Symplectic geometry is a central topic of current research in mathematics. Indeed, symplectic methods are key ingredients in the study of dynamical systems, differential equations, algebraic geometry, topology, mathematical physics and representations of Lie groups. This book is a true introduction to symplectic geometry, assuming only a general background in analysis and familiarity with linear algebra. It starts with the basics of the geometry of symplectic vector spaces. Then, symplectic manifolds are defined and explored. In addition to the essential classic results, such as Darboux's theorem, more recent results and ideas are also included here, such as symplectic capacity and pseudoholomorphic curves. These ideas have revolutionized the subject. The main examples of symplectic manifolds are given, including the cotangent bundle, K?hler manifolds, and coadjoint orbits. Further principal ideas are carefully examined, such as Hamiltonian vector fields, the Poisson bracket, and connections with contact manifolds. Berndt describes some of the close connections between symplectic geometry and mathematical physics in the last two chapters of the book. In particular, the moment map is defined and explored, both mathematically and in its relation to physics. He also introduces symplectic reduction, which is an important tool for reducing the number of variables in a physical system and for constructing new symplectic manifolds from old. The final chapter is on quantization, which uses symplectic methods to take classical mechanics to quantum mechanics. This section includes a discussion of the Heisenberg group and the Weil (or metaplectic) representation of the symplectic group. Several appendices provide background material on vector bundles, on cohomology, and on Lie groups and Lie algebras and their representations. Berndt's presentation of symplectic geometry is a clear and concise introduction to the major methods and applications of the subject, and requires only a minimum of prerequisites. This book would be an excellent text for a graduate course or as a source for anyone who wishes to learn about symplectic geometry.
Author(s): Rolf Berndt
Series: Graduate Studies in Mathematics 26
Publisher: American Mathematical Society
Year: 2000
Language: English
Pages: 195
Cover......Page 1
Title Page......Page 6
Copyright......Page 7
Contents......Page 8
Preface......Page 12
0.1. The Lagrange equations......Page 18
0.2. Hamilton's equations......Page 19
0.3. The Hamilton-Jacobi equation......Page 21
0.5. Hamilton's equations via the Poisson bracket......Page 23
0.6. Towards quantization......Page 24
1.1. Symplectic vector spaces......Page 26
1.2. Sympiectic morphisms and sympiectic groups......Page 31
1.3. Subspaces of sympiectic vector spaces......Page 34
1.4. Complex structures of real symplectic spaces......Page 41
2.1. Symplectic manifolds and their morphisms......Page 52
2.2. Darboux's theorem......Page 53
2.4. Kiihler manifolds......Page 62
2.5. Coadjoint orbits......Page 68
2.6. Complex projective space......Page 80
2.7. Symplectic invariants (a quick view)......Page 85
3.1. Preliminaries......Page 88
3.2. Hamiltonian systems......Page 91
3.3. Poisson brackets......Page 96
3.4. Contact manifolds......Page 102
4.1. Definitions......Page 110
4.2. Constructions and examples......Page 114
4.3. Reduction of phase spaces by the consideration of symmetry......Page 121
5.1. Homogeneous quadratic polynomials and .(2......Page 128
5.2. Polynomials of degree 1 and the Heisenberg group......Page 131
5.3. Polynomials of degree 2 and the Jacobi group......Page 137
5.4. The Groenewold-van Hove theorem......Page 141
5.5. Towards the general case......Page 145
A.l. Differentiable manifolds and their tangent spaces......Page 152
A.2. Vector bundles and their sections......Page 161
A.3. The tangent and the cotangent bundles......Page 163
A.4. Tensors and differential forms......Page 167
A.5. Connections......Page 175
B.l. Lie algebras and vector fields......Page 180
B.2. Lie groups and invariant wct.or fields......Page 182
B.3. One-parameter subgroups and the exponent map......Page 184
C.l. Cohomology of groups......Page 188
C.2. Cohomology of Lie a1gebras......Page 190
C.3. Cohomology of manifolds......Page 191
D.l. Linear representations......Page 194
D.2. Continuous and unitary representations......Page 196
D.3. On the construction of representations......Page 197
Bibliography......Page 202
Index......Page 206
Symbols......Page 210
