Abstract
As its title may suggest, this chapter contains exercises on the second part (of this book).
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- 1.
Even if H and K are autonomous, the composed Hamiltonian isotopy does not (in general) come from an autonomous Hamiltonian. Bonus question: When is it the case?
- 2.
In [18], you can find examples of manifolds that are symplectic but not complex.
- 3.
Replacing O by H and 7 by 3 would be an analogous (but more complicated) way in which to show that every oriented surface embedded in R 3 admits an almost complex structure.
- 4.
For this notion, basic results, and more, see for example [5].
- 5.
This means that the de Rham cohomology class of α is contained in the image of \(H^{1}(V;\mathbf{Z})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}H^{1}(V;\mathbf{R})\).
References
Audin, M.: Torus Actions on Symplectic Manifolds. Progress in Mathematics, vol. 93, Birkhäuser, Basel (2004). Revised and enlarged edition
Berger, M., Gostiaux, B.: Géométrie différentielle: variétés, courbes et surfaces. Presses Universitaires de France (1987)
Cannas da Silva, A.: Lectures on Symplectic Geometry. Lecture Notes in Math. Springer, Berlin (2001)
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Audin, M., Damian, M. (2014). Exercises for the Second Part. In: Morse Theory and Floer Homology. Universitext. Springer, London. https://doi.org/10.1007/978-1-4471-5496-9_14
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