Antipodal points on a round sphere have many minimizing geodesics

The point $p$ is not equal to $-p$: equality would give $p=0$, contrary to $\mathrm{\mid }p\mathrm{\mid }=1$. By [F1] and [F3], the round sphere satisfies all the geometric hypotheses of [F4]. Hence [F4], under [A1], supplies a minimizing geodesic $\eta :[0,1]\to S^n$ from $p$ to $-p$ with constant speed $L=d(p,-p)>0$ and length $L$.

Fix any unit $u\in T_p{S}^n$; such a vector exists because [F2] supplies $u_1$. By [F3] and [F5], ${\gamma }_u$ is a geodesic from ${\gamma }_u(0)=p\text{to}{\gamma }_u(\pi )=-p$ of length $\pi$. Therefore the definition of Riemannian distance gives $L=d(p,-p)\le \pi$. The calculation applies to every unit $u$.

For each $m\in \mathbb{N}$, define $u_m=\frac{u_1+m{u}_2}{\sqrt{1+m^2}}\mathrm{.}$ Orthonormality gives $\mathrm{\mid }{u}_m\mathrm{\mid }=1$. If $u_m=u_k$, comparison of the nonzero $u_1$ coefficients and then of the ratios of the $u_2$ and $u_1$ coefficients gives $m=k$. Hence $(u_m{)}_{m\in \mathbb{N}}$ is an infinite family of distinct unit tangent vectors. This construction uses the two fixed vectors from [F2], not a choice of a vector from each member of a family.

Apply the explicit great-circle formula [F3] to the nonconstant geodesic $\eta$, based at $t=0$. Its speed is $L$, so there is a unit $w\in T_p{S}^n$ such that $\eta (t)=\cos (Lt)p+\sin (Lt)w\mathrm{.}$ Taking the Euclidean inner product of the endpoint equality $-p=\eta (1)$ with $p$, and using $w\perp p$, gives $\cos L=-1$. Steps 1.1--1.2 put $L$ in $(0,\pi ]$. Cosine is strictly decreasing on $[0,\pi ]$ and $\cos \pi =-1$ by [F5], so $L=\pi$. Thus $d(p,-p)=\pi \mathrm{.}$

Since the unit vector in step 1.2 was arbitrary, steps 1.2 and 2.1 show that every ${\gamma }_u$ has length $\pi =d(p,-p)$ and is globally minimizing. In particular this holds for every $u_m$. Moreover [F5] gives ${\gamma }_{u_m}(\pi \mathrm{/}2)=u_m\mathrm{.}$ The distinctness in step 1.3 therefore makes these curves distinct. This is an explicit infinite collection of minimizing half-great-circles with the same two endpoints and proves the claimed failure of uniqueness.

The lower-dimensional cases $n=0,1$ lie outside the quantified claim: the construction of an infinite family in step 1.3 specifically requires the two orthonormal tangent directions that [F2] obtains from $n\ge 2$. The zero-distance case cannot occur because $p\mathrm{\ne }-p$ and the Riemannian distance is a metric; the parameter endpoints $0,\pi \mathrm{/}2,\pi$ were evaluated explicitly. No empty-manifold case arises because $p$ is given, and there is no iff assertion. Assumption [A1] is spent exactly in step 1.1 through Hopf--Rinow and nowhere in the explicit family.