Lipschitz minimality of the multiplication maps of unit complex, quaternion and octonion numbers

TL;DR

This paper proves that multiplication maps of unit complex, quaternion, and octonion numbers are uniquely minimal in Lipschitz constant within their homotopy classes, highlighting their optimal geometric properties.

Contribution

It establishes the uniqueness of Lipschitz minimality for these multiplication maps, extending understanding of geometric optimality in algebraic structures.

Findings

Multiplication maps are unique Lipschitz minimizers in their homotopy classes.

Other natural maps like Hopf fibrations also exhibit this minimality.

Supports the conjecture for all Riemannian submersions of compact homogeneous spaces.

Abstract

We prove that the multiplication maps $\mathbb{s}^n \times \mathbb{s}^n \rightarrow \mathbb{s}^n$ ($n = 1, 3, 7$) for unit complex, quaternion and octonion numbers are, up to isometries of domain and range, the unique Lipschitz constant minimizers in their homotopy classes. Other geometrically natural maps, such as projections of Hopf fibrations, have already been shown to be, up to isometries, the unique Lipschitz constant minimizers in their homotopy classes, and it is suspected that this may hold true for all Riemannian submersions of compact homogeneous spaces.