A brief introduction to symplectic and contact manifolds (Q2818799)

The book presents the fundamentals of symplectic and contact geometry in a quick but smooth way, suitable for second-year graduate students. It is divided in 7 chapters and 2 appendices. Chapter 1 presents the linear theory: symplectic vector spaces and the symplectic group. Chapter 2 introduces symplectic manifolds. In this chapter, after providing several examples, the authors state the Darboux theorem and present Weinstein's proof of it based on the Moser path method. They also discuss various groups of symplectomorphisms, together with their main properties, and properties of their tautological action, including the Boothby theorem on the \(p\)-transitivity of the action of the symplectomorphism group (with proof), and a theorem of Banyaga stating that the symplectomorphism group essentially determines the symplectic geometry (without proof). Chapter 2 ends with a quick discussion on Lagrangian submanifolds, including Weinstein normal form of Lagrangian embeddings, and the interaction with (almost) complex geometry. NEWLINENEWLINEChapter 3 introduces the Poisson bracket and Hamiltonian systems on symplectic manifolds. It states the Arnold-Liouville Theorem on the existence of action-angle coordinates for completely integrable Hamiltonian systems (without proof) and briefly discusses Hamiltonian diffeomorphisms. Chapter 4 deals shortly with Lie groups and Lie group actions, including the Hamiltonian Noether Theorem stating that the momentum map of a Hamiltonian Lie group action preserving an Hamiltonian system is a constant of the motion, and the Marsden-Weinstein symplectic reduction Theorem. Chapter 5 introduces contact structures and contact forms. The general philosophy here is that ``\textit{Every theorem in symplectic geometry may be formulated as a contact geometry theorem and an assertion in contact geometry may be translated in the language of symplectic geometry}'' [\textit{V. I. Arnold}, Contact geometry and wave propagation. Lectures given at the University of Oxford in November and December 1988 under the sponsorship of the International Mathematical Union. Genève: Univ. de Genève, L'Enseignement Mathématique (1989; Zbl 0694.53001)]. NEWLINENEWLINEAfter several examples, the authors discuss the relationship between contact and symplectic geometry. Then they state Weinstein's conjecture that the Reeb field of a contact form on a compact manifold \(M\) with \(H^1 (M, \mathbb R) = 0\) should possess at least one periodic orbit, and prove the Boothby-Wang Theorem on the existence of a prequantization bundle of an integral symplectic manifold. This chapter does also contain a proof of the contact Darboux Theorem, and ends with a short discussion on the contactomorphism group and its main properties, and on metric issues in contact geometry. NEWLINENEWLINEThe book contains several helpful exercises, and Chapter 6 collects the solutions to part of them. Chapter 7 contains more advanced material on \(C^0\)-symplectic and contact topology, particularly Banyaga's contributions to the subject. Appendix A is a short review on differential forms, de Rham cohomology, and the Hodge-de Rham decomposition. Finally, Appendix B contains an unpublished paper by Banyaga and Molino on complete integrability in contact geometry. The authors define completely integrable contact structures (CICS): a contact analogue of Duistermaat's generalized Lagrangian fibrations, and prove existence of action-angle coordinates with singularities, paralleling a result of Eliasson in symplectic geometry. They also define characteristic invariants of CICS and prove that two CICS are isomorphic iff they share the same characteristic invariants. Finally they prove the contact analogues of the Atiyah-Guillemin-Sternberg convexity theorem, and the Delzant realization theorem.