The study of CR manifolds lie at the intersection of three main
mathematical disciplines, partial differential equations, complex
analysis in several complex variables, and differential geometry.
While the complex analysis and PDEs aspect have been intensly
studied in the last fifty years, much effort has been recently made
to understand the differential geometric side of the subject.This
monograph provides a unified presentation of several differential
geometric aspects in the theory of CR manifolds and tangential
Cauchy-Riemann equations. It presents topics from the
Tanaka-Webster connection, a key contributor to the birth of
pseudohermitian geometry, to the major differential geometric
acheivements in the theory of CR manifolds, such as Fefferman's
metric, pseudo-Einstein structures and the Lee conjecture, CR
immersions, subelliptic harmonic maps as a local manifestation of
pseudoharmonic maps from a CR manifold, Yang-Mills fields on CR
manifolds, to name several. It also aims at explaining how certain
results from analysis are employed in CR geometry. results and
stimulating unproved statements and comments referring to the most
recent aspects of the theory, this monograph is suitable for
researchers and graduate students in differential geometry, complex
analysis, and PDEs.
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