Locally Isometric Embeddings of Quotients of the Rotation Group Modulo Finite Symmetries

TL;DR

This paper develops a general framework for constructing locally isometric embeddings of quotient manifolds of the rotation group, specifically tailored for applications in crystallography, material science, and biochemistry, improving upon previous methods.

Contribution

It introduces a unified approach to embed quotient manifolds of SO(3) modulo finite symmetry groups, ensuring isometry for all crystallographic groups, extending prior embedding techniques.

Findings

Provides a generic construction for embeddings of SO(3)/S.

Ensures isometric embeddings for all crystallographic symmetry groups.

Generalizes previous embedding methods to broader classes of quotient manifolds.

Abstract

The analysis of manifold valued data using embedding based methods is linked to the problem of finding suitable embeddings. In this paper we are interested in embeddings of quotient manifolds $\mathrm{SO}(3)/\mathcal{S}$ of the rotation group modulo finite symmetry groups. Data on such quotient manifolds naturally occur in crystallography, material science and biochemistry. We provide a generic framework for the construction of such embeddings which generalizes the embeddings constructed in arXiv:1701.01579. The central advantage of our larger class of embeddings is that it comprises isometric embeddings for all crystallographic symmetry groups.