Conjugate antipodes on the round sphere

By [F4] the supplied curve $\gamma (t)={\exp }_p(tv)$ is the affinely parametrized geodesic with $\gamma (0)=p$ and $\dot{\gamma }(0)=v$, and by [F5] it is unit speed, so $T:=\dot{\gamma }$ satisfies $g(T,T)=1$, $T(\pi R)\mathrm{\ne }0$, and $D_{t}T=0$ by the geodesic equation in [F9]. By [F1] the cut time of $v$ is $c_p(v)=\pi R$, so the radial endpoint $\gamma (\pi R)$ is the cut-time endpoint; by [F3] it belongs to $\mathrm{Cut}(p)$, and by [F2] $\mathrm{Cut}(p)=\{-p\}$. Hence $\gamma (\pi R)=-p$, the antipode of $p$. The interval $[0,\pi R]$ is nondegenerate because $R>0$.

By [F6] and [F7] the curvature is $K=1\mathrm{/}{R}^2$, so the constant-curvature scalar solution is $s_K(t)=R\sin (t\mathrm{/}R)$. Hence $s_K(0)=0$, and by [F10], $s_K(\pi R)=R\sin \pi =0$ while $s_K(\pi R\mathrm{/}2)=R\sin (\pi \mathrm{/}2)=R\mathrm{\ne }0$.

Let $J$ be any Jacobi field along $\gamma$ and put $\phi (t)=g(J(t),T(t))$. Since $D_{t}T=0$, two applications of the product rule [F9] give ${\phi }^{\mathrm{\prime }\mathrm{\prime }}=g(D_t^{2}J,T)\mathrm{.}$ By the Jacobi equation [F17], $D_t^{2}J=-R(J,T)T$, and the constant-curvature formula [F7] gives $R(J,T)T=K(g(T,T)J-g(J,T)T)\mathrm{.}$ Pairing with $T$ and using $g(T,T)=1$ yields $g(R(J,T)T,T)=K(\phi -\phi )=0$, so ${\phi }^{\mathrm{\prime }\mathrm{\prime }}=0$ and $\phi (t)=at+b$ is affine.

Conversely, fix $w\in v^{\perp }$ and let $E$ be the unique parallel field along $\gamma$ with $E(0)=w$ [F11]. Since $(g(E,T){)}^{\mathrm{\prime }}=g(D_{t}E,T)+g(E,D_{t}T)=0$ by [F9] and $g(E(0),T(0))=g(w,v)=0$, the field $E$ is normal. The normal classification [F8] therefore makes $J_w(t)=s_K(t)E(t)$ a Jacobi field with $J_w(0)=0$, and step 1.2 gives $J_w(\pi R)=s_K(\pi R)E(\pi R)=0$, so $J_w\in {\mathcal{K}}_{\gamma }(0,\pi R)$. The assignment $w\mapsto J_w$ is linear and injective: if $J_w=0$, then evaluating at $\pi R\mathrm{/}2$ gives $s_K(\pi R\mathrm{/}2)E(\pi R\mathrm{/}2)=0$, and since $s_K(\pi R\mathrm{/}2)\mathrm{\ne }0$, the parallel field $E$ vanishes at one point, hence $E=0$ by uniqueness in [F11] and $w=E(0)=0$. Therefore $\dim {\mathcal{K}}_{\gamma }(0,\pi R)\ge \dim v^{\perp }$ by [F15].

Suppose instead that $J\in {\mathcal{K}}_{\gamma }(0,\pi R)$. By step 2.1, $\phi =g(J,T)$ is affine, and $\phi (0)=g(0,T(0))=0$ while $\phi (\pi R)=g(0,T(\pi R))=0$; since $\pi R>0$, an affine function with two distinct zeros vanishes identically, so $\phi \equiv 0$. In particular $g(D_{t}J(0),T(0))={\phi }^{\mathrm{\prime }}(0)=0$, that is $W:=D_{t}J(0)\in v^{\perp }$. By [F12], $J$ is the unique Jacobi field with $J(0)=0$ and $D_{t}J(0)=W$, so the map $J\mapsto D_{t}J(0)$ is an injective linear map from ${\mathcal{K}}_{\gamma }(0,\pi R)$ into $v^{\perp }$, and $\dim {\mathcal{K}}_{\gamma }(0,\pi R)\le \dim v^{\perp }$ by [F15].

The linear functional $w\mapsto g_p(v,w)$ on $T_{p}M$ has kernel $v^{\perp }$ by [F14] and is surjective because $g_p(v,v)>0$ gives $g_p(v,\lambda v)=\lambda g_p(v,v)$ for every $\lambda \in \mathbb{R}$. Since $\dim T_{p}M=n$ by [F13], rank--nullity [F15] gives $\dim v^{\perp }=n-1$. Combining the two bounds of steps 2.2 and 3.1 yields $\dim {\mathcal{K}}_{\gamma }(0,\pi R)=n-1$. By the conjugacy definition [F16], the antipode $\gamma (\pi R)=-p$ is conjugate to $p$ along $\gamma$, with multiplicity $n-1$.

Boundary and choice audit. The sphere is nonempty and $p$ is supplied, so no empty case arises; $n\ge 2$ excludes the zero- and one-dimensional spheres, where the constant-curvature normal classification [F8] would not apply with positive curvature; the parameter interval $[0,\pi R]$ is nondegenerate and $\gamma$ is a nonconstant unit-speed geodesic, so no constant-geodesic or degenerate-segment case occurs. The zero Jacobi field is excluded from witnessing conjugacy by [F16], and the endpoint-vanishing space is computed exactly, not merely bounded. Exactly the inherited ${\mathrm{A}\mathrm{C}}_{\omega }$ of [A1] is assumed. This example asserts conjugacy at the antipode and its multiplicity; it makes no if-and-only-if claim about geodesics other than the exhibited radial one.