1. ## geodesic torsion

let a(t) be a regular curve which lies on the unit sphere in R^3

a(t) . a(t) = 1 for all t

i want to show that the geodesic torsion Tg(t) vanishes for such a curve.

i can use the fact that a(t) can be interpreted as a unit normal N(t) to the surface along he curve.

can anyone help me with this question? thanks

2. Let's see... Reparametrize for arclength, and let the torsion $\tau\neq 0$. By continuity, $\tau\neq 0$ over an interval $I$.

If the Frenet-Serret frame is $\{t,\eta,b\}$, then $b$ is also the position vector. We therefore have $b''=k\eta, \ b'=-\tau\eta, \
\eta'=-kt+\tau b$
where $k$ is the curvature of $a$. These last equations give $k\eta=(-\tau\eta)'$ or $\eta'+\psi\eta=0$, where $\psi=1+\tau'/k$.

This DE is linear and thus solvable throughout I. Solve to get $\eta=ce^{\int\psi}$, for some constant vector $c$. This means $b'=c\int ke^{\int\psi}$, and the parametrization gives $|b'|=\left|c\int ke^{\int\psi}\right|=1$, differentiating which gives $ke^{\int\psi}=0$. This implies $k=0$ over I, a contradiction as the curve lies on the sphere.