Relating curvature and torsion of a connection to those of a curve

Question

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.

Answer 1

This is not really an answer, but I think it might help you look at things a little bit differently. It is not at all clear that Cartan chose the word 'torsion' to describe the 'translation' component of the curvature because it was related to the torsion of a curve in flat space or had anything to do with developing maps associated to what are now called "Cartan connections". In fact, I rather suspect that this is a red herring. I think he chose the term because the geometric picture that he had in mind was connected with something physically 'twisting'.

The first article that he wrote that uses the word 'torsion' in this sense seems to be Sur une généralization de la notion de courbure de Riemann et les espaces à torsion (C. R. Acad. Sci. 174 (1922), 593–595). In that article he tries to give the feature of holonomy associated to his generalized 'connection' (a word he does not use in this article) that corresponds to translation a physical meaning, at least in $3$ dimensions. He explicitly compares 'torsion' (twisting(?)) to 'tension ou pression' (tension or pressure). For example, the first use of the word 'torsion' in the body of the article is in the sentence On a ansi une image géométrique d'un milieu matèrial continu en équilibre sous la seule action de ses forces élastiques, mais dans le cas où ces forces se manifesteraient sur chaque élément de surface, non seulement par une forçe unique (tension ou pression), mais par un couple (torsion).

In his later articles, he generalized this notion of torsion to apply to spaces with affine, conformal, or projective connections; and he kept the word 'torsion' to describe similar features in all of them, but I don't think that there is any place in Cartan (at least, I'm not aware of one) in which he tries to relate this notion of 'torsion' specifically to the notion of 'torsion' that one encounters in the standard theory of curves in Euclidean $3$-space.

Later, when I have time, I'll try to say something more about Cartan's notion of 'generalized spaces with torsion', which seems to be somewhat poorly understood these days. That may clarify why there isn't any apparent connection with the more classical notion of torsion of space curves.

Answer 2

The curvatures of a curve are all extrinsic. There is no intrinsic curvature of a curve since this is always a 2-form with values in a subgroup of endomorphimss of the tangent bundle. The analogon of the curvatures of a curve is the second fundamental form of a submanifold, and its covariant derivatives with respect to the the metric of the ambient space.