PT Unknown
AU Kliemann, M
TI The Thickness of Left-Invariant Metrics on Compact Connected Lie Groups
PY 2019
PU Christian-Albrechts-Universität zu Kiel
WP https://macau.uni-kiel.de/receive/diss_mods_00024826
LA en
DE Lie groups; thickness; maximal tori; diameter; carnot-caratheodory metrics; geometry; vogel decomposition; representation theory; Liegruppen; Dichte; maximale Tori; Durchmesser; Carnot-Caratheodory-Metriken; Geometrie; Vogelzerlegung; Darstellungstheorie
AB The subject of the present thesis is to study the thickness of left-invariant riemannian metrics Q on compact connected d-dimensional Lie groups G with respect to the existence of globally or locally thinnest metrics. This is a variation of a problem posed by Berger concerning the identification of particularly good riemannian metrics on compact manifolds. 

If the group in question is abelian, the question of the existence of a thinnest metric may be answered in the affirmative. By contrast, the existence of arbitrarily thin metrics on all non-abelian groups is proved using the theory of Carnot–Carathéodory metrics. 

Furthermore, it is shown that the bi-invariant metrics of certain simple Lie groups with finitely many antipodes may be continuously deformed into thinner metrics by shrinking a maximal torus. The affected class of Lie groups consists of the simply-connected Lie groups of the infinite families A n , B n and D n , the simply-connected exceptional Lie groups E 6 and E 7 as well as SO(2n) for n ≥ 2. The existence of locally thinnest metrics on these groups remains unsettled. 

Finally, it is proved that the bi-invariant metrics of the Lie group PSU(n + 1) for n ≥ 1 are locally optimal metrics, meaning that their thickness does not decrease under any deformations except (possibly) those contained in a certain zero set. It remains unknown whether the bi-invariant metrics are locally thinnest metrics, except in the affirmative case of PSU(2). The problematic zero set is described as a subset of deformations that preserve the volume of every maximal torus up to first order, or, equivalently, deformations whose inertia operator is perpendicular to the second Cartan power of the Lie algebra.
PI Kiel
ER