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This is from Kristopher Tapp's "Differential Geometry of Curves and Surfaces", Exercise 6.5 titled "Jacobi's Theorem"

Exercise 6.5 (Jacobi's Theorem). Let $\gamma:[a,b]\to \mathbb{R}^3$ be a simple closed unit-speed space curve with nowhere-vanishing curvature. For all $t\in [a,b]$, let $\{\textbf{t}(t),\textbf{n}(t),\textbf{b}(t)\}$ denote Frenet frame at $\gamma(t)$. Notice that $t\mapsto \textbf{n}(t)$ is a regular parametrized curve on the sphere $S^2$. If this curve on $S^2$ is reparametrized by arc lenght, prove that its total geodesic curvature equals zero. If it is simple, conclude that it divides $S^2$ into two regions of equal area.

My Process: I tried to find geodesic curvature $k_g$ of in terms of curvature and torsion of $\gamma$. To find $k_g$, I need to find unit tangent $\textbf{T}$ and geodesic normal $\textbf{N}_g$ because $\dfrac{D\textbf{T}}{ds}=k_g\textbf{N}_g$ and $\dfrac{D\textbf{N}_g}{ds}=-k_g\textbf{T}$. I know that $\textbf{T}(s)=\dfrac{n'(s)}{|n'(s)|}=\dfrac{-k\textbf{t}(s)+\tau\textbf{b}(s)}{\sqrt{k^2+\tau^2}}$ but do not know how to calculate $\textbf{N}_g$.

lecdabster
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  • Have you differentiated $\mathbf T$? Be careful, though, to remember that the Frenet frame of $\gamma$ are functions of $t$, not $s$, so what you're writing is a bit questionable. – Ted Shifrin May 18 '22 at 16:18
  • Where did that equation come from? Moreover, you don't get to assume $\mathbf n$ is arclength-parametrized if you're assuming $\gamma$ is. You have got to make chain rule adjustments, as always. To emphasize: You cannot reparametrize $\mathbf n$ and still work with the Frenet equations for $\gamma$. – Ted Shifrin May 18 '22 at 16:29
  • @TedShifrin I forgot to write that I changed the variable $\gamma$ of $s$. In question, it is $t$ but I am confused when it is unit speed and not $s$. – lecdabster May 19 '22 at 10:44

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