I'm currently trying to relate two descriptions of the curvature and torsion of a connection and am running into some confusion.

I know that an affine connection $A$ on an $n$-dimensional manifold $M$ can be split into two parts $A = \omega + e$, where $\omega$ takes values in the Lie algebra of rotations $\mathfrak{so}(n)$ while $e$ takes values in the Lie algebra of translations $\mathfrak{t}(n)$.

The curvature form $\Omega = d\omega + \omega \wedge \omega$ can then be obtained from the $\mathfrak{so}(n)$ part, and the torsion form $\theta = de + \omega \wedge e$ from the $\mathfrak{t}(n)$ part.

However, I am also aware that Cartan also related the curvature and torsion of an affine connection on $M$ to the older idea of curvature and torsion of curves in $M$ (I assume this is the origin of the word 'torsion' for this quantity?). If you take a curve from $M$ and develop it in a flat Euclidean space, then the curvature and torsion of the connection on $M$ both induce 'extra' curvature and torsion in the developed curve. I think that the modern version of this development would be a horizontal lift of the curve in $M$ into the principal bundle over $M$.

I'm struggling to see how these two ideas fit together. In particular, a curve in $\mathbb{R}^n$ has $n-1$ of these Frenet-Serret invariants, not just curvature and torsion.

What I would like to understand is why only the first two invariants of the curve appear in the affine connection, seeing as there are $n-1$ of these for a curve in $\mathbb{R}^n$. And what does the torsion of a curve have to do with the translation group?

I would really appreciate any help on understanding this, or any reference suggestions.