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.

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