= Solution
The <length of a curve> is $L(\alpha)=\int_a^b\|\alpha'(t)\|dt$. Its <arc length> parameter is $s(t)=\int_a^t\|\alpha'(u)\|du$. Regularity gives $s'(t)>0$, so the inverse function theorem gives a smooth inverse and $\beta(s)=\alpha(t(s))$ has $\|\beta'(s)\|=1$.
Reparametrize a length minimizer to constant speed on $[a,b]$ and use the <energy functional> $E=\tfrac12\int_a^b\|\alpha'\|^2dt$. By <Cauchy-Schwarz inequality>, every competitor has $E\geq L^2/(2(b-a))$, with equality for constant speed, so this parametrization also minimizes $E$. For a fixed-endpoint <variation> with tangential <vector field> $V$, differentiation and <integration by parts> give
$$
\delta E=\int_a^b\langle\alpha',V'\rangle dt
=-\int_a^b\langle\alpha'',V\rangle dt
=-\int_a^b\langle\nabla_t\alpha',V\rangle dt.
$$
The boundary term is zero. Arbitrary smooth tangential $V$ of <compact> interior support are realizable by the stated <variation> fact. Taking $V$ to be a nonnegative cutoff times $\nabla_t\alpha'$ proves $\nabla_t\alpha'=0$. Hence \b[a length minimizer is a <geodesic> after constant-speed reparametrization]. The original parameter need not be affine: arbitrary varying-speed parametrizations preserve length but have nonzero covariant <acceleration>. This qualification is necessary for the literal wording.
For distinct nonantipodal $p,q$ on the punctured unit <sphere>, there is a unique shorter <great circle> arc, of length $d=\arccos(p\cdot q)<\pi$. A minimizing <curve> exists \b[exactly when that shorter arc avoids the removed north pole]. If it avoids the pole it attains the spherical lower bound. If it passes through the pole, arbitrarily small smooth detours have lengths tending to $d$, but equality would force the unique shorter arc, which is unavailable. Thus the infimum is not attained; the longer <great circle> arc is not a substitute minimizer.
For antipodal $p,q$ in the punctured <sphere>, one can choose a <great circle> semicircle avoiding the north pole; it attains length $\pi$. Thus \b[every admissible antipodal pair has a minimizer]. If identical endpoints are included despite the earlier distinctness stipulation, the infimum is zero, but no smooth regular <curve> attains it; only a nonregular constant <curve> does.
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