In 1907 Poincare showed that two real hypersurfaces in
are in gen-
eral biholomorphically inequivalent, [Po]. That is, given two real analytic
submanifolds of real dimension three in C there usually is no biholomor-
2 2
phism of one open set in C to another open set in C that takes a piece
of the first submanifold onto the second. He then raised the question of
finding the invariants that distinguish one real hypersurface from another.
This basic question was first completely answered by Cartan, in [Ca2]. Car-
tan remarked "Je reprends la question directement comme application de
ma methode generate d'equivalence. La resolution complete du probleme de
Poincare me conduit a des notions geometriques nouvelles ... ". A second
solution was given by Moser in 1973. In joint work with Chern [CM] this
was generalized, along with Cartan's original solution, to dimensions greater
than two. (At about this time, Tanaka in [Tanl] and [Tan2] gave a different
extension of Cartan's work to higher dimensions.)
The study of the basic problem for CR structures primarily rests on these
two works. Thus, although most introductions to an area of mathematics are
a synthesis from many sources, reflecting how mathematics usually develops,
the present one, to a surprising degree, is not. Rather, it is in large measure
an exposition of the papers of Cartan and of the joint paper of Chern and
For Cartan one needs to know something of his general method of equiv-
alences and of the structure of the Lie group SU(2, 1). So we have tried
to provide this necessary background before going over the main construc-
tion. This background is also important for the part of the Chern-Moser
paper that extends Cartan's work to higher dimension. For the rest of the
Chern-Moser paper the problem is somewhat different. Here the approach is
more straightforward but also more technically difficult. We do the lowest-
dimensional case in detail. This should also make the higher-dimensional
case more accessible.
An exception to this focus on the above two works is Chapter 1. Here
we give the basic definitions and properties and draw from many sources
in the literature and in the "folklore." Chapter 2 uses simple facts about
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