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An introduction to symplectic geometry. Transl. from the German by Michael Klucznik. (English) Zbl 0986.53028

Graduate Studies in Mathematics. 26. Providence, RI: American Mathematical Society (AMS). xvi, 195 p. (2001).
This book is a standard introduction to symplectic geometry, with special emphasis in geometric quantization. To motivate the study of symplectic geometry, the author discusses in a preliminary section (Chapter 0) some rudiments of symplectic mechanics. So, he introduces the Euler-Lagrange equations, Hamilton equations (obtained using the Legendre transformation), Hamilton-Jacobi theory, and Poisson brackets. He also presents some preliminary aspects of quantization of classical mechanical systems. In the following chapters, the author presents symplectic algebra (Chapter 1), a good introduction to symplectic manifolds paying special attention to Kähler manifolds and including some initial results in symplectic invariants (Chapter 2), Hamiltonian vector fields including Poisson brackets and contact manifolds (Chapter 3), momentum maps and symplectic reduction (Chapter 4), and, finally, he studies the procedure of geometric quantization, including a proof of the Groenewold-van Hove theorem (Chapter 5). The book is completed with four appendices. Appendices A and B give a brief and clear introduction to differentiable manifolds, Lie groups and Lie algebras. Appendix C is devoted to study some cohomology theories in Lie groups, Lie algebras and manifolds, and Appendix D treats with the theory of representation of groups.
The book is very well-written, and contains the essential facts on symplectic geometry and symplectic mechanics. It is highly recommendable for graduate students in Mathematics and Physics.

MSC:

53Dxx Symplectic geometry, contact geometry
53-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
70Hxx Hamiltonian and Lagrangian mechanics
81S10 Geometry and quantization, symplectic methods

Citations:

Zbl 0943.53002
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