Length of curve on $(S^2,\tilde g)$

Question

Under the polar coordinate, the unit sphere is $$ S^2=\{(\sin\theta\cos\varphi,\sin\theta\sin\varphi,\cos\theta)\in \mathbb R^3:\theta\in[0,\pi],\varphi\in[0,2\pi] \} $$ consider a non-induced metric, for example, $$ \tilde g =\begin{pmatrix} f(\theta,\varphi) &0 \\ 0 & h(\theta,\varphi) \end{pmatrix} $$ And a curve on $S^2$ $$ r(t)=(\cos t,\sin t \cos K, \sin t \sin K), t\in[0,\pi] $$ then how to calculate the length of $r(t)$? The $K>0$ is constant.

What I try: Let $$ r(\theta,\varphi)= (\sin\theta\cos\varphi,\sin\theta\sin\varphi,\cos\theta) $$ then, $$ \langle r_{\theta},r_{\theta}\rangle = f(\theta,\varphi) \\ \langle r_{\varphi},r_{\varphi}\rangle = h(\theta,\varphi) $$ Besides, I have $$ r'(t)=(-\sin t, \cos t \cos K, \cos t\sin K) $$ But I don't know how to use $r_\theta,r_\varphi$ to present $r'(t)$. Therefore, I can't calculate the $|r'(t)|_{\tilde g}$.

Answer 1

For the standard metric $$ g=d\theta\otimes d\theta+\sin^2\theta\,d\varphi\otimes d\varphi $$ you calculate the length of a curve $t\mapsto (\theta(t),\varphi(t))$ by $$ \int_0^t\sqrt{\dot\theta^2+(\sin^2\theta)\,\dot\varphi^2}\,dt\,. $$ For your metric $\tilde g=f\,d\theta\otimes d\theta+h\,d\varphi\otimes d\varphi$ the length is accordingly $$ \int_0^t\sqrt{f\,\dot\theta^2+h\,\dot\varphi^2}\,dt\,. $$ If the curve you are considering is given in ambient space $\mathbb R^3$ by $$ r(t)=\begin{pmatrix}\cos t\\\sin t\cos K\\\sin t\sin K\end{pmatrix} $$ then you find its $S^2$ coordinates $(\theta(t),\varphi(t))$ from \begin{align} \sin\theta\cos\varphi&=\cos t\,,\\ \sin\theta\sin\varphi&=\sin t\cos K\,,\\ \cos\theta&=\sin t\sin K\,. \end{align} The solution for $\theta,\varphi$ does not look pretty but is straightforward.