Supervisors info:
Διονύσιος Λάππας Αναπλ. Καθηγητής ΕΚΠΑ (επιβλέπων), Ιάκωβος Ανδρουλιδάκης Επίκ. Καθηγητής ΕΚΠΑ, Αντώνιος Μελάς Καθηγητής ΕΚΠΑ
Summary:
We study Lie group actions on symplectic manifolds. A symplectic manifold is a
pair (Μ,ω), where M is a differentiable manifold and ω is a non-degenarate,
closed 2-form on M. We are interested in symplectic group actions, which are
these actions which preserve the form ω. Moreover, we mainly deal with a
special subset of these actions, the so called hamiltonian actions. These are
closely related with Classical mechanics, as their name indicates. For the
definition of a hamiltonian action we have to introduce the notion of the
moment or momentum map.
The first chapter is an intoductory chapter to Symplectic Geometry. We study
symplectic vector spaces and then symplectic manifolds.The central theorem of
this chapter is the Darboux's Theorem. It is a fundamental theorem for
Symplectic Geometry. As a consequnce of this theorem, all symplectic manifolds
of equal dimension are locally symplectomorphic (isomorphic in the symplectic
category). So, in contrast with Riemannian Geometry, in Symplectic Geomerty
there are no local invariants. We are going to give two proofs of the Darboux's
Theorem.
In the second chapter we study Lie group actions on general differentiable
manifolds. After a brief introduction to Lie group theory, we deal with free
and proper actions. The basic theorem for these actions is that the orbit space
admits differentiable structure in a natural way. We study also proper actions
with no further assumptions.
In chapter number three we study free, proper hamiltonian actions. We prove the
Marsden-Weinstein-Meyer regular reduction theorem, that under the above
mentioned assumptions the orbit space is a symplectic manifold. The reduction
method is a generalisation of a well known method from physics. If you have a
constant of motion, you can decrease the number of the equations of motion.
The fourth chapter contains the proof of the Atiyah-Guillemin-Sternberg
convexity theorem. The image of the moment map, for a hamiltonian action of the
torus on a compact, connected symplectic manifold, is a convex polytope.
Finally we study singular symplectic reduction: reduction when the action is
proper and no necessarilly free. The orbit space in this occasion isn't a
manifold in general, but it has a decomposition in symplectic pieces.
Keywords:
Hamiltonian group actions, Moment map, Symplectic reduction, Convexity theorem, Symplectic strata
