**Graduate Studies in Mathematics**

Volume: 107;
2009;
671 pp;
Hardcover

MSC: Primary 58; 53; 22; 55;

Print ISBN: 978-0-8218-4815-9

Product Code: GSM/107

List Price: $89.00

Individual Member Price: $71.20

**Electronic ISBN: 978-1-4704-1170-1
Product Code: GSM/107.E**

List Price: $89.00

Individual Member Price: $71.20

#### Supplemental Materials

# Manifolds and Differential Geometry

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*Jeffrey M. Lee*

Differential geometry began as the study of curves and surfaces using
the methods of calculus. In time, the notions of curve and surface were
generalized along with associated notions such as length, volume, and
curvature. At the same time the topic has become closely allied with
developments in topology. The basic object is a smooth manifold, to
which some extra structure has been attached, such as a Riemannian
metric, a symplectic form, a distinguished group of symmetries, or a
connection on the tangent bundle.

This book is a graduate-level introduction to the tools and structures
of modern differential geometry. Included are the topics usually found
in a course on differentiable manifolds, such as vector bundles,
tensors, differential forms, de Rham cohomology, the Frobenius theorem
and basic Lie group theory. The book also contains material on the
general theory of connections on vector bundles and an in-depth chapter
on semi-Riemannian geometry that covers basic material about Riemannian
manifolds and Lorentz manifolds. An unusual feature of the book is the
inclusion of an early chapter on the differential geometry of
hypersurfaces in Euclidean space. There is also a section that derives
the exterior calculus version of Maxwell's equations.

The first chapters of the book are suitable for a one-semester course on
manifolds. There is more than enough material for a year-long course on
manifolds and geometry.

#### Table of Contents

# Table of Contents

## Manifolds and Differential Geometry

- Cover Cover11 free
- Title page iii4 free
- Contents v6 free
- Preface xi12 free
- Differentiable manifolds 116 free
- The tangent structure 5570
- Immersion and submersion 127142
- Curves and hypersurfaces in Euclidean space 143158
- Lie groups 189204
- Fiber bundles 257272
- Tensors 307322
- Differential forms 345360
- Integration and Stokes’ theorem 391406
- De Rham cohomology 441456
- Distributions and Frobenius’ theorem 467482
- Connections and covariant derivatives 501516
- Riemannian and semi-Riemannian geometry 547562
- The language of category theory 637652
- Topology 643658
- Some calculus theorems 647662
- Modules and multilinearity 649664
- Bibliography 663678
- Index 667682 free
- Back Cover Back Cover1690

#### Readership

Graduate students and research mathematicians interested in differential geometry.

#### Reviews

This book is
certainly a welcome addition to the literature. As noted, the author has an
on-line supplement, so the interested reader can follow up on the development
of further topics and corrections. One cannot begin to imagine the Herculean
amount of work that went into producing a volume of this size and scope, over
660 pages!

-- Mathematical Reviews