opposite to the flow of $X$, and lastly for $\sqrt{t}$ time opposite to the flow of $Y$. This actually gives a loop because $X$ and $Y$ are commuting.

Then $R(X_0, Y_0) = \lim_{t\to 0}\frac{P_{\gamma_t}-I}{t}$

where $P_{\gamma_t}:T_pM\to T_pM$ is the parallel transport along $\gamma_t$.

It is remarked in the document that this can be proved in a similar manner as one proves the Lie bracket of two vector fields is the Lie derivative of one with respect to the other.

Since $X$ and $Y$ are commuting, we may assume that they are coordinate vector fields. But when trying to write down the coordinate expression for $[\nabla_X, \nabla_Y]$, I end up with a horrible mess featuring the Cristoffel coefficients and their derivatives. I am not able to see how to connect this to the parallel transport.


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