Riemannian distance is defined by the length of a unique shortest curve

For a piecewise $C^1$ path $\gamma =(x,y)$ in $P$ between the prescribed points, its Euclidean speed is $\sqrt{x^{\mathrm{\prime }2}+y^{\mathrm{\prime }2}}$. On each smooth piece this is at least $x^{\mathrm{\prime }}$. Integrating and telescoping the endpoint differences gives $L(\gamma )\ge x(b)-x(a)=2$; Newton–Leibniz applies to the continuously differentiable coordinates on every closed piece.

For any piecewise $C^1$ path in $S^1$, subdivide its parameter interval so that each piece lies in one open arc admitting a smooth angle coordinate. Such a finite subdivision exists: the inverse images of these arcs cover the compact interval; a finite subcover has a positive Lebesgue number, so sufficiently short equal subintervals refine it. Refine further at the original smooth-piece endpoints. Choose an angle value at the initial point and successively add integer multiples of $2\pi$ to each local angle so adjacent values agree at the joining parameter. This gives a continuous piecewise $C^1$ angle $\theta$ with $\gamma =(\cos \theta ,\sin \theta )$. Differentiation gives speed $\mathrm{\mid }{\theta }^{\mathrm{\prime }}\mathrm{\mid }$.

For $0<\epsilon <1$, travel along the horizontal axis from $(-1,0)$ to $(-\epsilon ,0)$, along the upper semicircle of radius $\epsilon$, and along the axis from $(\epsilon ,0)$ to $(1,0)$. All three pieces avoid the origin. Their lengths are $1-\epsilon$, $\pi \epsilon$, and $1-\epsilon$: the semicircle parametrization $(\epsilon \cos t,\epsilon \sin t)$ has speed $\epsilon$ for $0\le t\le \pi$, with reversal giving the required direction. Additivity and reversal yield $2+(\pi -2)\epsilon$. Consequently the infimum is $2$.

If an admissible path had length $2$, then the integral of $\sqrt{x^{\mathrm{\prime }2}+y^{\mathrm{\prime }2}}-x^{\mathrm{\prime }}$ would be zero. This function is continuous and nonnegative on each smooth piece, so it vanishes on each piece (a positive value would give a positive integral on a small interval). Thus $y^{\mathrm{\prime }}=0$ and $x^{\mathrm{\prime }}\ge 0$ there. Newton–Leibniz and continuity at the subdivision points imply $y\equiv 0$. The intermediate value theorem gives a parameter with $x=0$, contradicting avoidance of the origin. Hence the infimum on $P$ is not attained.

For the antipodal endpoints, take $\theta (a)=0$. Then $\theta (b)=(2k+1)\pi$ for some integer $k$. Integrating $\mathrm{\mid }{\theta }^{\mathrm{\prime }}\mathrm{\mid }$ and applying Newton–Leibniz on the pieces gives $L\ge \mathrm{\mid }\theta (b)-\theta (a)\mathrm{\mid }\ge \pi$. The paths $t\mapsto (\cos t,\sin t)$ and $t\mapsto (\cos t,-\sin t)$ for $0\le t\le \pi$ have speed $1$, length $\pi$, and distinct images. Both attain the distance. This proves failure of uniqueness as well as the failure of existence in step 3.1.