- Author: Anahita Eslami Rad
- Series: Latin American Mathematics Series
- Publisher: Springer
- Publication Date: 04/11/2024
- Number of Pages: 178
- Format: Electronic Book
- Price: $64.99
- ISBN: 978-3-031-56225-9
- Category: textbook
[Reviewed by Bill Satzer , on 11/13/2024]
This introduction to symplectic and contact geometry is intended for advanced mathematics undergraduates, graduate students and other interested researchers with some background in differential geometry, manifold theory, and a bit of Lie theory and algebraic topology.
Symplectic and contact geometry had their origins in classical mechanics but have developed broadly beyond that. This book stays fairly close to basic aspects of the subject, in particular those that developed in the context of classical mechanics. Symplectic geometry and symplectic topology have advanced considerably in the last few decades with many new results that are not addressed here. The author notes that she chose topics that emphasize the relationships between symplectic geometry, Lie theory, classical mechanics and contact geometry.
The book begins with symplectic forms on vector spaces (defined as non-degenerate anti-symmetric bilinear forms) and the associated linear algebra and proceeds to the linear symplectic group and its Lie algebra. The author then moves on to symplectic manifolds, providing examples (with the cotangent bundle of a manifold as the primary one) and noting that – by cohomological arguments – while every symplectic manifold must be even-dimensional, not every even-dimensional manifold can be symplectic. (For example, even-dimensional spheres are not symplectic manifolds.)
Contact manifolds are odd-dimensional counterparts of even-dimensional symplectic manifolds. Consequently, contact geometry is associated with $2n$-dimensional symplectic geometry that lead to contact manifolds of odd dimensions $2n+1$ and $2n-1$. They arise in $2n+1$ dimensions, for example, as the natural space for $2n$-dimensional Hamiltonian systems that also explicitly depend on time, or in $2n-1$ dimensions as constant-energy hypersurfaces for Hamiltonian systems.$$
This book is called a concise introduction, and many topics are treated quickly. Some subjects are touched on only very briefly (cohomology classes and almost complex structures, for example), and probably deserve more detail and some examples.
An assortment of more advanced topics are treated in the last chapter. These include an application to geometric optics, Legendrian knots in contact 3-manifolds, and a bit about surgery on contact manifolds.
This is an excellent introduction to the subject for those with sufficient background. It incorporates many topics in relatively few pages, but it is complete, rigorous and thorough.
Bill Satzer (bsatzer@gmail.com), now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications. He did his PhD work in dynamical systems and celestial mechanics.