Related to this question, https://mathoverflow.net/questions/380565/can-differential-geometry-aid-in-comparing-the-close-contour-integrals-of-fz , I am now seeking advice about how to best approach the following, somewhat more general, Differential Geometry problem involving closed planar curves:
- start from a closed planar curve $\gamma_1$ (regular, but not necessarily simple), described through its Whewell equation $\varphi(s)$ (i.e.: the tangent vector as a function of arc-length $s$).
- Transform $ \gamma_1 \, \rightarrow \, \gamma_2 \;\;$ through the tranformation: $\; \varphi (s) \, \rightarrow \, \varphi (s) + \theta(s) \;$.
- Whereby $\theta(s) $ monotonically increases from $\, \theta(0)=0 \,$ up to $\pi$ (though never crossing it over to values $>\pi$) to then monotonically come back to $\, \theta(L)=0 \,$ ($\, L \,$ representing the curve total length),
- No additional properties are specified for $\theta(s)$ .
It is straightforward to observe how the total curvature of $\gamma_2$ is conserved identical to the total curvature of $\gamma_1$ (as the rotation $\theta(s)$ always folds back to $0$ without ever having crossed over to values $>\pi$). Moreover, by definition also the total length is conserved.
I am asking myself whether it is true that such a transformation cannot possibly result in $\gamma_2$ also being a closed curve.
Heuristically, we might expect such a transformation to "open up" the originally closed curved $\gamma_1$ so that the resulting curve $\gamma_2$ would instead be an open curve. However, such kind of intuitions are often misleading, and I am hence searching for ways to more rigorously approach such kind of problems. Are there some already known Differential Geometry theorems which may help me in a rigorous study of this problem ? Or perhaps help in identifying candidate additional restrictions on $\theta(s)$ that would make it true ?
Advice on text books addressing this sort of problems is also welcome.