Sphere graph charts, surface density, and a finite partition

Calculus for the graph expressions. For $t,s>0$, $\mathrm{\mid }\sqrt{t}-\sqrt{s}\mathrm{\mid }=\mathrm{\mid }t-s\mathrm{\mid }\mathrm{/}(\sqrt{t}+\sqrt{s})\le \mathrm{\mid }t-s\mathrm{\mid }\mathrm{/}\sqrt{s}$, proving continuity at $s$. Rationalizing the difference quotient gives $(\sqrt{\cdot }{)}^{\mathrm{\prime }}(s)=1\mathrm{/}(2\sqrt{s})$. By the product and quotient rules in [F3], induction shows that every derivative of $\sqrt{t}$ is a constant times an integer power of $\sqrt{t}$, hence exists and is continuous for $t>0$. Put $q(y)=1-{\sum }_j{y}_j^2$. Applying the chain rule on each coordinate line gives ${\mathrm{\partial }}_j\sqrt{q(y)}=-y_j\mathrm{/}\sqrt{q(y)}$. Repeated coordinate product and quotient rules show that every ordered partial of $\sqrt{q}$ and of its reciprocal is a finite sum of polynomial numerators divided by positive integer powers of $\sqrt{q}$. All these expressions are continuous on $B$, where $q>0$; thus the graph functions and density are smooth in the sense of [F3].

A rank-one determinant. For every $t\ge 0$ and $v\in {\mathbb{R}}^m$ with $m\ge 1$, $\det (I+tv{v}^T)=1+t\mathrm{\mid }v{\mathrm{\mid }}^2$. Indeed the $j$-th column of $I+tv{v}^T$ is $e_j+t{v}_{j}v$, so multilinearity in the columns [F7] expands the determinant over subsets $S$ of the column indices: the term for $S$ is $\left({\prod }_{j\in S}t{v}_j\right)\det (c^{(S)})$, where $c^{(S)}$ has column $v$ in the slots $S$ and $e_j$ elsewhere. If $\mathrm{\mid }S\mathrm{\mid }\ge 2$ two columns are equal to $v$, so the determinant vanishes [F7]; the term $S=\mathrm{\varnothing }$ is $\det I=1$; and for $S=\{j\}$ the matrix has determinant $v_j$ (expanding along the standard basis columns), giving $t{v}_j^2$. Summing gives $1+t{\sum }_j{v}_j^2=1+t\mathrm{\mid }v{\mathrm{\mid }}^2$.

The density. Here $h(y)=\epsilon \sqrt{1-\mathrm{\mid }y{\mathrm{\mid }}^2}$, so by [F3] $\mathrm{\nabla }h(y)=\epsilon \cdot \frac{-y}{\sqrt{1-\mathrm{\mid }y{\mathrm{\mid }}^2}}=-\epsilon (1-\mathrm{\mid }y{\mathrm{\mid }}^2{)}^{-1\mathrm{/}2}y,\mathrm{\mid }\mathrm{\nabla }h(y){\mathrm{\mid }}^2=\frac{\mathrm{\mid }y{\mathrm{\mid }}^2}{1-\mathrm{\mid }y{\mathrm{\mid }}^2},$ hence $\sqrt{1+\mathrm{\mid }\mathrm{\nabla }h(y){\mathrm{\mid }}^2}=(1-\mathrm{\mid }y{\mathrm{\mid }}^2{)}^{-1\mathrm{/}2}$ on $B$. By [F2] this is the chart density of the graph, and by [F1] the chart measure is the polar surface measure $\sigma$; the density is finite and strictly positive on $B$ because $1-\mathrm{\mid }y{\mathrm{\mid }}^2>0$ there.

The charts and the cover. Fix $i$ and $\epsilon$ and put ${\pi }_i(x)=(x_1,\dots ,\widehat{x_i},\dots ,x_n)$ for the coordinate projection. On the hemisphere $H_i^{\epsilon }=\{x\in S^{n-1}:\epsilon x_i>0\}$ the map ${\pi }_i$ is a left inverse of $X_i^{\epsilon }$: ${\pi }_i(X_i^{\epsilon }(y))=y$; conversely $X_i^{\epsilon }({\pi }_i(x))=x$ for $x\in H_i^{\epsilon }$, because the removed coordinate is recovered by $x_i=\epsilon \sqrt{1-\mathrm{\mid }{\pi }_i(x){\mathrm{\mid }}^2}$, which is exactly the defining equation of the sphere with the sign $\epsilon$. The coordinates of $X_i^{\epsilon }$ are smooth by step 1.1, and its inverse is coordinate projection, since $1-\mathrm{\mid }y{\mathrm{\mid }}^2>0$ on $B$ and the coordinates of a unit vector satisfy $\mathrm{\mid }{x}_i\mathrm{\mid }\le 1$; hence $X_i^{\epsilon }$ is a $C^{\mathrm{\infty }}$ chart of $S^{n-1}$ onto $H_i^{\epsilon }$. Every $x\in S^{n-1}$ satisfies ${\sum }_k{x}_k^2=1$, so some coordinate is nonzero; then $x\in H_i^{\mathrm{sgn}(x_i)}$ for that $i$, and the $2n$ hemispheres cover $S^{n-1}$.

The Hessian determinant. Differentiating the gradient of step 1.3 with [F3] gives ${\mathrm{\partial }}_i{\mathrm{\partial }}_{j}h(y)=-\epsilon [(1-\mathrm{\mid }y{\mathrm{\mid }}^2{)}^{-1\mathrm{/}2}{\delta }_{ij}+(1-\mathrm{\mid }y{\mathrm{\mid }}^2{)}^{-3\mathrm{/}2}{y}_i{y}_j],$ that is $D^{2}h(y)=-\epsilon a[I+a^{2}y{y}^T]$ with $a:=(1-\mathrm{\mid }y{\mathrm{\mid }}^2{)}^{-1\mathrm{/}2}>0$. Taking determinants and using step 1.2 with $t=a^2$, $v=y$, $\det D^{2}h(y)=(-\epsilon {)}^{n-1}{a}^{n-1}(1+a^2\mathrm{\mid }y{\mathrm{\mid }}^2)=(-\epsilon {)}^{n-1}{a}^{n-1}\cdot \frac{1}{1-\mathrm{\mid }y{\mathrm{\mid }}^2}=(-\epsilon {)}^{n-1}(1-\mathrm{\mid }y{\mathrm{\mid }}^2{)}^{-(n+1)\mathrm{/}2},$ because $1+a^2\mathrm{\mid }y{\mathrm{\mid }}^2=1+\mathrm{\mid }y{\mathrm{\mid }}^2\mathrm{/}(1-\mathrm{\mid }y{\mathrm{\mid }}^2)=1\mathrm{/}(1-\mathrm{\mid }y{\mathrm{\mid }}^2)$ and $a^{n-1}{a}^2=a^{n+1}$. The value is nonzero for every $y\in B$ since $1-\mathrm{\mid }y{\mathrm{\mid }}^2>0$.

The smooth structure and compactness. Each $X_i^{\epsilon }$ is a bijection of the open ball onto its image with smooth inverse and smooth transitions: on overlaps, the transition is ${\pi }_{i^{\mathrm{\prime }}}\circ X_i^{\epsilon }$, a coordinate selection of a smooth graph map, smooth by [F3]. The ambient coordinate map that deletes $x_i$ and appends $x_i-h_i^{\epsilon }({\pi }_i(x))$ has smooth inverse obtained by reinserting $h_i^{\epsilon }(y)+z$ in the $i$th slot; near each point of the hemisphere it carries the sphere to $z=0$. These are slice charts in [F6], so the sphere is an embedded smooth manifold with the displayed graph charts. Being the zero set of the continuous function $x\mapsto \mathrm{\mid }x{\mathrm{\mid }}^2-1$, the sphere is closed; it is bounded by $\mathrm{\mid }x\mathrm{\mid }=1$, so it is compact by [F4].

The finite partition. The $2n$ hemispheres $H_i^{\epsilon }$ are coordinate balls of the smooth manifold $S^{n-1}$ of step 3.1 and form a countable (indeed finite) open cover. By [F5] this cover admits a smooth partition of unity $({\chi }_i^{\epsilon }{)}_{i,\epsilon }$ subordinate to it: the supports are locally finite, $\mathrm{supp}{\chi }_i^{\epsilon }\subseteq H_i^{\epsilon }$, and ${\sum }_{i,\epsilon }{\chi }_i^{\epsilon }=1$ with every ${\chi }_i^{\epsilon }\ge 0$. Since $S^{n-1}$ is compact by step 3.1, every support, being closed in $S^{n-1}$, is compact; enumerating the $2n$ pairs $(i,\epsilon )$ as $1,\dots ,m$ with $m=2n$ gives the asserted functions, each compactly supported inside the image $H_i^{\epsilon }$ of the chart $X_i^{\epsilon }$.

Conclusion. Step 2.1 gives the $2n$ smooth graph charts and their hemisphere cover, step 1.3 computes the density $(1-\mathrm{\mid }y{\mathrm{\mid }}^2{)}^{-1\mathrm{/}2}$, step 2.2 computes $\det D^{2}h=(-\epsilon {)}^{n-1}(1-\mathrm{\mid }y{\mathrm{\mid }}^2{)}^{-(n+1)\mathrm{/}2}\mathrm{\ne }0$, and step 4.1 produces the finite smooth partition with compactly supported pieces. Each clause holds for every $n\ge 2$, with the case $n=2$ covered by the same computation ($m=n-1=1$ and the determinant is $-\epsilon (1-\mathrm{\mid }y{\mathrm{\mid }}^2{)}^{-3\mathrm{/}2}$).