I admit I am not a differential geometer (a probabilist actually). However recently I get interested and I would like to have more intuitions and insight of what is the Riemann curvature.

This is the way I see it so far (please correct me if I am wrong):

- We start from a connection $\nabla$.
- This defines a parallel transport along paths $\gamma$ that is a linear application on the vector bundle.
- If $\gamma$ is a loop, this application may be different to the identity.
- The Riemann curvature is different from the identity when $\gamma$ is a very small loop (at first order).

This kind of definition seems very similar to the one of the rotational (as presented in physics classes).

We start from a vector field (one form) $u$,

If $\gamma$ is loop, $I=\oint_{\gamma}u\cdot d\gamma$ can be different to $0$.

The rotation $\operatorname{rot}(u)$ is this value $I$ when $\gamma$ is a very small loop (at first order).

and we have the wonderful Stokes Theorem which for the $\operatorname{rot}$ follows very naturally from this definition (we glue small loops together to get a big loop) $$\oint_\gamma u \cdot d\gamma = \iint_\mathcal{S}\operatorname{rot}(u)\cdot d\sigma $$ with $\mathcal{S}$ a surface delimited by $\gamma$.

So here is my question: Does there exist an equivalent for the Riemann curvature? That is: can one calculate the parallel transport of $\nabla$ along a loop $\gamma$ from the Riemann curvature on $\mathcal{S}$?