Connections and Finsler geometry of the structure group of a JBalgebra
Abstract
We endow the BanachLie structure group $Str(V)$ of an infinite dimensional JBalgebra $V$ with a leftinvariant connection and Finsler metric, and we compute all the quantities of its connection. We show how this connection reduces to $G(\Omega)$, the group of transformations that preserve the positive cone $\Omega$ of the algebra $V$, and to $Aut(V)$, the group of Jordan automorphisms of the algebra. We present the cone $\Omega$ as an homogeneous space for the action of $G(\Omega)$, therefore inducing a quotient Finsler metric and distance. With the techniques introduced, we prove the minimality of the oneparameter groups in $\Omega$ for any symmetric gauge norm in $V$. We establish that the two presentations of the Finsler metric in $\Omega$ give the same distance there, which helps us prove the minimality of certain paths in $G(\Omega)$ for its leftinvariant Finsler metric.
 Publication:

arXiv eprints
 Pub Date:
 June 2022
 arXiv:
 arXiv:2206.09208
 Bibcode:
 2022arXiv220609208L
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Functional Analysis;
 Mathematics  Group Theory;
 Mathematics  Metric Geometry;
 22E65;
 58B20 (Primary) 53C22 (Secondary)
 EPrint:
 30 pages, v2 only minor typos corrected