Why is the local canonical form important

Question

I'm studying Differential Geometry of Curves and Surfaces, by Do Carmo, and there's a section right before he dives into surface theory, where he talks about the Local Canonical Form. I get that it provides a somewhat natural reference frame for the curves in question, because we can express them in terms of curvature, torsion and their derivatives. What I don't quite "understand" is why do we truncate the Taylor expansion in third order terms.It seems to me we could keep going and perhaps write down an exact expression for the Taylor series of the curve around a point, like we do for functions, but I haven't seen it done. Is it not possible for the general case, or is there no point to it? Maybe I'm missing the point entirely but I would surely appreciate some insight into where and how the Local Canonical Form fits in the theory.

Answer 1

Do Carmo's book is typically not bothered with regularity matters (but see Exr.7 on p.10) and takes any curve to be $C^\infty$. However a simple regularity count gives that the Frenet-Serret theory in the classical sense works for $C^3$ curves (that is, curves $\gamma:[a,b]\to\mathbb{R}^3$ all of whose components are three time continuously differentiable functions). Indeed, let's assume $\gamma$ is $C^k$ for $k\in\mathbb{Z}_{\geq1}$ whose minimum for the classical theory will be determined later. Then the arclength parameterization is a $C^k$ diffeomorphism, so we may assume that $\gamma$ is parameterized by arclength. Then $T(s)=\gamma'(s)$ is $C^{k-1}$ and $\kappa(s)=|T'(s)|$, which is $C^{k-2}$. Further assuming $\kappa$ never vanishes (which is equivalent to Do Carmo's "no singularity of order $1$ assumption"), we have that $N(s)=\dfrac{T(s)}{|T'(s)|}$ is also $C^{k-2}$ (recall that as long as chain rule is used in tandem with algebraic operations the composite function is as good as the worst composant). Finally $B(s)=T(s)\times N(s)$ is $C^{k-2}$ and $\tau(s)=-B'(s)\bullet N(s)$ is hence $C^{k-3}$. This gives that $k\geq3$ is sufficient.

I'm not giving an argument here, but a similar derivative count also gives that the Fundamental Theorem of the Local Theory of Curves in $\mathbb{R}^3$ on $p.19$ actually works for an anonymous $\kappa\in C^{k-2}$ nonvanishing and an anonymous $\tau\in C^{k-3}$ for $k\in\mathbb{Z}_{\geq3}$ and produces a unique $C^k$ curve (up to special Euclidean motions $\mathbb{R}^3\rtimes SO(3,\mathbb{R})$). (Put differently, to get a local canonical form that that defines the curve uniquely a Taylor expansion with three coefficients is not only sufficient but also necessary.)

So as long as curves in $\mathbb{R}^3$ goes, the Frenet-Serret theory applies to curves whose Taylor expansions may not be valid after the third coefficient (one can produce examples out of the examples in $f^{(n)}(x)$ exist $\forall x \in I$, an interval in $\mathbb{R}$ but $f^{(n+1)}(x)$ does not exist for any $x \in I$ or What function can be differentiated twice, but not 3 times?).

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Finally I should mention that it's not a coincidence that the curves in $\mathbb{R}^3$ to which the Frenet-Serret theory applies ought to be $C^3$ (and $\kappa$ nonvanishing). Indeed, the main idea of the Frenet-Serret theory is to produce an orthonormal basis out of the derivative vectors; so the dimension and the regularity of the curve are closely related. One can generalize the theory to $\mathbb{R}^d$ (to say the least), where the curve would need to be $C^d$ (again with a nondegeneracy condition) and this time a $d$-term Taylor expansion would be required to come up with a local canonical form in terms of generalized curvatures.