Preface This book is an introduction to the fundamentals of differential geometry (manifolds, flows, Lie groups and their actions, invariant theory, differential forms and de Rham cohomology, bundles and connections, Riemann mani- folds, isometric actions, symplectic geometry) which stresses naturality and functoriality from the beginning and is as coordinate-free as possible. The material presented in the beginning is standard but some parts are not so easily found in text books: Among these are initial submanifolds (2.13) and the extension of the Probenius theorem for distributions of nonconstant rank (the Stefan-Sussman theory) in (3.21) - (3.28). A quick proof of the Campbell-Baker-Hausdorff formula for Lie groups is in (4.29). Lie group actions are studied in detail: Palais' results that an infinitesimal action of a finite-dimensional Lie algebra on a manifold integrates to a local action of a Lie group and that proper actions admit slices are presented with full proofs in sections (5) and (6). The basics of invariant theory are given in section (7): The Hilbert-Nagata theorem is proved, and Schwarz's theorem on smooth invariant functions is discussed, but not proved. In the section on vector bundles, the Lie derivative is treated for natural vector bundles, i.e., functors which associate vector bundles to manifolds and vector bundle homomorphisms to local diffeomorphisms. A formula for the Lie derivative is given in the form of a commutator, but it involves the tangent bundle of the vector bundle. So also a careful treatment of tangent bundles of vector bundles is given. Then follows a standard presentation of differential forms and de Rham cohomoloy including the theorems of de Rham and Poincare duality. This is used to compute the cohomology of compact Lie groups, and a section on extensions of Lie algebras and Lie groups follows. IX
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