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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
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
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.
Addendum. The embedding problem via differential systems.
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
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.
Addenda. Invariant theory for the unitary group
Recovering the differential forms; the Gauss-Bonnet-Chern Theorem
A. Other topics in Differential Geometry
C. Journal articles