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Tesi etd-02092023-132505


Tipo di tesi
Tesi di laurea magistrale
Autore
SOPIO, CRISTIAN
URN
etd-02092023-132505
Titolo
Milnor theorems for sub-Finsler Lie groups
Dipartimento
MATEMATICA
Corso di studi
MATEMATICA
Relatori
relatore Prof. Le Donne, Enrico
Parole chiave
  • bi-invariant metrics
  • Finsler metrics
  • geodesics
  • length minimiser
  • Lie groups
  • one-parameter subgroups
  • sub-Finsler metrics
Data inizio appello
24/02/2023
Consultabilità
Completa
Riassunto
In 1976 Milnor proved two theorems regarding bi-invariant Riemannian metrics on Lie groups.
He stated that such groups are isomorphic to a Euclidean direct product of a compact group and an additive vector group. Moreover, he characterised geodesics in those groups, i.e., every one-parameter subgroup is a Riemannian geodesic if and only if the metric is bi-invariant.

In this work we generalise those theorems for bi-invariant sub-Finsler Lie groups. We stress that we only consider continuous norms without any additional assumptions on their smoothness and convexity.
We observe that if we take a sub-Finsler Lie group and assume either the bi-invariance of the metric or that every one-parameter subgroup is geodesic, the metric structure on the group is actually Finsler.

We show that every bi-invariant sub-Finsler Lie group is a direct product of a compact group and an additive vector group (as in the Riemannian case).
Concerning geodesics, in the sub-Finsler case we were not able to conclude that one-parameter subgroups are length minimisers if and only if the metric is bi-invariant. Nevertheless, every length minimiser satisfies the Hamilton equation for geodesics and we prove that every one-parameter subgroup solves the geodesic equation if and only if the norm is bi-invariant.
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