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

Carlos Beltrán, Damir Ferizović, Pedro R. López-Gómez

TL;DR

The paper provides constructive, measure-preserving mappings from the unit cube $(0,1)^d$ to the $d$-dimensional unit ball and to the compact rank one symmetric spaces $\mathbb{S}^n, \mathbb{RP}^n, \mathbb{CP}^n, \mathbb{HP}^n, \mathbb{OP}^2$. The approach combines a radial map on $\mathbb{R}^d$ with an exponential map on the target manifold, yielding explicit formulas like $\Phi_{\mathbb{B}^d}=\varphi_{\mathbb{B}^d}\circ\Phi_{\mathbb{R}^d}$ and $\varphi_{\mathcal{M}}(x)=x\rho(\|x\|)/\|x\|$ where $\rho$ solves a weighted integral equation involving the volume density. The results extend to finite products and to fiber bundles under suitable Jacobian conditions, and a Hopf-fibration construction provides explicit measure-preserving maps for odd spheres. Together, these results enable efficient, uniform discretizations and sampling on highly symmetric spaces with provable measure preservation, with potential applications in cartography, graphics, PDEs, and communications. The framework highlights the role of the incomplete gamma function and related integrals in balancing volume elements under nonlinear mappings.

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.

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

TL;DR

The paper provides constructive, measure-preserving mappings from the unit cube to the -dimensional unit ball and to the compact rank one symmetric spaces . The approach combines a radial map on with an exponential map on the target manifold, yielding explicit formulas like and where solves a weighted integral equation involving the volume density. The results extend to finite products and to fiber bundles under suitable Jacobian conditions, and a Hopf-fibration construction provides explicit measure-preserving maps for odd spheres. Together, these results enable efficient, uniform discretizations and sampling on highly symmetric spaces with provable measure preservation, with potential applications in cartography, graphics, PDEs, and communications. The framework highlights the role of the incomplete gamma function and related integrals in balancing volume elements under nonlinear mappings.

Abstract

We construct measure-preserving mappings from the -dimensional unit cube to the -dimensional unit ball and the compact rank one symmetric spaces, namely the -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 -dimensional unit cube to product spaces and fiber bundles under certain conditions.
Paper Structure (21 sections, 12 theorems, 110 equations, 2 figures, 2 tables)

This paper contains 21 sections, 12 theorems, 110 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction and main results
  2. Notation
  3. The compact rank one symmetric spaces
  4. Main results
  5. Structure of the paper
  6. A technical result
  7. Measure-preserving mappings from the unit cube to the compact rank one symmetric spaces
  8. The unit sphere $\mathbb{S}^n$
  9. The real projective space $\mathbb{RP}^n$
  10. The complex projective space $\mathbb{CP}^n$
  11. The quaternionic projective space $\mathbb{HP}^n$
  12. The Cayley plane $\mathbb{OP}^2$
  13. Measure-preserving mappings from the unit cube to fiber bundles
  14. The smooth coarea formula
  15. Auxiliary computations
  16. ...and 6 more sections

Key Result

Lemma 1.1

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

Figures (2)

  • Figure 1: The measure-preserving mapping $\Phi_{\mathbb{S}^2}=\mathord{\operatorname{exp}}_{\mathbb{S}^2}\circ\varphi_{\mathbb{S}^2}\circ\Phi_{\mathbb{R}^2}\colon((0,1)^2,\operatorname{unif})\to(\mathbb{S}^2,\operatorname{unif})$ transforms points on $(0,1)^2$ into points on $\mathbb{S}^2$. For an initial collection of $784$ uniformly distributed points on $(0,1)^2$, we show the different steps from the unit square to the sphere.
  • Figure 2: The measure-preserving mapping $\Phi_{\mathbb{S}^2}=\mathord{\operatorname{exp}}_{\mathbb{S}^2}\circ\varphi_{\mathbb{S}^2}\circ\Phi_{\mathbb{R}^2}\colon((0,1)^2,\operatorname{unif})\to(\mathbb{S}^2,\operatorname{unif})$ transforms uniform grids in $(0,1)^2$ into uniform grids in $\mathbb{S}^2$. We show the image of a grid in $(0,1)^2$ formed by 1369 cells.

Theorems & Definitions (28)

  • Lemma 1.1
  • proof
  • Proposition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • proof
  • Example 2.2: 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 $(\mathbb{R}^d,\mu_{\text{stereo}})$
  • Example 2.4: A measure-preserving mapping from $(\mathbb{R}^d,\mu_{c=1})$ to $(\mathbb{B}^d,\operatorname{unif})$
  • ...and 18 more