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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