Measure-preserving mappings from the unit cube to some symmetric spaces

TL;DR

This paper constructs measure-preserving mappings from the unit cube to various symmetric spaces, including spheres and projective spaces, and provides a method for generating such mappings to product spaces and fiber bundles.

Contribution

It introduces explicit constructions of measure-preserving maps from the unit cube to symmetric spaces and extends to product spaces and fiber bundles.

Findings

Constructed measure-preserving maps to spheres and projective spaces.

Provided a procedure for mappings to product spaces and fiber bundles.

Applicable to a range of symmetric spaces and complex structures.

Abstract

We construct measure-preserving mappings from the $d$-dimensional unit cube to the $d$-dimensional unit ball and the compact rank one symmetric spaces, namely the $d$-dimensional sphere, the real, complex, and quaternionic projective spaces, and the Cayley plane. We also give a procedure to generate measure-preserving mappings from the $d$-dimensional unit cube to product spaces and fiber bundles under certain conditions.

Tables

Table 1. Table 1 . The volume density in the cross es is ω p ​ ( q ) = Ω ​ ( r ) subscript 𝜔 𝑝 𝑞 Ω 𝑟 \omega_{p}(q)=\Omega(r) , where r = d R ​ ( p , q ) 𝑟 subscript 𝑑 𝑅 𝑝 𝑞 r=d_{R}(p,q) is the Riemannian distance. In this table, we show r d − 1 ​ Ω ​ ( r ) superscript 𝑟 𝑑 1 Ω 𝑟 r^{d-1}\Omega(r) , where d = dim ℝ ( ℳ ) 𝑑 subscript dimension ℝ ℳ d=\dim_{\mathbb{R}}(\mathcal{M}) , for the cross es . We also include the diameter D = diam ⁡ ( ℳ ) 𝐷 diam ℳ D=\operatorname{diam}(\mathcal{M}) , the volume V = vol ⁡ ( ℳ ) 𝑉 vol ℳ V=\operatorname{vol}(\mathcal{M}) , and the exponential map exp ℳ subscript exp ℳ \mathord{\operatorname{exp}}_{\mathcal{M}} . Table taken from [ BeltranDelaTorreLizarte2022 ] .

$ℳ$	$d$	$D$	$V$	${\exp }_{ℳ}\left(v\right)$	$r^{d-1}\Omega \left(r\right)$
${𝕊}^n$	$n$	$\pi$	$\frac{2{\pi }^{\left(n+1\right)/2}}{\Gamma \left(\frac{n+1}{2}\right)}$	$\left(\begin{array}{c}\frac{v}{\parallel v\parallel }\sin \parallel v\parallel \\ \cos \parallel v\parallel \end{array}\right)$	${\sin }^{n-1}r$
$ℝ{\mathbb{P}}^n$	$n$	$\pi /2$	$\frac{{\pi }^{\left(n+1\right)/2}}{\Gamma \left(\frac{n+1}{2}\right)}$	$\left(\begin{array}{c}\frac{v}{\parallel v\parallel }\tan \parallel v\parallel \\ 1\end{array}\right)$	${\sin }^{n-1}r$
$ℂ{\mathbb{P}}^n$	$2n$	$\pi /2$	$\frac{{\pi }^n}{n!}$	$\left(\begin{array}{c}\frac{v}{\parallel v\parallel }\tan \parallel v\parallel \\ 1\end{array}\right)$	${\sin }^{2n-1}r\cos r$
$ℍ{\mathbb{P}}^n$	$4n$	$\pi /2$	$\frac{{\pi }^{2n}}{\left(2n+1\right)!}$	$\left(\begin{array}{c}\frac{v}{\parallel v\parallel }\tan \parallel v\parallel \\ 1\end{array}\right)$	${\sin }^{4n-1}r{\cos }^{3}r$
$𝕆{\mathbb{P}}^2$	$16$	$\pi /2$	$\frac{{\pi }^8}{1320\Gamma \left(8\right)}$	$\left(\begin{array}{c}\frac{v}{\parallel v\parallel }\tan \parallel v\parallel \\ 1\end{array}\right)$	${\sin }^{15}r{\cos }^{7}r$

Table 2. Table 2 . Summary of the manifolds for which we have a closed formula for the measure-preserving mapping Φ ℳ subscript Φ ℳ \Phi_{\mathcal{M}} , where Φ ℝ d subscript Φ superscript ℝ 𝑑 \Phi_{\mathbb{R}^{d}} is as in Equation   1 . The computations are straightforward from our main results; see Appendix B .

$ℳ$	${\Phi }_{ℳ}={\exp }_{ℳ}\circ {\phi }_{ℳ}\circ {\Phi }_{{ℝ}^d}:({(0,1)}^d,\mathrm{unif})\to (ℳ,\mathrm{unif})$
${𝔹}^n$	$\frac{{\Phi }_{{ℝ}^n}\left(x\right)}{\parallel {\Phi }_{{ℝ}^n}\left(x\right)\parallel }{\left(\frac{\gamma (\frac{n}{2},\frac{{\parallel {\Phi }_{{ℝ}^n}\left(x\right)\parallel }^2}{2})}{\Gamma \left(\frac{n}{2}\right)}\right)}^{1/n}$
${𝕊}^1$	$(-\sin 2\pi x,-\cos 2\pi x)$
${𝕊}^2$	$(\frac{{\Phi }_{{ℝ}^2}\left(x\right)}{\parallel {\Phi }_{{ℝ}^2}\left(x\right)\parallel }2e^{-{\parallel {\Phi }_{{ℝ}^2}\left(x\right)\parallel }^2/4}\sqrt{1-e^{-{\parallel {\Phi }_{{ℝ}^2}\left(x\right)\parallel }^2/2}},2e^{-{\parallel {\Phi }_{{ℝ}^2}\left(x\right)\parallel }^2/2}-1)$
$ℝ{\mathbb{P}}^1$	$(-\cot \pi x,1)$
$ℝ{\mathbb{P}}^2$	$(\frac{{\Phi }_{{ℝ}^2}\left(x\right)}{\parallel {\Phi }_{{ℝ}^2}\left(x\right)\parallel }\sqrt{e^{{\parallel {\Phi }_{{ℝ}^2}\left(x\right)\parallel }^2}-1},1)$
$ℂ{\mathbb{P}}^1$	$(\frac{{\Phi }_{{ℝ}^2}\left(x\right)}{\parallel {\Phi }_{{ℝ}^2}\left(x\right)\parallel }\sqrt{e^{{\parallel {\Phi }_{{ℝ}^2}\left(x\right)\parallel }^2/2}-1},1)$
$ℂ{\mathbb{P}}^n$	$(\frac{{\Phi }_{{ℝ}^{2n}}\left(x\right)}{\parallel {\Phi }_{{ℝ}^{2n}}\left(x\right)\parallel }{\left(-1+\frac{1}{1-{\left(\frac{1}{\left(n-1\right)!}\gamma (n,\frac{{\parallel {\Phi }_{{ℝ}^{2n}}\left(x\right)\parallel }^2}{2})\right)}^{1/n}}\right)}^{1/2},1)$

Equations

$$d{\mu }_c(x)=\frac{1}{(2\pi c{)}^{d/2}}{e}^{-\Vert x{\Vert }^2/(2c)}d\lambda (x).$$

$$\begin{array}[]{cccc}\Phi_{\mathbb{R}^{d}}\colon&(0,1)^{d}&\longrightarrow&\mathbb{R}^{d},\\[2.0pt] &\displaystyle\mathchoice{\begin{pmatrix}x_{1}\\ \vdots\\ x_{d}\end{pmatrix}}{\begin{pmatrix}x_{1}\\ \vdots\\ x_{d}\end{pmatrix}}{}{}&\longmapsto&\displaystyle\mathchoice{\begin{pmatrix}\sqrt{2}\operatorname{erf}^{-1}(2x_{1}-1)\\ \vdots\\ \sqrt{2}\operatorname{erf}^{-1}(2x_{d}-1)\end{pmatrix}}{\begin{pmatrix}\sqrt{2}\operatorname{erf}^{-1}(2x_{1}-1)\\ \vdots\\ \sqrt{2}\operatorname{erf}^{-1}(2x_{d}-1)\end{pmatrix}}{}{},\end{array}$$

$$\mathrm{erf}(t)=\frac{2}{\sqrt{\pi }}{\int }_0^t{e}^{-s^2}ds.$$

$$d{\mu }_{\omega }(x)=\omega (\Vert x\Vert )d\lambda (x).$$

$$1={\int }_{x\in B^d(0,R)}d{\mu }_{\omega }(x)={\int }_{x\in B^d(0,R)}\omega (\Vert x\Vert )d\lambda (x)=\frac{2{\pi }^{d/2}}{\Gamma (\frac{d}{2})}{\int }_0^R\omega (s)s^{d-1}ds.$$

$$\begin{array}[]{cccc}\mathord{\operatorname{exp}}_{\mathcal{M}}\colon&B^{d}(0,D)&\longrightarrow&\mathcal{M},\\[2.0pt] &v&\longmapsto&\mathord{\operatorname{exp}}_{(0,\dotsc,0,1)}(v),\end{array}$$

$$\gamma (t,x)={\int }_0^x{s}^{t-1}{e}^{-s}ds.$$

$$\varphi_{\mathbb{B}^{d}}(x)=\frac{x}{\lVert x\rVert}\bigg{(}\frac{\gamma\big{(}\frac{d}{2},\frac{\lVert x\rVert^{2}}{2}\big{)}}{\Gamma(\frac{d}{2})}\bigg{)}^{1/d}.$$

$$\int_{0}^{\rho}\omega(s)s^{d-1}\,ds=\frac{1}{2\pi^{d/2}}\gamma\bigg{(}\frac{d}{2},\frac{r^{2}}{2}\bigg{)}.$$

$$((0,1{)}^d,\mathrm{unif})$$

$$\begin{array}[]{cccc}\Phi_{\mathcal{M}_{1}\times\mathcal{M}_{2}}\colon&((0,1)^{\dim(\mathcal{M}_{1})+\dim(\mathcal{M}_{2})},\operatorname{unif})&\longrightarrow&(\mathcal{M}_{1}\times\mathcal{M}_{2},\operatorname{unif}),\\[2.0pt] &(x,y)&\longmapsto&(\Phi_{\mathcal{M}_{1}}(x),\Phi_{\mathcal{M}_{2}}(y)),\end{array}$$

$$\begin{array}[]{cccc}\Phi_{E}\colon&((0,1)^{\dim(E)},\operatorname{unif})&\longrightarrow&(E,\operatorname{unif}),\\[2.0pt] &(y,z)&\longmapsto&\Psi_{\Phi_{B}(y)}(\Phi_{F}(z)),\end{array}$$

$$G(\rho)=\frac{1}{2\pi^{d/2}}\gamma\bigg{(}\frac{d}{2},\frac{r^{2}}{2c}\bigg{)}.$$

$$\omega (\rho ){\rho }^{d-1}{\rho }^{\prime }(r)=\frac{r^{d-1}}{(2\pi c{)}^{d/2}}{e}^{-r^2/(2c)}.$$

$$\mathrm{Jac}\phi (x)=$$

$$=$$

$$\stackrel{\text{eqref}eq:aux1}{=}$$

$${\mu }_c(\mathcal{A})=$$

$$=$$

$$\omega (\rho )=\frac{e^{-{\rho }^2/(2b)}}{(2\pi b{)}^{d/2}}.$$

$$G(\rho)=\int_{0}^{\rho}\omega(s)s^{d-1}\,ds=\frac{1}{(2\pi b)^{d/2}}\int_{0}^{\rho}s^{d-1}e^{-s^{2}/(2b)}\,ds=\frac{1}{2\pi^{d/2}}\gamma\bigg{(}\frac{d}{2},\frac{\rho^{2}}{2b}\bigg{)}.$$

$$\frac{1}{2\pi^{d/2}}\gamma\bigg{(}\frac{d}{2},\frac{\rho^{2}}{2b}\bigg{)}=\frac{1}{2\pi^{d/2}}\gamma\bigg{(}\frac{d}{2},\frac{r^{2}}{2c}\bigg{)},$$

$$\phi (x)=x\sqrt{\frac{b}{c}}$$

$$d{\mu }_{\text{stereo}}(x)=\frac{1}{\mathrm{vol}({\mathbb{S}}^d)}\frac{2^d}{(1+\Vert x{\Vert }^2{)}^d}d\lambda (x).$$

$$\omega (\rho )=\frac{1}{\mathrm{vol}({\mathbb{S}}^d)}\frac{2^d}{(1+{\rho }^2{)}^d}=\frac{\Gamma (\frac{d+1}{2})}{2{\pi }^{(d+1)/2}}\frac{2^d}{(1+{\rho }^2{)}^d}.$$

$$G(\rho )={\int }_0^{\rho }\omega (s)s^{d-1}ds=\frac{2^{d-1}\Gamma (\frac{d+1}{2})}{{\pi }^{(d+1)/2}}{\int }_0^{\rho }\frac{s^{d-1}}{(1+s^2{)}^d}ds.$$

$$G(\rho )=\frac{2\Gamma (\frac{3}{2})}{{\pi }^{3/2}}{\int }_0^{\rho }\frac{s}{(1+s^2{)}^2}ds=\frac{{\rho }^2}{2\pi (1+{\rho }^2)}.$$

$$\frac{\rho^{2}}{2\pi(1+\rho^{2})}=\frac{1}{2\pi}\gamma\bigg{(}1,\frac{r^{2}}{2}\bigg{)},$$

$${\rho }^2=\frac{1}{e^{-r^2/2}}-1=e^{r^2/2}-1.$$

Full text

The first and third authors have been supported by grant PID2020-113887GB-I00 funded by MCIN/AEI/ 10.13039/501100011033. The second author thankfully acknowledges support by the Methusalem grant of the Flemish Government. The third author has also been supported by grant PRE2021-097772 funded by MCIN/AEI/ 10.13039/501100011033 and by “ESF Investing in your future”.

1. 1 Introduction and main results
2. 2 A technical result
3. 3 Measure-preserving mappings from the unit cube to the compact rank one symmetric spaces
4. 4 Measure-preserving mappings from the unit cube to fiber bundles
5. A The smooth coarea formula
6. B Auxiliary computations

1. Introduction and main results

Given two measure spaces $(\Omega_{1},\Sigma_{1},\mu_{1})$ and $(\Omega_{2},\Sigma_{2},\mu_{2})$, a bijective mapping $\varphi\colon\Omega_{1}\to\Omega_{2}$ is said to be measure preserving if both $\varphi$ and $\varphi^{-1}$ are measurable mappings and moreover $\mu_{2}(A)=\mu_{1}(\varphi^{-1}(A))$ for every $A\in\Sigma_{2}$; or, equivalently, if $\mu_{1}(A)=\mu_{2}(\varphi(A))$ for every $A\in\Sigma_{1}$. In this work, we look for measure-preserving smooth diffeomorphisms between Riemannian manifolds.

The problem of finding measure-preserving mappings from one manifold to another has applications in cartography, computer graphics, medical imaging, signal processing, or, more generally, in any area that requires good discretizations of a certain space. Thus, when looking for uniform collections of points or uniform grids (that is, grids all of whose cells have the same volume) on a manifold $\mathcal{M}$, a frequent approach consists in generating collections or grids with that property on a simpler, easily discretizable space such as the unit cube, and then transporting them to $\mathcal{M}$ through a measure-preserving mapping. In this sense, most of the research has been carried out for two-dimensional and three-dimensional manifolds (see [ShirleyChiu1997, HolhosRosca2014, HolhosRosca2016, HolhosRosca2019, Holhos2017, RoscaPlonka2011, RoscaMorawiecDeGraef2014] and references therein). In [RoscaMorawiecDeGraef2014], the authors also obtain a mapping from the $n$-dimensional sphere of radius $r$ in $\mathbb{R}^{n+1}$ to the $n$-dimensional ball of radius $R$ in $\mathbb{R}^{n}$ which generalizes the equal-area Lambert mapping.

Measure-preserving mappings are also relevant in the theory of partial differential equations on Lipschitz domains (see [GriepentrogHoepnerKaiserRehberg2008]), in the generation of low-discrepancy points (see, for example, [BrauchartDick2012, DeMarchiElefante2018, Alexa2022, FerizovicHofstadlerMastrianni2022, Ferizovic2022]), and, more recently, they have been used to generate projective constellations for noncoherent communications over single-input-multiple-output (simo) channels; see [NgoDecurningeGuillaudYang2020], where the authors construct a measure-preserving mapping from the unit square to the complex projective line $\mathbb{CP}^{1}$, or [CuevasAlvarezVizosoBeltranSantamariaTucekPeters2022] for the higher dimensional case. However, to the best of our knowledge, there are no constructive procedures to generate measure-preserving mappings from the $d$-dimensional unit cube to the $d$-sphere and to the remaining projective spaces.

1.1. Notation

In this paper, $\lambda$ denotes the Lebesgue measure in $\mathbb{R}^{d}$, and $B^{d}(0,R)$ denotes the open ball of radius $R\in(0,\infty]$ in $\mathbb{R}^{d}$ (if $R=\infty$, this means just $\mathbb{R}^{d}$). When $R=1$, we denote it by $\mathbb{B}^{d}$. We will call $(0,1)^{d}$ the (open) unit cube.

We denote the measure associated to the normal distribution $\mathcal{N}(0,c)$ in $\mathbb{R}^{d}$ by $\mu_{c}$, that is,

[TABLE]

It is well known that the mapping

[TABLE]

is measure preserving from $((0,1)^{d},\lambda)$ to $(\mathbb{R}^{d},\mu_{c=1})$, where $\operatorname{erf}^{-1}$ is the inverse of the error function $\operatorname{erf}\colon\mathbb{R}\to(-1,1)$ given by

[TABLE]

Given any continuous function $\omega\colon(0,R)\to(0,\infty)$, we consider the associated measure in $B^{d}(0,R)$ given by the weight function $\omega(\lVert x\rVert)$ and denote it by $\mu_{\omega}$, that is,

[TABLE]

We will always assume that $\mu_{\omega}$ is a probability measure, i.e.,

[TABLE]

Finally, if we have a Riemannian manifold $\mathcal{M}$ (including the case that $\mathcal{M}$ is the unit cube with the standard structure, the usual $d$-sphere, or any open set of a compact Riemannian manifold), we denote by $\mathrm{unif}$ the uniform measure in $\mathcal{M}$ according to its volume form. For example, $(\mathbb{B}^{d},\operatorname{unif})$ is the unit ball endowed with the Lebesgue measure normalized to have volume $1$, which can be denoted in our previous notation by $\mu_{\omega=1/\operatorname{vol}(\mathbb{B}^{d})}$.

1.2. The compact rank one symmetric spaces

The compact rank one symmetric spaces (crosses) are the $n$-sphere $\mathbb{S}^{n}$ and the real, complex, quaternionic, and octonionic projective spaces $\mathbb{RP}^{n},\mathbb{CP}^{n}$, $\mathbb{HP}^{n}$, and $\mathbb{OP}^{2}$. These spaces, which were classified by É. Cartan, are examples of locally harmonic Blaschke manifolds; in fact, Lichnerowicz’s conjecture claims that the crosses are the only Riemannian manifolds of this kind. They are also the only compact connected two-point homogeneous Riemannian manifolds. See [Besse1978] for more information about these spaces.

Let $\mathcal{M}$ be a cross and let $d$, $D$, and $V$ be, respectively, its real dimension, its diameter (that is, the maximum Riemannian distance between two points in $\mathcal{M}$), and its volume. The exponential map based on the north pole

[TABLE]

is a diffeomorphism onto $\mathcal{M}\setminus\mathcal{X}$, where $\mathcal{X}$ is a measure zero set (just a point in the case $\mathcal{M}=\mathbb{S}^{n}$ and a hyperplane for the projective spaces). Moreover, the Jacobian of $\mathord{\operatorname{exp}}_{\mathcal{M}}$ is known as the volume density and has the form $\Omega(\lVert v\rVert)$ for a certain function $\Omega$. As a consequence, we have the following lemma:

Lemma 1.1**.**

Let $\mathcal{M}$ be a cross. Then, the exponential map $\mathord{\operatorname{exp}}_{\mathcal{M}}$ is a measure-preserving mapping from $(B^{d}(0,D),\mu_{\omega=\Omega/V})$ to $(\mathcal{M},\operatorname{unif})$.

Proof.

Since $\mathord{\operatorname{exp}}_{\mathcal{M}}$ is a smooth diffeomorphism, both $\mathord{\operatorname{exp}}_{\mathcal{M}}$ and its inverse are measurable mappings. Now, let $A\subseteq(0,1)^{d}$ be a measurable set. Applying the change of variables theorem to $\mathord{\operatorname{exp}}_{\mathcal{M}}$ we readily get the result. ∎

Table 1 summarizes the dimension, the diameter, the volume, the exponential map, and the volume density of these classical spaces.

1.3. Main results

Let $\gamma$ denote the incomplete gamma function:

[TABLE]

Our first main result is the following proposition, which yields a measure-preserving mapping from the unit cube to the unit ball:

Proposition 1.2**.**

Let $\varphi_{\mathbb{B}^{d}}\colon(\mathbb{R}^{d},\mu_{c=1})\to(\mathbb{B}^{d},\operatorname{unif})$ be the mapping given by

[TABLE]

Then, the mapping $\Phi_{\mathbb{B}^{d}}=\varphi_{\mathbb{B}^{d}}\circ\Phi_{\mathbb{R}^{d}}\colon((0,1)^{d},\operatorname{unif})\to(\mathbb{B}^{d},\operatorname{unif})$ is measure preserving.

The next main result provides a procedure to generate measure-preserving mappings from the unit cube to each cross.

Theorem 1.3**.**

Let $\mathcal{M}$ be a cross and let $\varphi_{\mathcal{M}}\colon(\mathbb{R}^{d},\mu_{c=1})\to(B^{d}(0,D),\mu_{\omega=\Omega/V})$ be the mapping given by $\varphi_{\mathcal{M}}(x)=x\rho(\lVert x\rVert)/\lVert x\rVert$, where $\rho=\rho(r)$ is the unique solution to

[TABLE]

Then, the mapping $\Phi_{\mathcal{M}}=\mathord{\operatorname{exp}}_{\mathcal{M}}\circ\varphi_{\mathcal{M}}\circ\Phi_{\mathbb{R}^{d}}\colon((0,1)^{d},\operatorname{unif})\to(\mathcal{M},\operatorname{unif})$ is measure preserving.

The following conmutative diagram illustrates the construction described in Theorem 1.3:

[TABLE]

It follows straightforwardly from the definition of measure-preserving mapping that, given two Riemannian manifolds $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$, and two measure-preserving mappings $\Phi_{\mathcal{M}_{1}}\colon(0,1)^{\dim(\mathcal{M}_{1})}\to\mathcal{M}_{1}$ and $\Phi_{\mathcal{M}_{2}}\colon(0,1)^{\dim(\mathcal{M}_{2})}\to\mathcal{M}_{2}$, the mapping

[TABLE]

where $x\in(0,1)^{\dim(\mathcal{M}_{1})}$ and $y\in(0,1)^{\dim(\mathcal{M}_{2})}$, is also measure preserving. As a consequence, since by Theorem 1.3 we have measure-preserving mappings from the unit cube to any cross, we also have a constructive procedure to generate measure-preserving mappings from the unit cube to any finite product of crosses. In this work we generalize this property to the case of fiber bundles:

Theorem 1.4**.**

Let $E$, $B$, and $F$ be Riemannian manifolds, where we assume that the measures in $E$, $B$, and $F$ are normalized to have unit volume, and let $F\hookrightarrow E\overset{\pi}{\to}B$ be a smooth fiber bundle such that $\operatorname{NJac}\pi(x)$ is constant for every $x\in E$. Let $\Phi_{B}\colon((0,1)^{\dim(B)},\operatorname{unif})\to(B,\operatorname{unif})$ and $\Phi_{F}\colon((0,1)^{\dim(F)},\operatorname{unif})\to(F,\operatorname{unif})$ be measure-preserving mappings. Let $\Psi_{y}\colon(F,\operatorname{unif})\to(\pi^{-1}(y),\operatorname{unif})$ be a measure-preserving mapping for every $y\in B$ such that the mapping $\xi\colon(B\times F,\operatorname{unif})\to(E,\operatorname{unif})$ given by $\xi(y,z)=\Psi_{y}(z)$ is measurable. Then, $\xi$ is measure preserving and hence the mapping

[TABLE]

where $y\in(0,1)^{\dim(B)}$ and $z\in(0,1)^{\dim(F)}$, is measure preserving.

1.4. Structure of the paper

In Section 2, we prove our main technical result, which yields a procedure to generate measure-preserving mappings from $(\mathbb{R}^{d},\mu_{c})$ to $(B^{d}(0,R),\mu_{\omega})$, and we prove Proposition 1.2. In Section 3, we prove Theorem 1.3 and we construct measure-preserving mappings from the unit cube to each cross. In Section 4, we prove Theorem 1.4 and we show an alternative procedure to construct measure-preserving mappings from the unit cube to odd-dimensional spheres using the Hopf fibration. Appendix A is devoted to the smooth coarea formula, a technical tool. Finally, in Appendix B we present some auxiliary computations.

2. A technical result

Recall that $\omega\colon(0,R)\to(0,\infty)$ is any continuous function satisfying (2).

Theorem 2.1**.**

Let $G(\rho)=\int_{0}^{\rho}\omega(s)s^{d-1}\,ds$, and let $\rho=\rho(r)$ be the unique solution to

[TABLE]

Then, the mapping $\varphi\colon(\mathbb{R}^{d},\mu_{c})\to(B^{d}(0,R),\mu_{\omega})$ given by $\varphi(x)=x\rho(\lVert x\rVert)/\lVert x\rVert$ is measure preserving.

Proof.

First, note that $\omega(\rho)>0$ implies that $G$ is an increasing function with $G(0)=0$. Moreover, Equation 2 implies that $G(R)=\Gamma(d/2)/2\pi^{d/2}$, which means that $\rho$ is a well-defined bijection with $\rho(r\to\infty)\to R$. The inverse of $\varphi$ is easily computed: $\varphi^{-1}(y)=y\rho^{-1}(\lVert y\rVert)/\lVert y\rVert$.

Computing the derivative with respect to $r$ at both sides of (3), we get

[TABLE]

Now, let $f(r)=\rho(r)/r$ and compute the Jacobian of $\varphi(x)=xf(\lVert x\rVert)$ by choosing an orthonormal basis $v_{1}^{x},\ldots,v_{d}^{x}$ at $x\in\mathbb{R}^{d}$, with $v_{1}^{x}=x/\lVert x\rVert$. A straightforward computation shows that $D\varphi(x)$ preserves the orthogonality of that basis and yields

[TABLE]

Then, given any measurable set $\mathcal{A}\subseteq\mathbb{R}^{d}$, we can check that the measure of $\mathcal{A}$ in $(\mathbb{R}^{d},\mu_{c})$ equals that of $\varphi(\mathcal{A})$ in $(B^{d}(0,R),\mu_{\omega})$ using the change of variables theorem: if we denote by $\chi_{\mathcal{A}}$ the characteristic function of $\mathcal{A}$, then

[TABLE]

which proves the theorem. ∎

Example 2.2** (A measure-preserving mapping from $(\mathbb{R}^{d},\mu_{c})$ to $(\mathbb{R}^{d},\mu_{b})$).**

In this case, we have $R=\infty$ and

[TABLE]

Following Theorem 2.1, let

[TABLE]

We have to obtain $\rho$ from

[TABLE]

which is obviously solved by $\rho(r)=r\sqrt{b/c}$. Thus, we conclude that

[TABLE]

defines a measure-preserving mapping from $(\mathbb{R}^{d},\mu_{c})$ to $(\mathbb{R}^{d},\mu_{b})$.

Example 2.3** (A measure-preserving mapping from $(\mathbb{R}^{d},\mu_{c=1}))$ to $(\mathbb{R}^{d},\mu_{\text{stereo}})$).**

Let $\mu_{\text{stereo}}$ be the measure that makes the stereographic projection a measure-preserving mapping, that is,

[TABLE]

In this case, we have $c=1$, $R=\infty$, and

[TABLE]

We compute

[TABLE]

If $d=2$,

[TABLE]

Therefore, we have to obtain $\rho$ from

[TABLE]

that is,

[TABLE]

Hence, we conclude that

[TABLE]

defines a measure-preserving mapping from $(\mathbb{R}^{2},\mu_{c=1})$ to $(\mathbb{R}^{2},\mu_{\text{stereo}})$.

Example 2.4** (A measure-preserving mapping from $(\mathbb{R}^{d},\mu_{c=1})$ to $(\mathbb{B}^{d},\operatorname{unif})$).**

In this case, we have $c=1$, $R=1$, and

[TABLE]

We easily compute

[TABLE]

and we have to obtain $\rho$ from

[TABLE]

concluding that, following the notation of Proposition 1.2,

[TABLE]

defines a measure-preserving mapping from $(\mathbb{R}^{d},\mu_{c=1})$ to $(\mathbb{B}^{d},\operatorname{unif})$.

Proof of Proposition 1.2.

Immediate from Example 2.4 and the fact that $\Phi_{\mathbb{R}^{d}}$ is measure preserving. ∎

3. Measure-preserving mappings from the unit cube to the compact rank one symmetric spaces

After Theorem 2.1, the proof of Theorem 1.3 is now straightforward:

Proof of Theorem 1.3.

Immediate from Theorem 2.1, Lemma 1.1, and the fact that $\Phi_{\mathbb{R}^{d}}$ is measure preserving. ∎

We can now generate measure-preserving mappings from the unit cube to all the crosses following Theorem 2.1: it suffices to consider $\mathord{\operatorname{exp}}_{\mathcal{M}}\circ\varphi_{\mathcal{M}}\circ\Phi_{\mathbb{R}^{d}}$, where, according to Theorem 2.1, the mapping $\varphi_{\mathcal{M}}\colon(\mathbb{R}^{d},\mu_{c=1})\to(B^{d}(0,D),\mu_{\omega=\Omega/V})$ can be computed, to some extent, explicitly. We do the computations for the different choices of $\mathcal{M}$ in the next few subsections. Recall that, for each cross $\mathcal{M}$, we denote its real dimension by $d$, its diameter by $D$, and its volume by $V$ (see Table 1).

3.1. The unit sphere $\mathbb{S}^{n}$

In this case, we have $d=n$, $D=\pi$, and

[TABLE]

Corollary 3.1**.**

The mapping $\varphi_{\mathbb{S}^{n}}\colon(\mathbb{R}^{n},\mu_{c=1})\to(B^{n}(0,\pi),\mu_{\omega=\Omega/V})$ given by $\varphi_{\mathbb{S}^{n}}(x)=x\rho(\lVert x\rVert)/\lVert x\rVert$ is measure preserving if $\rho=\rho(r)$ satisfies

[TABLE]

As a consequence, the mapping $\Phi_{\mathbb{S}^{n}}=\mathord{\operatorname{exp}}_{\mathbb{S}^{n}}\circ\varphi_{\mathbb{S}^{n}}\circ\Phi_{\mathbb{R}^{n}}\colon((0,1)^{n},\operatorname{unif})\to(\mathbb{S}^{n},\operatorname{unif})$ is measure preserving. For $n=1$ we have

[TABLE]

and so

[TABLE]

For $n=2$ we can compute $\rho(r)$ explicitly:

[TABLE]

and hence

[TABLE]

Proof.

From Theorem 2.1 we just need to check that

[TABLE]

which is equivalent to the formula in the corollary. The case $n=2$ reads

[TABLE]

which is equivalent to the last claim in the corollary. ∎

Note that the integral in the left-hand side of the expression in Corollary 3.1 is an incomplete beta function:

[TABLE]

Hence, it is not possible to obtain a closed expression for $\Phi_{\mathbb{S}^{n}}$ when $n>2$. In Section 4 we consider a different approach that provides measure-preserving mappings with closed expressions for odd-dimensional spheres.

Figures 1 and 2 illustrate the measure-preserving mapping obtained in Corollary 3.1 for the particular case of the two-dimensional sphere $\mathbb{S}^{2}$.

3.2. The real projective space $\mathbb{RP}^{n}$

In this case, we have $d=n$, $D=\pi/2$, and

[TABLE]

Corollary 3.2**.**

The mapping $\varphi_{\mathbb{RP}^{n}}\colon(\mathbb{R}^{n},\mu_{c=1})\to(B^{n}(0,\pi/2),\mu_{\omega=\Omega/V})$ given by $\varphi_{\mathbb{RP}^{n}}(x)=x\rho(\lVert x\rVert)/\lVert x\rVert$ is measure preserving if $\rho=\rho(r)$ satisfies

[TABLE]

As a consequence, the mapping $\Phi_{\mathbb{RP}^{n}}=\mathord{\operatorname{exp}}_{\mathbb{RP}^{n}}\circ\varphi_{\mathbb{RP}^{n}}\circ\Phi_{\mathbb{R}^{n}}\colon((0,1)^{n},\operatorname{unif})\to(\mathbb{RP}^{n},\operatorname{unif})$ is measure preserving. For $n=1$ we have

[TABLE]

and so

[TABLE]

For $n=2$ we can compute $\rho(r)$ explicitly:

[TABLE]

and hence

[TABLE]

Proof.

From Theorem 2.1 we just need to check that

[TABLE]

which is equivalent to the formula in the corollary. The case $n=2$ reads

[TABLE]

which is equivalent to the last claim in the corollary. ∎

3.3. The complex projective space $\mathbb{CP}^{n}$

In this case, we have $d=2n$, $D=\pi/2$, and

[TABLE]

Corollary 3.3**.**

The mapping $\varphi_{\mathbb{CP}^{n}}\colon(\mathbb{R}^{2n},\mu_{c=1})\to(B^{2n}(0,\pi/2),\mu_{\omega=\Omega/V})$ given by $\varphi_{\mathbb{CP}^{n}}(x)=x\rho(\lVert x\rVert)/\lVert x\rVert$ is measure preserving if $\rho=\rho(r)$ satisfies

[TABLE]

As a consequence, the mapping $\Phi_{\mathbb{CP}^{n}}=\mathord{\operatorname{exp}}_{\mathbb{CP}^{n}}\circ\varphi_{\mathbb{CP}^{n}}\circ\Phi_{\mathbb{R}^{2n}}\colon((0,1)^{2n},\operatorname{unif})\to(\mathbb{CP}^{n},\operatorname{unif})$ is measure preserving.

Proof.

From Theorem 2.1 we just need to check that

[TABLE]

which is equivalent to

[TABLE]

and the corollary follows. ∎

3.4. The quaternionic projective space $\mathbb{HP}^{n}$

In this case, we have $d=4n$, $D=\pi/2$, and

[TABLE]

Corollary 3.4**.**

The mapping $\varphi_{\mathbb{HP}^{n}}\colon(\mathbb{R}^{4n},\mu_{c=1})\to(B^{4n}(0,\pi/2),\mu_{\omega=\Omega/V})$ given by $\varphi_{\mathbb{HP}^{n}}(x)=x\rho(\lVert x\rVert)/\lVert x\rVert$ is measure preserving if $\rho=\rho(r)$ satisfies

[TABLE]

As a consequence, the mapping $\Phi_{\mathbb{HP}^{n}}=\mathord{\operatorname{exp}}_{\mathbb{HP}^{n}}\circ\varphi_{\mathbb{HP}^{n}}\circ\Phi_{\mathbb{R}^{4n}}\colon((0,1)^{4n},\operatorname{unif})\to(\mathbb{HP}^{n},\operatorname{unif})$ is measure preserving.

Proof.

From Theorem 2.1 we just need to check that

[TABLE]

which is equivalent to the formula in the corollary. ∎

3.5. The Cayley plane $\mathbb{OP}^{2}$

In this case, we have $d=16$, $D=\pi/2$, and

[TABLE]

Corollary 3.5**.**

The mapping $\varphi_{\mathbb{OP}^{2}}\colon(\mathbb{R}^{16},\mu_{c=1})\to(B^{16}(0,\pi/2),\mu_{\omega=\Omega/V})$ given by $\varphi_{\mathbb{OP}^{2}}(x)=x\rho(\lVert x\rVert)/\lVert x\rVert$ is measure preserving if $\rho=\rho(r)$ satisfies

[TABLE]

As a consequence, the mapping $\Phi_{\mathbb{OP}^{2}}=\mathord{\operatorname{exp}}_{\mathbb{OP}^{2}}\circ\varphi_{\mathbb{OP}^{2}}\circ\Phi_{\mathbb{R}^{16}}\colon((0,1)^{16},\operatorname{unif})\to(\mathbb{OP}^{2},\operatorname{unif})$ is measure preserving.

Proof.

From Theorem 2.1 we just need to check that

[TABLE]

which is equivalent to the formula in the corollary. ∎

In Table 2 we show the cases for which we have a closed expression for the measure-preserving mapping $\Phi_{\mathcal{M}}$. In addition, in the next section we present an approach that will allow us to obtain measure-preserving mappings with explicit expressions for any odd-dimensional sphere.

4. Measure-preserving mappings from the unit cube to fiber bundles

In this section we show how to construct measure-preserving mappings from the unit cube to the total space $E$ of the smooth fiber bundle $F\hookrightarrow E\overset{\pi}{\to}B$, where the total space $E$, the base space $B$, and the fiber $F$ are Riemannian manifolds, assuming that we have measure-preserving mappings from the corresponding unit cubes to $B$ and $F$.

To prove Theorem 1.4 we need the following lemma. The main technical tool used in its proof is the smooth coarea formula, an integral formula due to Federer [Federer1969] and Howard [Howard1993] that generalizes the change of variables formula and Fubini’s theorem (see Appendix A). We refer the interested reader to [Beltran2011, Section 2].

Lemma 4.1**.**

Let $E$, $B$, and $F$ be finite-volume Riemannian manifolds, and let $F\hookrightarrow E\overset{\pi}{\to}B$ be a smooth fiber bundle such that $\operatorname{NJac}\pi(x)$ is constant for every $x\in E$. Let $\Psi_{y}\colon F\to\pi^{-1}(y)$ be a measure-preserving mapping for every $y\in B$ and consider the mapping

[TABLE]

If $\xi$ is measurable, then it is measure preserving. Moreover, if the measures in $E$, $B$, and $F$ are normalized to have unit volume, then $\operatorname{NJac}\pi(x)=1$ for every $x\in E$.

Proof.

Without loss of generality, assume that the measures in $E$, $B$, and $F$ are normalized. We first check that $\operatorname{NJac}\pi(x)=1$ for every $x\in E$. Since $F\hookrightarrow E\overset{\pi}{\to}B$ is a smooth fiber bundle, we know that $\pi$ is a submersion and hence we can apply the smooth coarea formula. Therefore,

[TABLE]

and so $C=1$. Now we prove that $\xi$ is measure preserving. Let $\mathcal{A}\subseteq E$ be a measurable set. We have to prove that

[TABLE]

Using again the smooth coarea formula together with the fact that $\operatorname{NJac}\pi(x)=1$ for all $x\in E$, we have

[TABLE]

Note that

[TABLE]

Therefore, since

[TABLE]

the lemma follows. ∎

Proof of Theorem 1.4.

From Lemma 4.1 we have that the mapping $\xi\colon B\times F\to E$ given by $\xi(y,z)=\Psi_{y}(z)$ is measure preserving. Since $\Phi_{E}=\xi\circ\Phi_{B\times F}$, and both mappings are measure preserving, the theorem follows. ∎

Example 4.2** (The Hopf fibration).**

Consider $\mathbb{S}^{1}\subset\mathbb{C}$ and $\mathbb{S}^{2n+1}\subset\mathbb{C}^{n+1}$. Recall that the (complex) Hopf fibration $\mathbb{S}^{1}\hookrightarrow\mathbb{S}^{2n+1}\overset{h}{\to}\mathbb{CP}^{n}$ is given by

[TABLE]

The fiber of each $[y]=[y_{1}:\dotsb:y_{n+1}]\in\mathbb{CP}^{n}$ is a unit circle in $\mathbb{S}^{2n+1}$ given by

[TABLE]

For each $[y]\in\mathbb{CP}^{n}$, we choose a unit norm representative $y$ smoothly out of a lower-dimensional set, and, thinking of the elements of $\mathbb{S}^{1}$ as unimodular complex numbers, we consider the mapping

[TABLE]

which is an isometry and hence it is measure preserving. Therefore, by Theorem 1.4, the mapping

[TABLE]

where $y\in\mathbb{C}^{n}\cong\mathbb{R}^{2n}$ and $t\in\mathbb{R}$ (and recall that we are assuming that the representative of $\Phi_{\mathbb{CP}^{n}}(y)$ has unit norm) is measure preserving. Note that we have explicit expressions for both $\Phi_{\mathbb{S}^{1}}$ and $\Phi_{\mathbb{CP}^{n}}$, and hence for $\Phi_{\mathbb{S}^{2n+1}}^{h}$:

[TABLE]

For the particular case of $\mathbb{S}^{3}$ we have

[TABLE]

since $\lVert\Phi_{\mathbb{CP}^{1}}(y)\rVert=e^{\lVert\Phi_{\mathbb{R}^{2}}(y)\rVert^{2}/4}$. Note that we are considering $\Phi_{\mathbb{R}^{2}}(y)\in\mathbb{C}$ through the canonical isomorphism $\mathbb{R}^{2}\cong\mathbb{C}$ given by $(a,b)\mapsto a+bi$.

Appendix A The smooth coarea formula

Let $\mathcal{M},\mathcal{N}$ be Riemannian manifolds. Given a smooth mapping $\varphi\colon\mathcal{M}\to\mathcal{N}$, let $D\varphi(x)\colon T_{x}\mathcal{M}\to T_{\varphi(x)}\mathcal{N}$ denote the differential mapping, where $T_{x}\mathcal{M}$ is the tangent space to $\mathcal{M}$ at $x\in\mathcal{M}$ and $T_{\varphi(x)}\mathcal{N}$ is the tangent space to $\mathcal{N}$ at $\varphi(x)\in\mathcal{N}$.

Definition A.1** (Normal Jacobian).**

Let $\mathcal{M}$ and $\mathcal{N}$ be Riemannian manifolds and let $\varphi\colon\mathcal{M}\to\mathcal{N}$ be a $C^{1}$ surjective map. Let $n=\dim(\mathcal{N})$ be the real dimension of $\mathcal{N}$. For every point $x\in\mathcal{M}$ such that the differential mapping $D\varphi(x)$ is surjective, let $v_{1}^{x},\dotsc,v_{n}^{x}$ be an orthogonal basis of $(\ker(D\varphi(x)))^{\perp}$. Then we define the normal Jacobian of $\varphi$ at $x$, written as $\operatorname{NJac}\varphi(x)$, as the volume in the tangent space $T_{\varphi(x)}\mathcal{N}$ of the parallelepiped spanned by $D\varphi(x)(v_{1}^{x}),\dotsc,D\varphi(x)(v_{n}^{x})$. In the case that $D\varphi(x)$ is not surjective, we define $\operatorname{NJac}\varphi(x)=0$.

Theorem A.2** (Smooth coarea formula).**

Let $\mathcal{M}$ and $\mathcal{N}$ be two Riemannian manifolds of dimension $m$ and $n$, respectively, where $m\geq n$. Let $\varphi\colon\mathcal{M}\to\mathcal{N}$ be a smooth surjective map such that the differential mapping $D\varphi(x)$ is surjective for almost all $x\in\mathcal{M}$. Let $\psi\colon\mathcal{M}\to\mathbb{R}$ be an integrable mapping. Then, the following equalities hold:

[TABLE]

Note that if $m=n$ and $\varphi$ is a diffeomorphism we recover the classical change of variables theorem.

Appendix B Auxiliary computations

In this appendix, we show the explicit computations leading to the formulas in Table 2.

B.1. Explicit expression of $\Phi_{\mathbb{B}^{n}}$

Recall from Proposition 1.2 that we have

[TABLE]

Hence,

[TABLE]

B.2. Explicit expression of $\Phi_{\mathbb{S}^{1}}$

Although we could simply define $\Phi_{\mathbb{S}^{1}}(x)=e^{i2\pi x}$, let us find the expression of this mapping using the general procedure. In this case, we have

[TABLE]

Following Corollary 3.1, to find $\varphi_{\mathbb{S}^{1}}$ we have to obtain $\rho$ from

[TABLE]

Since in this case $n=1$, we have

[TABLE]

Hence,

[TABLE]

Therefore,

[TABLE]

Since both $\operatorname{erf}$ and $\operatorname{erf}^{-1}$ are odd functions, the absolute values cancel each other and so

[TABLE]

Recall from Table 1 that the exponential map $\mathord{\operatorname{exp}}_{\mathbb{S}^{1}}\colon(-\pi,\pi)\to\mathbb{S}^{1}$ is given by

[TABLE]

Hence,

[TABLE]

B.3. Explicit expression of $\Phi_{\mathbb{S}^{2}}$

Recall from Corollary 3.1 that we have

[TABLE]

Let us compute first $\mathord{\operatorname{exp}}_{\mathbb{S}^{2}}\circ\varphi_{\mathbb{S}^{2}}$. Recall from Table 1 that

[TABLE]

Hence,

[TABLE]

Due to the parity of the sine and the cosine, we can rewrite the previous expression as

[TABLE]

To further simplfy these expressions, note that, for $-1<x<1$,

[TABLE]

and

[TABLE]

Therefore,

[TABLE]

Hence, we conclude that

[TABLE]

B.4. Explicit expression of $\Phi_{\mathbb{RP}^{1}}$

In this case, we have

[TABLE]

Following Corollary 3.2, to find $\varphi_{\mathbb{RP}^{1}}$ we have to obtain $\rho$ from

[TABLE]

Since in this case $n=1$, we have

[TABLE]

Hence,

[TABLE]

Therefore,

[TABLE]

Since both $\operatorname{erf}$ and $\operatorname{erf}^{-1}$ are odd functions, the absolute values cancel each other and so

[TABLE]

Recall from Table 1 that the exponential map $\mathord{\operatorname{exp}}_{\mathbb{RP}^{1}}\colon(-\pi/2,\pi/2)\to\mathbb{RP}^{1}$ is given by

[TABLE]

Hence,

[TABLE]

Since the tangent function is odd, we can simplify the previous expression as follows:

[TABLE]

B.5. Explicit expression of $\Phi_{\mathbb{RP}^{2}}$

Recall from Corollary 3.2 that we have

[TABLE]

Let us compute $\mathord{\operatorname{exp}}_{\mathbb{RP}^{2}}\circ\varphi_{\mathbb{RP}^{2}}$. Recall from Table 1 that

[TABLE]

Hence,

[TABLE]

Since the tangent function is odd, we can simplify the previous expression as follows:

[TABLE]

Note that

[TABLE]

Hence,

[TABLE]

and we conclude that

[TABLE]

B.6. Explicit expression of $\Phi_{\mathbb{CP}^{n}}$

Recall from Corollary 3.3 that we have

[TABLE]

Recall from Table 1 that

[TABLE]

Let us compute $\mathord{\operatorname{exp}}_{\mathbb{CP}^{n}}\circ\varphi_{\mathbb{CP}^{n}}$. As for the case of $\mathbb{RP}^{2}$, the parity of the tangent function implies that

[TABLE]

Note that, in our range,

[TABLE]

Hence,

[TABLE]

and we conclude that

[TABLE]