User lkeer - MathOverflow

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

2013-02-18T15:28:16Z

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.

What theorem of Liouville's is Gian-Carlo Rota referring to here?

2011-04-22T16:16:08Z

I am very curious about this remark in Lesson Four of Rota's talk, Ten Lessons I Wish I Had Learned Before I Started Teaching Differential Equations:

"For second order linear differential equations, formulas for changes of dependent and independent variables are known, but such formulas are not to be found in any book written in this century, even though they are of the utmost usefulness.

"Liouville discovered a differential polynomial in the coefficients of a second order linear differential equation which he called the invariant. He proved that two linear second order differential equations can be transformed into each other by changes of variables if and only if they have the same invariant. This theorem is not to be found in any text. It was stated as an exercise in the first edition of my book, but my coauthor insisted that it be omitted from later editions."

Does anyone know where to find this theorem?

Comment by lkeer

2013-02-20T09:30:20Z

Thank you! I have that paper and should have thought to go back to it. As you suggest, I was misled by the idea of Cartan connections, which I didn't realise came later. I really thought that Cartan did make that link to torsion of a curve in 'Riemannian Geometry in an Orthogonal Frame', but looking back through it I'm not so sure where I got this idea from.

Comment by lkeer

2013-02-18T20:50:28Z

Sorry, I realise I haven't been very clear here. I do understand that the curvatures of a curve are extrinsic, but I think that there is still a link to the curvature of a connection. The modern analogue of Cartan's development in Euclidean space would I think be the horizontal lift of a curve in $M$ to a curve in the principal bundle over $M$. E.g. given a connection with torsion and no curvature, the horizontal lift of a circle in $\mathbb{R}^2$ would be a helix.

Comment by lkeer

2011-04-23T07:21:37Z

Thank you very much, I will look up this reference.