Topology of misorientation spaces
type: Item
instance of: scholarly article
title: Topology of misorientation spaces
author: A. A. Aizenberg
author: D. V. Gugnin
published in: Proceedings of the Steklov Institute of Mathematics
publication date: 2024-11-05T00:00:00Z
review text: ``Let \(G_1\) and \(G_2\) be finite subgroups of \(SO(3)\). The double quotients of the form \(X(G_1, G_2) = G_1 \backslash SO(3)/ G_2\) were introduced in materials science under the name misorientation spaces. In this article, we review several known results that allow one to study the topology of misorientation spaces.''\N\N``Definition 1.1: The topological orbit space \(X(G_1,G_2) = SO(3)/(G_1 \times G_2) = G_1 \backslash SO(3)/ G_2\) is called the \textit{misorientation space} of the pair of groups \(G_1\), \(G_2\)'', where the two-sided actions on \(SO(3)\) are defined by \((g_1, g_2)A = g_1 \cdot A \cdot g_2\) for \((g_1, g_2) \in G_1 \times G_2\) and \(A \in SO(3)\).\N\N``In the case when \(G_1\) and \(G_2\) are crystallographic groups, we compute the fundamental groups \(\pi_1(X(G_1,G_2))\) and apply the elliptization theorem to describe these spaces. Many misorientation spaces are homeomorphic to \(S^3\) by Perelman's theorem. However, we explicitly describe the topological types of several misorientation spaces without relying on Perelman's theorem.''\N\NThe full classification of finite subgroups of \(SO(3)\) is known. These are cyclic, dihedral, tetrahedral, octahedral and icosahedral groups which are respectively denoted by \( \mathbb Z_m\), \(\hbox{D}_m\), \(T\), \(O\) and \(I\). The misorientation spaces \(X(G_1,G_2)\) have been studied well if one of the acting groups is trivial. In this case, the remaining action, say \(G_1\), by the left multiplication on \(SO(3)\) is free, thus the orbit space is an elliptic manifold whose universal covering space is \(S^3 \cong Sp(1)\), which is the double cover of \(SO(3)\).\N\NPossible manifolds for \(X(G_1, 1)\) are:\N\begin{itemize}\N\item If \(G_1\) is also trivial, then \(X(1, 1)\) is \(SO(3) \cong \mathbb R\hbox{P}^3\).\N\item If \(G_1 = \mathbb Z_m\), then \(X(\mathbb Z_m, 1)\) is the lens space \(L(2m, 1) \cong S^3/\mathbb Z_{2m}\).\N\item If \(G_1 = \hbox{D}_2 = Z_2 \times \mathbb Z_2\), then \(X(\hbox…
reviewed by: Ryo Ohashi
Mathematics Subject Classification ID: 57M60
Mathematics Subject Classification ID: 20H15
Mathematics Subject Classification ID: 57R18
Mathematics Subject Classification ID: 57S17
Mathematics Subject Classification ID: 82D25
Mathematics Subject Classification ID: 57M12
Mathematics Subject Classification ID: 57S25
Mathematics Subject Classification ID: 57R60
Mathematics Subject Classification ID: 13A50
Mathematics Subject Classification ID: 51F15
zbMATH DE Number: 7939059
zbMATH Keywords: misorientation space
zbMATH Keywords: mathematical crystallography
zbMATH Keywords: point crystallographic group
zbMATH Keywords: finite group action
zbMATH Keywords: orbit space
zbMATH Keywords: elliptic manifold
MaRDI profile type: Publication
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DOI: 10.1134/S0081543824020019
description: scientific article; zbMATH DE number 7939059