This book provides a lucid and comprehensive introduction to the differential geometric study of partial differential equations. It is the first book to present substantial results on local solvability of general and, in particular, nonlinear PDE systems without using power series techniques. The book describes a general approach to systems of partial differential equations based on ideas developed by Lie, Cartan and Vessiot. The most basic question is that of local solvability, but the methods used also yield classifications of various families of PDE systems. The central idea is the exploitation of singular vector field systems and their first integrals. These considerations naturally lead to local Lie groups, Lie pseudogroups and the equivalence problem, all of which are covered in detail.
This book will be a valuable resource for graduate students and researchers in partial differential equations, Lie groups and related fields.

Preface
1
Introduction and summary
2
PDE systems, pfaffian systems and vector field systems
2.1
ODE systems, vector fields and 1-parameter groups
2.2
First order PDE systems in one dependent variable, pfaffian equations and contact transformations
2.3 Jet bundles and contact pfaffian systems
2.4 The theorem of Frobenius
2.5 Mayer''s blowing-up method for proving the Frobenius theorem
3
Cartan''s local existence theorem
3.1
Involutions and characters
3.2
From involutions to complete systems
3.3
How general is the general solution
3.4
Cauchy characteristics
3.5
Maximal involutions and integrable vector field systems
4
Involutivity and the prolongation theorem
4.1
Independence condition and involutivity
4.2
Prolongations
4.3
Explanation of the prolongation theorem
5
Drach''s classification, second order PDEs in one dependent variable, and Monge characteristics
5.1
The classification of Drach
5.2
Second order PDEs in one unknown and their singular vector fields
5.3
Monge characteristic subsystems
6
Integration of vector field .systems V satisfying dim V'' = dim V
1
6.1
Maximal involutions
6.2
Complete subsystems
6.3
The generalized contact bracket
6.4
Reduction to a canonical form and systems of contact coordinates
6.5
How to find all maximal complete subsystems of
6.6
Contact transformations and Lie pseudogroups
6.7
Explicitly integrable systems Higher order contact transformations
7.1
Lie''s rectification theorem for first order PDE systems in one dependent variable
7.2
Backlund''s theorems
7.3
Contact prolongations of local diffeomorphisms
8
Local Lie groups
8.1
The parameter group and its structure constants
8.2
The left- and right-invariant parameter groups
8.3
Left- and right-invariant vector fields and their dual Maurer-Cartan forms
8.4
One-parameter subgroups and the exponential mapping
8.5
The first and second fundamental theorems
8.6
The third fundamental theorem
8.7
Local transformation groups
9
Structural classification of 3-dimensional Lie algebras over the complex numbers
9.1
The classification
9.2
Realizations as transformation groups
10
Lie equations and Lie vector field systems
10.1 Characterization of ODE systems with fundamental solutions
10.2 Lie vector field systems associated to, Lie groups
11
Second order PDEs in one dependent and two independent variables
11.1 Second order PDEs and associated vector field systems
11.2 Monge systems
11.3 A connection with line geometry
11.4 Darboux''s method for hyperbolic PDEs
12
Hyperbolic PDEs with Monge systems admitting two or three first integrals
12.1 First integrals of the first order
12.2 Two first integrals for each Monge system
12.3 How to find integral manifolds
12.4 Integrable systems
12.5 Two first integrals for one Monge system and three for the other
12.6 Three first integrals for each Monge system
13
Classification of hyperbolic Goursat equations
13.1 Goursat equations which are associated to 2-dimensional Lie groups
13.2 Goursat equations associated to the projective group in one variable
14
Cartan''s theory of Lie pseudogroups
14.1 The first fundamental theorem
14.2 The second fundamental theorem
14.3 The third fundamental theorem
14.4 The stability group and its Lie algebra
14.5 Getting rid of inessential invariants
14.6 Normal prolongations
15
The equivalence problem
15.1 The simplest case: e-structures
15.2 The general equivalence problem of Cartan
15.3 Examples
16
Parabolic PDEs and associated PDE systems
16.1 Parabolic PDEs for which the Monge system admits at least two first integrals
16.2 Pfaffian systems of three and two dimensions in five variables
16.3 Systems of two PDEs having a Cauchy characteristic vector field
17
The equivalence problem for general 3-dimensional pfafiian systems in five variables
17.1 The naturally associated homogeneous polynomial F of degree four in two variables
17.2 F has four simple roots
17.3 F vanishes identically
17.4 F has a root of multiplicity 4
17.5 F has one triple and one simple root
17.6 F has two double roots
18
Involutive second order PDE systems in one dependent and three independent variables, solved by the method of Monge
18.1 Preliminaries
18.2 Structural classification
18.3 A single PDE
18.4 Two PDEs
18.5 Three PDEs
18.6 Four and five PDEs
18.7 How to go further
Bibliography
Index

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