Let us denote a unit sphere by $S$ and assume that $\gamma: [0,1] \rightarrow S$ is a continuos and differentiable function.
Let us parametrize $S$ by spherical coordinates $(a, b)$ and assume the riemannian metric on $S$ is given by $\mathrm{d}s^2=\mathrm{d}a^2 + \sin^2(a)\mathrm{d}b^2$.
To find out the length of $\gamma$ we need to evaluate $\int_{0}^1\mathrm{d}s\vert_{\gamma(t)}\,\mathrm{d}t$.
Question
Circle is also a Riemannian manifold. Can we calculate the length of a curve on a unit circle using above formula ? What is the $ds^2$ for a unit circle that is parameterized is only by one parameter $a$?