Preface
The roots of the project to write this book originated in the early nineteen–
nineties, when several streams of mathematical ideas met after being developed
more or less separately for several decades. This resulted in an amazing interac-
tion between active groups of mathematicians working in several directions. One
of these directions was the study of conformally invariant differential operators re-
lated, on the one hand, to Penrose’s twistor program (M.G. Eastwood, T.N. Bailey,
R.J. Baston, C.R. Graham, A.R. Gover, and others) and, on the other hand, to
hypercomplex analysis (V. Souˇ cek, F. Sommen, and others). It turned out that,
via the canonical Cartan connection or equivalent data, conformal geometry is just
one instance of a much more general picture. Over the years we noticed that the
foundations for this general picture had already been developed in the pioneering
work of N. Tanaka. His work was set in the language of the equivalence problem
and of differential systems, but independent of the much better disseminated de-
velopments of the theory of differential systems linked to names like S.S. Chern,
R. Bryant, and M. Kuranishi. While Tanaka’s work did not become widely known,
it was further developed, in particular, by K. Yamaguchi and T. Morimoto, who
put it in the setting of filtered manifolds and applied it to the geometric study
of systems of PDE’s. A lot of input and stimulus also came from Ch. Fefferman’s
work in complex analysis and geometric function theory, in particular, his parabolic
invariant theory program and the relation between CR–structures and conformal
structures. Finally, via the homogeneous models, all of these studies have close
relations to various parts of representation theory of semisimple Lie groups and Lie
algebras, developed for example by T. Branson and B. Ørsted.
Enjoying the opportunities offered by the newly emerging International Erwin
Schr¨ odinger Institute for Mathematical Physics (ESI) in Vienna as well as the long
lasting tradition of the international Winter Schools “Geometry and Physics” held
every year in Srn´ ı, Czech Republic, the authors of this book started a long and
fruitful collaboration with most of the above mentioned people. Step by step, all of
the general concepts and problems were traced back to old masters like Schouten,
Veblen, Thomas, and Cartan, and a broad research program led to a conceptual
understanding of the common background of the various approaches developed
more recently. In the late nineties, the general version of the invariant calculus
for a vast class of geometrical structures extended the tools for geometric analy-
sis on homogeneous vector bundles and the direct applications of representation
theory expanded in this way to situations involving curvatures. In particular, the
celebrated Bernstein–Gelfand–Gelfand resolutions were recovered in the realm of
general parabolic geometries by V. Souˇ cek and the authors and the cohomological
substance of all these constructions was clarified.
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