# 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.

## Key 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.