This book is an excellent introduction to the field of differential geometry with a strong emphasis on Lie groups. It stresses naturality, functoriality and a coordinate-free approach. Coordinate formulas are, however, always derived for extra information. The first chapter introduces basic notions of differentiable manifolds, vector fields and flows. The next 80 pages are devoted to Lie groups and their actions, ending with invariant theory for polynomials (the Hilbert-Nagata theorem) and smooth functions (the Schwartz theorem). The third chapter builds up the theory of vector bundles, integration on manifolds and de Rham cohomology, along with Poincaré duality. The fourth chapter on general bundles and connections starts with a thorough treatment of the Frölicher-Nijenhuis bracket, which is used to express any kind of curvature and second Bianchi identity, even for fiber bundles (without structure groups). The author proves that every fiber bundle admits the so-called complete connection, which allows one to define parallel transport along any curve, and treats their holonomy groups. The principal bundles, associated bundles and principal and induced connections are dealt with in detail. The chapter finishes with characteristic classes and jets.
A chapter on Riemannian geometry starts with a careful treatment of connections to geodesic structures to sprays to connectors and back to connections, going via the second and third tangent bundles. The Jacobi flow on the second tangent bundle is a new aspect coming from this point of view. Isometric immersions and Riemannian submersions are treated in analogy to each other. The sixth chapter treats homogeneous Riemann manifolds, the beginning of symmetric space theory and polar actions. The final chapter covers symplectic geometry and classical mechanics, completely integrable Hamiltonian systems and Poisson manifolds. The emphasis is on group actions, momentum mappings and reductions. The careful reader will gain a working knowledge in a wide range of topics of modern coordinate-free differential geometry. A prerequisite for using the book is a good knowledge of undergraduate analysis and linear algebra. The book grew out of the author’s three decades of experience in the field and meets the highest standards. One could hardly find a better book for graduate students and researchers interested in modern differential geometry.