Normal coordinates on the round sphere

Let $v\in T_p{S}^n$. If $v=0$, [F3] gives the constant geodesic $\gamma (t)=p$. If $v\mathrm{\ne }0$, put $r=\mathrm{\mid }v\mathrm{\mid }$ and $u=v\mathrm{/}r$; then $p,u$ are orthonormal and [F3] gives the geodesic ${\gamma }_v(t)=\cos (rt)p+\sin (rt)u\mathrm{.}$ It is defined for every real $t$, so $v$ lies in the exponential domain. Evaluating at time one as in [F1] yields the two displayed branches for ${\exp }_p(v)$.

By [F2], there is a normal domain $D_0$ at $p$. Its intersection $D_0\cap B_{\pi }(0_p)$ is open and star-shaped about zero, and restricting the diffeomorphism ${\exp }_p:D_0\to {\exp }_p(D_0)$ gives a diffeomorphism on that intersection, so at least one normal domain lies in $B_{\pi }(0_p)$. Now let $D\subseteq B_{\pi }(0_p)$ be any normal domain at $p$ and let $v={\sum }_i{v}^i{e}_i\in D$. The definition in [F2] gives $x_e({\exp }_p(v))=E_e^{-1}(v)=(v^1,\dots ,v^n),$ which proves the asserted formula for every such $D$.

For $v\mathrm{\ne }0$, with $r=\mathrm{\mid }v\mathrm{\mid }$, step 1.1 gives $\mathrm{\mid }{\exp }_p(v)-p\mathrm{\mid }\le \mathrm{\mid }\cos r-1\mathrm{\mid }+\mathrm{\mid }\sin r\mathrm{\mid }\mathrm{.}$ As $v\to 0$, one has $r\to 0$, and [L1] makes the right side tend to zero. Thus the nonzero branch converges to $p={\exp }_p(0)$, proving the asserted continuous value without assigning a value to $v\mathrm{/}\mathrm{\mid }v\mathrm{\mid }$ at zero.

Suppose $v,w\in B_{\pi }(0_p)$ and ${\exp }_p(v)={\exp }_p(w)$. Put $r=\mathrm{\mid }v\mathrm{\mid }$ and $s=\mathrm{\mid }w\mathrm{\mid }$. Taking the Euclidean inner product with $p$ in the formula of step 1.1 gives $\cos r=\cos s$, because $v,w\perp p$. Since $r,s\in [0,\pi )$ and cosine is strictly decreasing there by [L2], $r=s$. If this common value is zero, then $v=w=0$. If it is positive, [L3] gives $\sin r>0$, and equality of the components perpendicular to $p$ gives $(\sin r\mathrm{/}r)v=(\sin r\mathrm{/}r)w$, hence $v=w$. This proves injectivity on the open ball, including all zero/nonzero combinations.

The vectors $\pi e_1$ and $-\pi e_1$ are distinct because $n\ge 1$ and $e_1$ is a unit vector. Step 1.1 and [L3]--[L4] give ${\exp }_p(\pi e_1)=-p={\exp }_p(-\pi e_1)\mathrm{.}$ Thus radius $\pi$ is the first boundary at which the antipodal collision can occur: step 2.2 excludes collisions at smaller radii, while the displayed pair realizes one at radius $\pi$. The source ball is open, so its endpoint is not silently included. Dimension zero is outside the stated $n\ge 1$ claim; there the tangent space is the singleton $\{0_p\}$ and no antipodal-direction pair exists. An empty sphere has no supplied $p$. The explicit calculations and displayed collision pair make no choices; ${\mathrm{A}\mathrm{C}}_{\omega }$ is used exactly through the inherited exponential/normal-neighborhood framework recorded in [A1]--[F2].