Define $S^2 \subset \mathbb{R^3}$ be the unit sphere. Suppose that $\alpha :\mathbb{R} \to S^2$ is a differentiable curve parametrized by arc-length.

a) Show that $\kappa(s)$, the curvature of $\alpha$ is non-vanishing.

b) Suppose also that $\alpha$ is $l$-periodic for some $l \in \mathbb{R}$ (i.e. for all $s \in \mathbb{R}$, $\alpha (s+l)=\alpha(s)$). Let $\tau(s)$ be the torsion of $\alpha$. Prove that $$\int_0^l \, \tau(s)\, ds=0.$$

Part a is easy to prove by using the fact that the curve is unit speed. However, I don't know how to solve part b.

  • 1
    $\begingroup$ The proof is there p.231 problem 3114 $\endgroup$ – Semsem Oct 7 '14 at 9:04

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