Abstract
Infinite dimensional differential geometry is usually not taught in university courses but it is perfectly developed in a theoretical and global way. However, the local writing and the tensorial calculus do not seem to be promoted by comparison to global results. Hilbert manifolds, i.e. manifolds with Hilbert space as representation space, seem to potentially be able to play a role intheoretical physics. Indeed, quantum mechanics uses Hilbert spaces and general relativity is built on riemannian geometry. This motivates the study of Hilbert manifolds that is achieved in this master thesis. A detailed presentation of the general concepts of infinite dimensional differential geometry is realized. Moreover, this work is closed interested to infinite dimensional local writing and tensorial calculus. Most tensorial formulae used in differential and riemannian geometry are developed in the hilbertian case. Finally, an attempt of application to quantum mechanics is presented. That attempt reveals a deep problem : quantum mechanics is deeply linear while differential geometry is naturally non linear. This master thesis is both a bibliographic study and an exploratory research.
Date of Award  27 Jun 2016 

Original language  French 
Awarding Institution 

Supervisor  Andre FUZFA (Supervisor), Nicolas Franco (Jury), DOMINIQUE LAMBERT (Jury) & Valerie HENRY (Jury) 
Keywords
 Hilbert manifold
 infinite dimensional
 differential geometry
 Quantum Mechanics
 vector field
 local representation
 tangent space
 Hilbert space
 state vector