Essay on Symplectic Geometry
The essay should not be longer than 10 pages and should be typed on computer. Moreover,
it should contain the following:
1. Define what a symplectic manifold is and give examples. In particular, show that the
cotangent space T∗
M of any manifold M is in a natural way a symplectic manifold.
2. State and prove the Theorem of Darboux (which says that any symplectic manifold
of dimension 2n is locally isomorphic to the symplectic manifold T∗
Rn
= Rn
×
(Rn
)∗
.)
3. Define what a Lagrangian submanifold of a symplectic manifold is. Give examples
of Langrangian submanifolds of T∗
M: When is the image of a section of T∗
M
a Lagrangian submanifold of T∗
M? What about the (so-called) conormal bundle
inside T∗
M of a submanifold X ⊂ M?
4. What is the Poisson bracket of a symplectic manifold?
References
[1] Agrikola, I. and Friedrich, T.: Global analysis : differential forms in analysis,
geometry and physics, Chapter 7, Transl. by A. Nestke. American Mathematical
Society, 2002.
[2] Cannas da Silva, A.: Lectures on Symplectic Geometry, Springer, 2008.
Also available at: https://people.math.ethz.ch/ acannas/Papers/lsg.pdf
[3] Michor, P.: Topics in Differential Geometry, Chapter VII ,American Mathematical
Society, 2008.
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