Why is the curvature function of a closed curve periodic?

Question

I'm trying to read through this paper, but I'm having trouble making connections in the last sentence of the first page:

"Therefore, the curvature of the curve $\kappa(s)$ is also a periodic function whose period $\rho_k$ satisfies $n\rho_k=L$ for some natural number $n$."

How does this sentence follow from the previous, which states that $\rho$ can be taken as $L$, the length of $\gamma(s)$ in one period?

Answer 1

The curvature $\kappa(s)$ of a unit-speed curve $\gamma(s)$ is simply $\kappa(s) = |\gamma''(s)|.$ If $\gamma(s) = \gamma(s+\rho),$ then we also have $\gamma'(s) = \gamma'(s+\rho);$ this property of the derivative follows easily either from the definition of the derivative or from the chain rule. Applying the same property again we find$$\gamma''(s) = \gamma''(s+\rho).$$ Thus taking the norm of both sides we conclude $\kappa(s)=\kappa(s+\rho);$ i.e. the curvature is periodic with period $\rho$. This is not necessarily the minimum period, however; but the minimum period must divide any other period, so we have $\rho_k = n\rho$ for some integer $n$.

Answer 2

The arc length period is $\rho $ where $ \rho_n = n \rho = L $ when position,slope, curvature have this period repeated as valid for isometric natural/intrinsic equations.

Here

$$ a \,\kappa = a \frac{d\phi}{ds}= \sin (s/L) $$

Instead of period $2 \pi$ period here we have arc period $L.$