Round sphere model geometry

The tangential projection of the ambient derivative is the Levi-Civita connection. Write $p$ for the position field on ${\mathbb{R}}^{n+1}$, so that $D_{X}p=X$ for every smooth ambient field $X$, and extend tangent fields on $S_R^n$ smoothly to the ambient space locally. By [F1] the tangent space at $x\in S_R^n$ is $x^{\perp }$, and the $g$-orthogonal projection of an ambient vector $w$ onto $x^{\perp }$ along $\mathbb{R}x$ is $w-\frac{⟨w,x⟩}{⟨x,x⟩}x=w-\frac{⟨w,x⟩}{R^2}x\mathrm{.}$ Differentiating the identically vanishing function $⟨Y,p⟩$ in the direction $X$ gives $0=⟨D_{X}Y,p⟩+⟨X,Y⟩$, so the projection of $D_{X}Y$ is ${\mathrm{\nabla }}_X^{S}Y:=D_{X}Y-\frac{⟨D_{X}Y,p⟩}{R^2}p=D_{X}Y+\frac{⟨X,Y⟩}{R^2}p\mathrm{.}$ The assignment ${\mathrm{\nabla }}^S$ is a connection, read off from the corresponding properties of $D$ recorded in [F3]; it is torsion-free because ${\mathrm{\nabla }}_X^{S}Y-{\mathrm{\nabla }}_Y^{S}X=D_{X}Y-D_{Y}X=[X,Y]$ and the correction terms cancel; and it is metric compatible because $⟨{\mathrm{\nabla }}_X^{S}Y,Z⟩+⟨Y,{\mathrm{\nabla }}_X^{S}Z⟩=⟨D_{X}Y,Z⟩+⟨Y,D_{X}Z⟩=X⟨Y,Z⟩,$ the two correction terms vanishing since $Y,Z\perp p$. By the uniqueness clause of [F3], ${\mathrm{\nabla }}^S$ is the Levi-Civita connection of $g$.

The geodesic equation on the sphere. Let $\gamma$ be a smooth curve in $S_R^n$. Differentiating the identity $⟨\gamma ,\gamma ⟩=R^2$ twice gives $⟨\gamma ,{\gamma }^{\mathrm{\prime }\mathrm{\prime }}⟩=-\mathrm{\mid }{\gamma }^{\mathrm{\prime }}{\mathrm{\mid }}^2$. Projecting ${\gamma }^{\mathrm{\prime }\mathrm{\prime }}$ as in step 1.1, the curve $\gamma$ is a geodesic exactly when ${\gamma }^{\mathrm{\prime }\mathrm{\prime }}-\frac{⟨{\gamma }^{\mathrm{\prime }\mathrm{\prime }},\gamma ⟩}{R^2}\gamma =0,\text{that is}{\gamma }^{\mathrm{\prime }\mathrm{\prime }}=-\frac{\mathrm{\mid }{\gamma }^{\mathrm{\prime }}{\mathrm{\mid }}^2}{R^2}\gamma \mathrm{.}$ Since $\mathrm{\mid }{\gamma }^{\mathrm{\prime }}\mathrm{\mid }$ is constant along a geodesic by [F4], a nonconstant geodesic of $S_R^n$ satisfies ${\gamma }^{\mathrm{\prime }\mathrm{\prime }}=-c^2\gamma \mathrm{/}{R}^2$ with $c=\mathrm{\mid }{\gamma }^{\mathrm{\prime }}\mathrm{\mid }>0$, and a constant curve is a geodesic by [F4].

The maximal geodesics. Fix $p\in S_R^n$ and $v\in T_p{S}_R^n$; by [F1], $⟨p,v⟩=0$. For $v\mathrm{\ne }0$ put $\theta (t)=\mathrm{\mid }v\mathrm{\mid }t\mathrm{/}R$ and $\gamma (t)=\cos \theta (t)p+\frac{R}{\mathrm{\mid }v\mathrm{\mid }}\sin \theta (t)v\mathrm{.}$ Then $⟨\gamma (t),\gamma (t)⟩=R^2{\cos }^2\theta +R^2{\sin }^2\theta =R^2$, so $\gamma$ takes values in $S_R^n$; moreover $\gamma (0)=p$, ${\gamma }^{\mathrm{\prime }}(0)=v$ and, by [F10], ${\gamma }^{\mathrm{\prime }\mathrm{\prime }}=-(\mathrm{\mid }v{\mathrm{\mid }}^2\mathrm{/}{R}^2)\gamma$. Step 2.1 therefore makes $\gamma$ a geodesic, and it is defined on all of $\mathbb{R}$. By uniqueness of the maximal geodesic in [F4], it is the maximal geodesic with initial data $(p,v)$; for $v=0$ the same conclusion holds for the constant curve. Evaluating at $t=\pi R\mathrm{/}\mathrm{\mid }v\mathrm{\mid }$ gives $\gamma (\pi R\mathrm{/}\mathrm{\mid }v\mathrm{\mid })=-p$, and the image is $S_R^n\cap \mathrm{span}\{p,v\}$: the orthonormal pair $p\mathrm{/}R,v\mathrm{/}\mathrm{\mid }v\mathrm{\mid }$ parametrizes that circle, and $\mathrm{\mid }v\mathrm{\mid }t\mathrm{/}R$ ranges over all real angles.

Path connectedness. Let $x,y\in S_R^n$. If $y=x$, the constant curve joins them. If $y\mathrm{\ne }\pm x$, put $\theta =\arccos (⟨x,y⟩\mathrm{/}{R}^2)\in (0,\pi )$ by [F10] and $u=(y-\cos \theta x)\mathrm{/}(R\sin \theta )$; then $⟨x,u⟩=0$ and $\mathrm{\mid }y-\cos \theta x{\mathrm{\mid }}^2=R^2-2\cos \theta ⟨x,y⟩+{\cos }^2\theta R^2=R^2{\sin }^2\theta ,$ so $u\in T_x{S}_R^n$ is a unit vector, and step 3.1 gives ${\gamma }_{x,u}(R\theta )=y$. If $y=-x$, choose any unit $u\in T_x{S}_R^n$, which is possible because $x^{\perp }\cong {\mathbb{R}}^n\mathrm{\ne }\{0\}$; step 3.1 gives ${\gamma }_{x,u}(\pi R)=-x=y$. Hence every pair of points is joined by a continuous curve, and $S_R^n$ is path connected, hence connected.

Metric completeness and compactness. By step 3.1 every maximal geodesic of $S_R^n$ is defined on all of $\mathbb{R}$, so $(S_R^n,g)$ is geodesically complete. It is nonempty, connected by step 4.1 and boundaryless by [F1], so Hopf–Rinow [F5] makes it metrically complete. Being a closed and bounded subset of ${\mathbb{R}}^{n+1}$, it is also compact by [F9].

The distance formula. Let $x,y\in S_R^n$ and put $\theta =\arccos (⟨x,y⟩\mathrm{/}{R}^2)\in [0,\pi ]$, which is well defined by the Cauchy–Schwarz bound in [F10]. Step 4.1 constructs a curve from $x$ to $y$ of length $R\theta$ — constant in the case $y=x$, an arc ${\gamma }_{x,u}$ of unit speed over a time interval of length $R\theta$ in the other cases — so $d_g(x,y)\le R\theta$ by [F6]. Conversely, $S_R^n$ is complete by step 5.1, so [F5] provides $v\in T_x{S}_R^n$ with ${\exp }_x(v)=y$ and $\mathrm{\mid }v{\mathrm{\mid }}_{g_x}=d_g(x,y)$; put $c:=\mathrm{\mid }v{\mathrm{\mid }}_{g_x}$. If $c=0$ then $x=y$ and $\theta =0$, so $c=R\theta$. If $c>0$, step 3.1 applied to the initial datum $(x,v\mathrm{/}c)$ describes the unit-speed minimizing geodesic $t\mapsto {\exp }_x(tv\mathrm{/}c)$ on $[0,c]$; evaluating at $t=c$ and comparing with $y={\exp }_x(v)$ gives $⟨x,y⟩=R^2\cos (c\mathrm{/}R)$, hence $\cos (c\mathrm{/}R)=\cos \theta$. Since $c\mathrm{/}R\ge 0$ and $\theta \in [0,\pi ]$, the solutions of $\cos s=\cos \theta$ on $[0,\mathrm{\infty })$ are $s=\theta +2k\pi$ and $s=2\pi -\theta +2k\pi$, $k\ge 0$, whose smallest element is $\theta$; therefore $c\ge R\theta$. Combining both inequalities gives $d_g(x,y)=R\theta =R\arccos (⟨x,y⟩\mathrm{/}{R}^2)$.

Diameter and the antipodal pair. By step 6.1, $d_g(x,y)\le \pi R$ for all $x,y$, with equality $d_g(x,y)=\pi R$ exactly when $⟨x,y⟩=-R^2$. By the equality case of Cauchy–Schwarz, $\mathrm{\mid }⟨x,y⟩\mathrm{\mid }=R^2$ holds exactly for linearly dependent $x,y$, that is for $y=\pm x$; the negative sign is precisely $⟨x,y⟩=-R^2$. Equal points give distance $0$, so $\mathrm{diam}(S_R^n,g)=\pi R$, attained exactly at antipodal pairs.

Cut times, cut locus, injectivity radius and injectivity of the exponential. Let $v\in T_p{S}_R^n$ be a unit vector and $\gamma (t)={\exp }_p(tv)$, a unit-speed geodesic by step 3.1. Step 6.1 applied to the pair $(p,\gamma (t))$ gives $d_g(p,\gamma (t))=R\arccos \frac{⟨p,\gamma (t)⟩}{R^2}=R\arccos (\cos (t\mathrm{/}R)),$ because $⟨p,\gamma (t)⟩=R^2\cos (t\mathrm{/}R)$. For $0<t\le \pi R$ the inverse-cosine identity in [F10] gives $d_g(p,\gamma (t))=t$, so all these $t$ belong to the set defining $c_p(v)$. For $t>\pi R$, write $t=2\pi Rk+s$ with $k\ge 0$ and $0\le s<2\pi R$; then $\gamma (t)=\gamma (s)$ and $d_g(p,\gamma (t))\le \pi R<t$. Hence $\{t>0:d_g(p,{\exp }_p(tv))=t\}=(0,\pi R]$, and the supremum definition of [F8] gives $c_p(v)=\pi R$, with cut point ${\exp }_p(\pi Rv)=\gamma (\pi R)=-p$ by step 3.1. Consequently $\mathrm{Cut}(p)=\{-p\}$, and $\mathrm{inj}(p)=\inf \{c_p(v):\mathrm{\mid }v{\mathrm{\mid }}_g=1\}=\pi R$ by [F8]. Since $\{tv:0<t<c_p(v),\mathrm{\mid }v{\mathrm{\mid }}_g=1\}=B_0(\pi R)\setminus \{0\}$ and ${\exp }_p$ is a diffeomorphism there onto $S_R^n\setminus \{p,-p\}$ by [F8], while ${\exp }_p(0)=p$ is not in that image, ${\exp }_p$ is injective on the whole open ball $B_0(\pi R)$.

Boundary cases and choice. The cases $y=x$ and $y=-x$ of the distance formula, that is $\theta =0$ and $\theta =\pi$, were treated separately in step 4.1 and are re-derived in step 6.1; the endpoint $t=\pi R$ of the minimizing interval is included, and minimization fails only strictly beyond it. The zero vector and constant geodesics were handled in step 3.1, and the case $n\ge 2$ guarantees that the curvature statement of [F7] is not vacuous. The only selections are single minimizing geodesics supplied by [F5] and, in the antipodal case of step 4.1, one unit tangent vector; the inherited ${\mathrm{A}\mathrm{C}}_{\omega }$ of [A1] is used only through the cited suppliers, and no family of choices is made.