Let $S$ be a surface in $\mathbb{R}^{3}$ with the following property:

There is a uniform constant $M$ such that for every Frenet curve $\gamma(t)$, contained in $S$, $| \tau(t) | \leq M$, for all t, where $\tau$ is the torsion. Does this imply that $S$ is a part of a plane?

And what would be a possible generalization of the above for a submanifold of $\mathbb{R}^{n}$?

The motivation and comment:

We do not assume that $S$ is compact. Even in the compact case $S^{2}$, there is a Frenet curve $\gamma :(0, \infty) \rightarrow S^{2}$ such that $\tau(t)$ is unbounded. The reason: a necessary and sufficient condition for that a regular curve $\gamma$ lies on $S^{2}$ is that the pair $(\kappa, \tau)$, curvature and torsion, satisfies in certain differential equation. see Elements of diff geometry by A. Pressley. It can be easily shown that this differential equation has a solution which $\tau$ is unbounded. On the other hand for every smooth functions $\kappa(t), \tau(t)$, there is a Frenet curve for which the curvature and torsion are $\kappa (t), \tau (t) $, respectively . So there is a Frenet curve in $S^{2}$ which torsion is unbounded.

geodesiccurvature of the curve on the surface. From this formula, one sees that, unless the second fundamental form vanishes identically, there cannot be such a bound on $\tau$. $\endgroup$ – Robert Bryant Dec 21 '13 at 12:05