Let $γ : I → S^2$ be a regular curve in the 2-sphere. Let $\kappa_g$ denote the geodesic curvature. Regarding the curve $γ$ as a space curve $S^2 ⊂ \mathbb R^3$ and assuming it to be Frenet, calculate its curvature $κ$ and torsion $τ$ in terms of $\kappa_g$.
So we have $$\kappa_g = \langle T', \gamma \times T\rangle = \langle\gamma'', \gamma \times \gamma'\rangle$$
where the brackets denote the inner product, where $$T = \frac {d\gamma}{dt}, $$
and where the Frenet frame is the 3-dimensional orthonormal basis $(\gamma', \gamma \times \gamma', \gamma)$.
The curvature of a space curve is given by $$\kappa = \frac{\lVert \gamma \times \gamma'\rVert}{\lVert\gamma'\rVert^3}$$
The Torsion is given by $$\tau = \frac{\langle\gamma' \times \gamma'',\gamma'''\rangle}{\lVert\gamma' \times \gamma''\rVert^3}$$
So basically the objective is to write $\kappa$ and $\tau$ in terms of $\kappa_g$? I'm not really sure how it's possible to get something like $\lVert\gamma'\rVert^3$ out of the definition of $\kappa_g$.
On the other hand it sort of looks like we could obtain the numerator for $\tau$ by just differentiating $\kappa_g$ and noticing that both are of the form of a scalar triple product with the inner product between the two lower order derivatives and the highest order derivative with the numerator of $\kappa_g$ is one degree less than that of the numerator of $\tau$.
Can anyone let me know if I'm on the right track or perhaps suggest how I might acquire the entire expression(s) for $\kappa$ and $\tau$ from $\kappa_g$?
Best Answer
Here are a couple of explicit hints. For a theoretical problem, you (most) always want to assume arclength parametrization and avoid those cross-product expressions. I will use $T,N,B$ for the Frenet frame. Differentiation is with respect to arclength.
(1) If $\gamma$ is a curve on the unit sphere, show that $\gamma\cdot N = -1/\kappa$.
(2) By definition (since the unit normal to the sphere is the position vector), $\kappa N = \kappa_g (\gamma\times T) + \kappa_n\gamma$, and $\kappa_n = -1$. First, this tells you that $\kappa = \sqrt{\kappa_g^2+1}$.
(3) Next, differentiate this and precisely one term will involve $\tau B$. We can isolate $\tau$ by dotting with $B$. You should then solve for $\tau$ in terms of $\kappa_g'$ and $\kappa$.