**Spivak, A Comprehensive Introduction to Differential
Geometry**

**Volume 5.** 3rd edition. x + 467 pages.
Clothbound. 1999

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**Outline of Contents:**

10. And Now A Brief Message From Our Sponsor

1. FIRST ORDER PDE's

Linear first
order PDE's; characteristic curves; Cauchy problem for

free initial
curves. Quasi-linear first order PDE's; characteristic

curves;
Cauchy problem for free initial conditions; characteristic

initial
conditions. General first order PDE's; Monge cone; characteristic

curves of a
solution; characteristic strips; Cauchy problem for free

initial data;
characteristic initial data. First order PDE's in *n* variables.

2. FREE INITIAL MANIFOLDS FOR HIGHER
ORDER EQUATIONS

3. SYSTEMS OF FIRST ORDER PDE'S

4. THE CAUCHY-KOWALEWSKI THEOREM

5. CLASSIFICATION OF SECOND ORDER
PDE'S

Classification of semi-linear equations. Reduction to normal forms.

Classification of general second order equations.

6. THE PROTOTYPICAL PDE'S OF PHYSICS

The wave
equation; the heat equation; Laplace's equation. Elementary

properties.

7. HYPERBOLIC SYSTEMS IN TWO VARIABLES

8. HYPERBOLIC SECOND ORDER EQUATIONS IN TWO VARIABLES

First
reduction of the problem. New system of characteristic equations.

Characteristic initial data. Monge-Ampère equations.

9. ELLIPTIC SOLUTIONS OF SECOND ORDER EQUATIONS IN TWO
VARIABLES

Addenda. Differential systems; the Cartan-Kähler Theorem. An
elementary

maximum principal.

11. Existence and Non-Existence of Isometric Imbeddings

Non-imbeddability
theorems; exteriorly orthogonal bilinear forms;

index of nullity and index of
relative nullity. The Darboux equation.

Burstin-Janet-Cartan Theorem.

Addendum. The embedding problem via
differential systems.

12. Rigidity

Rigidity in higher
dimensions; type number. Bendings, warpings, and

infinitesimal bendings. Vector-valued
differential forms, the support

function, and Minkowski's formulas.
Infinitesimal rigidity of convex

surfaces. Cohn-Vossen's Theorem.
Minkowski's Theorem. Christoffel's

Theorem. Other problems, solved and
unsolved. Local problems;

the role of the asymptotic curves.
Other classical results. E. E. Levi's

Theorems and Schilt's Theorem.
Surfaces in the 3-sphere and hyperbolic

3 space. Rigidity for higher
codimension.

Addendum. Infinitesimal bendings of
rotation surfaces.

13. The Generalized Gauss-Bonnet Theorem

Historical remarks.

1. OPERATIONS ON BUNDLES

Bundle maps
and principal bundle maps; Whitney sums and induced

bundles; the
covering homotopy theorem.

2. GRASSMANNIANS AND UNIVERSAL
BUNDLES

3. THE PFAFFIAN

4. DEFINING THE EULER CLASS IN
TERMS OF A CONNECTION

The Euler
class. The class *C*. The Gauss-Bonnet-Chern Theorem.

5. THE CONCEPT OF
CHARACTERISTIC CLASSES

6. THE COHOMOLOGY OF
HOMOGENEOUS SPACES

The smooth
structure of homogeneous spaces. Invariant forms.

7. A SMATTERING OF CLASSICAL
INVARIANT THEORY

The Capelli
identities. The first fundamental theorem of invariant theory

for O(*n*)
and SO(*n*).

8. AN EASIER
INVARIANCE PROBLEM

9. THE COHOMOLOGY OF THE
ORIENTED GRASSMANNIANS

Computation
of the cohomology; Pontryagin classes. Describing the

characteristic classes in terms of a connection.

10. THE WEIL HOMOMORPHISM

11. COMPLEX BUNDLES

Hermitian
inner products, the unitary group, and complex Grassmanians.

The
cohomology of the complex Grassmanians; Chern classes.

Relations
between the Chern classes and the Pontryagin and Euler classes.

12. VALEDICTORY

Addenda. Invariant theory for the
unitary group

Recovering the differential forms;
the Gauss-Bonnet-Chern Theorem

for manifolds-with-boundary

BIBLIOGRAPHY

A.
Other topics in Differential Geometry

B. Books

C. Journal articles

**Stephanie Alexander, Bulletin of the AMS, volume 84,
number 1, January 1978
**

For the 5 volume set:

The *Comprehensive introduction* is probably best suited
for leisurely and enjoyable background reference by almost anyone interested in
differential geometry. Great care has been taken to make it accessible to
beginners, but even the most seasoned reader will find stimulating reading here
... The appeal of the book is due first of all to its choice of material, which
is guided by the liveliest geometric curiosity. In addition, Spivak has a clear,
natural and well-motivated style of exposition; in many places, his book unfolds
like a novel.

The *Comprehensive introduction* will be widely read and
enjoyed, and will surely become a standard reference for graduate courses in
differential geometry. Spivak is greatly to be thanked for this spontaneous,
exuberant and beautifully geometrical book.

Specifically for Volume 5:

Spivak has prepared a course on PDE for geometers ... its chosen topics are treated rigorously and in considerable generality. As always, Spivak emphasizes conceptually appealing proofs.

Several of the book's outstanding virtues are represented in
this treatment [of the relationship between characteristic classes and
curvature]: it is self-contained; it gives more than cursory attention to
classical invariant theory; and it prizes and imparts geometric insight.

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