The problem asks:
Let $\kappa(s)$ and $\tau(s)$ be the curvature and torsion of a curve parameterized by its arclength $s$. If
$$(\frac{1}{\kappa})^2 + [\frac{1}{\tau}\frac{d}{ds}(\frac{1}{\kappa})]^2 = constant$$
, then either this curve lies on a sphere or the its curvature $\kappa$ is constant. (Thinking in $\mathbb{R}^3$, assuming all "good conditions", like smooth curve, $\kappa \neq0$ etc.)
Since the equation looks like a inner product. I tried to construct a curve with $\frac{1}{\kappa}$ and $\frac{1}{\tau}\frac{d}{ds}(\frac{1}{\kappa})$ as its components. But it turns out that the resulting curve's curvature is not the given $\kappa$. I also tried differentiating the equations w.r.t $s$, but did not find any clues. Any ideas?
Update: although this question was marked duplicate, that post did not solve the problem completely. Here I posted my solution.
Updated 2: the following text is also posted to that post (math.SE). But be noticed that the two posts' problem differs subtlety, and my problem requires an extra step to prove.
Thanks to @Ted Shifrin and another post at math.SE, I solved this problem.
The key is to construct a vector $\alpha$ as in the link above:
We assume that $R$ is in a sphere. Then think of the vector $\alpha$ as exactly the vector that points from the centre of sphere to the curve $R$. So $\alpha$ is different from the original curve $R$ by just by a translation, hence they have the same tangent, curvature and torsion.
We can always express $\alpha$ as a linear combination (as suggested by Shifrin) of:
$$\alpha(s) = \lambda(s)T(s) + \mu(s)N(s)+\nu(s)B(s)$$
for some functions $\lambda$, $\mu$, $\nu$
(Note that $\alpha$ itself not necessarily arc length parametrised.)
Then since we assume that the $R$ is on the sphere, and that $\alpha$ is from the centre to the sphere, we require that $\alpha$ is tangent to its tangent vector (notice that if $\alpha$ does not point from the centre to the sphere, it will not be tangent to its tangent vector $T$), that is,
$$<\alpha, T>=0$$
By differentiating above, we will find the expression for $\alpha$ as shown in the post mentioned:
$$\alpha=-\frac{1}{\kappa}N-\left(\frac{1}{\kappa}\right)'\frac{1}{\tau}B,$$
We see clearly $\alpha$ is of constant length. If we can prove $r(s)= R(s) – \alpha(s)$ is a fixed point, then our proof is done.
To prove $r(s)$ is a fixed point, we prove that $\frac{d}{ds}r(s) =0$. This could be done by differentiating the original equation w.r.t $s$ (do a straight forward calculation and then compare the results of the two)
Best Answer
HINT: Write $\alpha(s) = \lambda(s)T(s) + \mu(s)N(s)+\nu(s)B(s)$ for some functions $\lambda$, $\mu$, $\nu$. How can you tell when $\alpha$ lies on a sphere centered at the origin? At a general point?