Frenet-Serret formulas in arbitrary dimensions

Question

Can anyone point me to a proof of the Frenet-Serret formulas for arbitrary (i.e. $N>3$) dimensions?

Answer 1

Assuming appropriate genericity of the curve (linear independence of all derivatives), it's really the same proof. Assuming an arclength parametrization $\alpha(s)$, set $\alpha'(s)=e_1(s)$. Now, assuming that you've defined $e_1,\dots,e_k$ with $e_1' = \kappa_1e_2$ $(\kappa_1>0), \dots, e_j'=-\kappa_{j-1}e_{j-1}+\kappa_{j}e_{j+1}$ for $2\le j\le k-1$, you write $e_k'=-\kappa_{k-1}e_{k-1}+\kappa_{k}e_{k+1}$ for $\kappa_{k}>0$ and an appropriate unit vector $e_{k+1}$. When you get to $k=N-1$, you can ordain that $e_1,\dots,e_N$ be a positively oriented orthonormal basis, and then $\kappa_{N-1}$ will have a sign. The skew-symmetry follows, as in the $\Bbb R^3$ case, by differentiating $e_i\cdot e_j = 0$, $i\ne j$, and $e_i\cdot e_i=1$.

I don't know your background, but a beautiful exposition of this and far more (if you are familiar with differential forms and Lie groups) is in Phillip Griffiths's paper "On Cartan's Method of Lie Groups and Moving Frames as Applied to Existence and Uniqueness Questions in Differential Geometry," Duke Math J. {\bf 41}, 1974. (However, I think his subscripts may get a bit messed up on this.)