This book treats the fundamentals of differential geometry: manifolds, flows, Lie groups and their actions, invariant theory, differential forms and de Rham cohomology, bundles and connections, Riemann manifolds, isometric actions, and symplectic and Poisson geometry.
The layout of the material stresses naturality and functoriality from the beginning and is as coordinate-free as possible. Coordinate formulas are always derived as extra information. Some attractive unusual aspects of this book are as follows:
This book gives the careful reader working knowledge in a wide range of topics of modern coordinate-free differential geometry in not too many pages. A prerequisite for using this book is a good knowledge of undergraduate analysis and linear algebra.
* Manifolds and Vector Fields
* Lie Groups and Group Actions
* Differential Forms and de Rham Cohomology
* Bundles and Connections
* Riemann Manifolds
* Isometric Group Actions or Riemann G-Manifolds
* Symplectic and Poisson Geometry
* List of Symbols