The mass metric of a particle constrained to the unit sphere in spherical coordinates

The map $F:{\mathbb{R}}^3\to \mathbb{R}$, $F(x)=⟨x,x{⟩}_2$, is smooth because it is a finite sum of coordinate squares and [L2] applies. For $h\in {\mathbb{R}}^3$, $F(x+h)-F(x)-2⟨x,h{⟩}_2=⟨h,h{⟩}_2=\mathrm{\parallel }h{\mathrm{\parallel }}_2^2$, whose ratio to $\mathrm{\parallel }h{\mathrm{\parallel }}_2$ tends to $0$; hence $D{F}_x(h)=2⟨x,h{⟩}_2$. At every $x\in F^{-1}(1)$, the vector $h=x\mathrm{/}2$ has $D{F}_x(h)=1$, so this differential is surjective. The fibre is nonempty since $e_3\in F^{-1}(1)$; [L4] therefore makes $S^2=F^{-1}(1)$ a smooth embedded submanifold of dimension $3-1=2$. Its inclusion is a smooth embedding by [L2], so [L1] applies and gives $G_q(u,v)=m⟨u,v{⟩}_2$ and positive definiteness at every point of the sphere.

The partial derivatives of $\mathrm{\Psi }$ are ${\mathrm{\partial }}_{\theta }\mathrm{\Psi }=(\cos \theta \cos \phi ,\cos \theta \sin \phi ,-\sin \theta )$ and ${\mathrm{\partial }}_{\phi }\mathrm{\Psi }=(-\sin \theta \sin \phi ,\sin \theta \cos \phi ,0)$. For $0<\theta <\pi$ these are linearly independent: a linear combination $\alpha {\mathrm{\partial }}_{\theta }\mathrm{\Psi }+\beta {\mathrm{\partial }}_{\phi }\mathrm{\Psi }=0$ has third component $-\alpha \sin \theta$ and first two components combining as $\cos \theta (\alpha \cos \phi ,\alpha \sin \phi )+\beta (-\sin \theta \sin \phi ,\sin \theta \cos \phi )=0$; the first relation gives $\alpha =0$ because $\sin \theta \mathrm{\ne }0$, and then $\beta =0$ because $\sin \theta \mathrm{\ne }0$. By [L2] this proves that $d\mathrm{\Psi }$ is injective.

The image of $\mathrm{\Psi }$ lies in $S^2$ by [L3]. For $x\in U$, the exclusion of the poles gives $-1<x_3<1$, so [L5] gives a unique $\theta \in (0,\pi )$ with $\cos \theta =x_3$ and $\sin \theta =\sqrt{1-x_3^2}>0$. The pair $(x_1,x_2)\mathrm{/}\sin \theta$ lies on the unit circle and is not $(1,0)$; hence [L5] gives a unique $\phi \in (0,2\pi )$ producing $x$. Conversely an angle in that interval cannot give $(1,0)$ by the same circle bijection. Thus $\mathrm{\Psi }:D\to U$ is bijective. The removed meridian is relatively closed, so $U$ is open in $S^2$. Smoothness follows from [L3] and the embedded-target criterion [L4]. Its differential is an isomorphism between two-dimensional tangent spaces by steps 1.1 and 2.1, so [L4] supplies smooth local inverses everywhere; bijectivity makes these restrictions of its single global inverse. Thus $\mathrm{\Psi }$ is a diffeomorphism onto $U$, and $(U,{\mathrm{\Psi }}^{-1})$ is a smooth chart.

The three inner products of the coordinate vector fields are $⟨{\mathrm{\partial }}_{\theta }\mathrm{\Psi },{\mathrm{\partial }}_{\theta }\mathrm{\Psi }{⟩}_2={\cos }^2\theta {\cos }^2\phi +{\cos }^2\theta {\sin }^2\phi +{\sin }^2\theta =1$, $⟨{\mathrm{\partial }}_{\theta }\mathrm{\Psi },{\mathrm{\partial }}_{\phi }\mathrm{\Psi }{⟩}_2=-\cos \theta \sin \theta \cos \phi \sin \phi +\cos \theta \sin \theta \sin \phi \cos \phi =0$ and $⟨{\mathrm{\partial }}_{\phi }\mathrm{\Psi },{\mathrm{\partial }}_{\phi }\mathrm{\Psi }{⟩}_2={\sin }^2\theta {\sin }^2\phi +{\sin }^2\theta {\cos }^2\phi ={\sin }^2\theta$, using ${\cos }^2+{\sin }^2=1$ for each angle.

Since $\mathrm{\Psi }$ is the parametrization of the inclusion chart, the mass metric in these coordinates is $G_{ij}=m⟨{\mathrm{\partial }}_i\mathrm{\Psi },{\mathrm{\partial }}_j\mathrm{\Psi }{⟩}_2$; by step 3.2 its matrix is $m\left(\begin{array}{cc}1& 0\\ 0& {\sin }^2\theta \end{array}\right)$, so for a coordinate velocity $(\dot{\theta },\dot{\phi })$ the kinetic energy is $T=\frac{1}{2}m({\dot{\theta }}^2+{\sin }^2\theta {\dot{\phi }}^2)$, and the metric is positive definite on the chart domain because $\sin \theta \mathrm{\ne }0$.