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Archivio digitale delle tesi discusse presso l’Università di Pisa

Tesi etd-09082020-114231


Tipo di tesi
Tesi di laurea magistrale
Autore
TIBERIO, DANIELE
URN
etd-09082020-114231
Titolo
Vanishing and non-vanishing geodesic distances in infinite dimensions
Dipartimento
MATEMATICA
Corso di studi
MATEMATICA
Relatori
relatore Prof. Magnani, Valentino
controrelatore Prof. Le Donne, Enrico
Parole chiave
  • geodesic distances
  • infinite dimensional manifolds
  • right-invariant Riemannian metrics
  • sub-Riemannian controllability
Data inizio appello
25/09/2020
Consultabilità
Non consultabile
Data di rilascio
25/09/2060
Riassunto
It is well known that in a connected and finite dimensional Riemannian manifold, the infimum among the lengths of all continuous and piecewise smooth curves connecting two points yields a distance, called geodesic distance. The objective of this thesis is to understand whether the analogous procedure leads to a distance also when the model space of the manifold is infinite-dimensional. Examples of vanishing geodesic distances for Fréchet manifolds were known. In this thesis we give a new result, presenting a simple example of vanishing geodesic distance in a separable Hilbert space. We also study the case of right-invariant Riemannian metrics on Lie groups, presenting a recent result by Bauer, Harms and Preston, which characterizes the vanishing of the associated geodesic distance in terms of the action of the group on a set. Finally, we present a recent generalization of the Chow-Rashevskii theorem to horizontal vector bundles on Hilbert manifolds, proved by Arguillére.
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