Synopses & Reviews
In this book the authors study the differential geometry of varieties with degenerate Gauss maps. They use the main methods of differential geometry, namely, the methods of moving frames and exterior differential forms as well as tensor methods. By means of these methods, the authors discover the structure of varieties with degenerate Gauss maps, determine the singular points and singular varieties, find focal images and construct a classification of the varieties with degenerate Gauss maps. The authors introduce the above mentioned methods and apply them to a series of concrete problems arising in the theory of varieties with degenerate Gauss maps. What makes this book unique is the authors' use of a systematic application of methods of projective differential geometry along with methods of the classical algebraic geometry for studying varieties with degenerate Gauss maps. This book is intended for researchers and graduate students interested in projective differential geometry and algebraic geometry and their applications. It can be used as a text for advanced undergraduate and graduate students.
Review
From the reviews: "The study of the Gauss map of algebraic varieties falls into the fields of the so-called projective-differential geometry. ... the authors show their mastery of the Cartan method, stating and proving the most interesting results. ... this book has a very ample and generally accurate bibliography ... . This provides an excellent reference for the interested researcher. ... this book is recommended for people interested in differential geometry and in projective differential geometry, wishing to learn the methods of E. Cartan." (Emilia Mezzetti, SIAM Review, Vol. 48 (1), 2006) "The present authors first aim at an extension of the motion of a variety to the real C8 -category and, secondly, embark on a theory for varieties with degenerate Gauss map, including several structure theorems and a lot of concrete situations. ... is of high interest for specialists who find in it a comprehensive study on degenerate Gauss maps in projective space if they are willing to look after necessary assumptions and foundations ... . The book has an ample bibliography of about 180 items." (Rolf Walter, Zentralblatt MATH, Vol. 1114 (16), 2007)
Synopsis
This book surveys the differential geometry of varieties with degenerate Gauss maps, using moving frames and exterior differential forms as well as tensor methods. The authors illustrate the structure of varieties with degenerate Gauss maps, determine the singular points and singular varieties, find focal images and construct a classification of the varieties with degenerate Gauss maps.
Table of Contents
Preface.- Foundational Material.- Varieties in Projective Spaces and Their Gauss Maps.- Basic Equations of Varieties with Degenerate Gauss Maps.- Main Structure Theorems.- Further Examples and Applications of the Theory of Varieties with Degenerate Gauss Maps.- Bibliography.- Symbols Frequently Used.- Author Index.- Subject Index.