A friend and I are trying to understand the construction on section 3.2 of this paper by Bernatzki and Ye. Suppose $\gamma$ is a smooth simple closed curve in $\mathbb{R}^3$. Then this paper claims there exist vector fields $v_i: \gamma \to \mathbb{R}^3$, $i = 1,2,3$, such that $v_i \perp \dot{\gamma}$ everywhere, and the three vectors $V_i = \int_{\gamma} v_i(s) ds$, $s$ being the arc-length parameter, form an orthonormal basis. Furthermore, we can make the support of the $v_i$ arbitrarily small. Why is this true?
We know that each $v_i$ can be written as a linear combination $f_iN + g_i B$, with $(T,N,B)$ being the Frenet frame, $f_i$ and $g_i$ being functions, but the integrals $\int f_iN + g_i B$ don't seem to have particularly nice forms, even after applying the Frenet-Serret equations and integration by parts. I think almost any choice of $v_i$ should yield $V_i$ forming a basis of $\mathbb{R}^3$, but there seems to be difficulty in proving this rigorously.
Best Answer
This is an unusual result, and I'm surprised the authors dismissed it as if it were standard and well-known. Here's the main idea. You can sort out the final details. Having the supports be very small is required and is the main trick, from my point of view. (Think delta functions, or smooth approximations thereto.)
For a closed space curve, it is well-known that the tangent indicatrix is "balanced" with respect to every direction. Indeed, assuming $\gamma$ is arclength parametrized (with length $L$), we have $\int_0^L T(s)\,ds = \int_0^L \gamma'(s)\,ds = 0$, and so $\int_0^L (T\cdot A)\,ds = 0$ for any vector $A$. The same principle applies to the vector field $\kappa N$ (but not to $N$ itself), since $T'=\kappa N$. In particular, if you fix $s_1$ and take $A=N(s_1)$, then since $(\kappa N)(s_1)\cdot A>0$ and the integral is $0$, by the intermediate value theorem there must be $s_2$ so that $(\kappa N)(s_2) \cdot A = 0$.
Let $\rho_1$ be a bump function with integral $1$ supported in an arbitrarily small neighborhood of $s_1$, and let $\rho_2$ similarly be supported in an arbitrarily small neighborhood of $s_2$. Let $v_1 = \rho_1 N$ and let $v_2 = \rho_2 N$. You can rescale to make $V_1$ and $V_2$ unit vectors; they won't quite be orthogonal, but I believe you can alter $v_2$ by adding a small multiple of $B$ to fix it.
Now, can we find $s_3$ so that $\alpha N(s_3)+\beta B(s_3)$ is orthogonal to both $N(s_1)$ and $N(s_2)$ for some scalars $\alpha,\beta$? If not, the normal plane to $\gamma$ at every $s$ never contains the vector $A=N(s_1)\times N(s_2)$. But this means that $T(s)\cdot A$ is always nonzero, contradicting our original observation that the tangent indicatrix is always balanced. So there is such an $s_3$, and once again we use our bump function trick to create $v_3$ and correct as necessary.
If you find some fatal flaws with this, let me know. And certainly let us know if you figure out a trivial solution, but I certainly didn't see one.