I’ve seen the following theorem asserted more than once in the literature for differential geometry:
For every real number $r$ there is a closed curve $C$ such that its total torsion is equal to $r$.
The total torsion of a curve $C$ in $\mathbf{R}^3$ is defined as the integral $$\int_C\tau\,ds$$ where $\tau$ is the torsion of the curve.
However, I have yet to come across a constructive proof of this fact. It appears to me to be a fact that many are aware of and refer to, and yet a proof of this in the existing literature has eluded me thus far.
I’m looking for a proof of this or a reference to where I can find such a proof as an answer. In particular, the closed curve should be $C^3$ and regular of order $2$. This is so that
- The unit tangent and normal vectors exist and are continuously differentiable, closed curves and linearly independent; and
- The curvature and torsion are defined and continuous from the differential equations of the Frenet frame $\mathbf{T}, \mathbf{N}, \mathbf{T}\times\mathbf{N}$.
In [1], the only thing that is said about this is the all too common
It is well known that…
In [2], the theorem is stated without proof and followed by
To prove this, we need only to consider curves made up of a circular helix with the ends joined by a plane curve as in Fig. 2.37, and to vary the pitch, the number of coils, and the right- or left-handedness of the helix.
My Thoughts
It is clear to me that the construction in the picture will not work, not in the sense that the curve will be $C^3$ and regular of order $2$. The plane in which the plane curve resides will also contain that curve’s unit tangent and normal vectors (attached to the curve). Since the helix’s unit normal points inward towards the axis of the helix, there is no way that the plane that contains this supposed plane curve can contain
- The endpoints of the helix,
- The unit tangents at both ends, and
- The unit normals at both ends.
So, how do we fix this?
I understand that the torsion of the helix is constant so we can make the adjustments that Hsiung says to make the total torsion of the helical arc any real number we like. However, we can’t close that arc up by a plane curve (so that the total torsion will not change). We have to find some other type of arc that closes up the curve and does not add to the total torsion.
References
[1] Qin, Yong-an; Li, Shi-jie, Total torsion of closed lines of curvature, Bull. Aust. Math. Soc. 65, No. 1, 73-78 (2002). ZBL1008.53007.
[2] Hsiung, Chuan-Chih, A first course in differential geometry, Cambridge, MA: International Press. xviii, 343 p. (1997). ZBL0948.53001.