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This book draws a colorful and widespread picture of global affine hypersurface theory up to the most recent state. Moreover, the recent development revealed that affine differential geometry - as differential geometry in general - has an exciting intersection area with other fields of interest, like partial differential equations, global analysis, convex geometry and Riemann surfaces.
The second edition of this monograph leads the reader from introductory concepts to recent research. Since the publication of the first edition in 1993 there appeared important new contributions, like the solutions of two different affine Bernstein conjectures, due to Chern and Calabi, respectively. Moreover, a large subclass of hyperbolic affine spheres were classified in recent years, namely the locally strongly convex Blaschke hypersurfaces that have parallel cubic form with respect to the Levi-Civita connection of the Blaschke metric. The authors of this book present such results and new methods of proof.
From Review for the first edition by R. Walter (Dortmund) in Zenralblatt MATH:
"This book draws a colorful and widespread picture of global affine hypersurface theory up to the most recent state. [. . .] Moreover, the recent development revealed that affine differential geometry - as differential geometry in general - has an exciting intersection area with other fields of interest, like partial differential equations, global analysis, convex geometry and Riemann surfaces (in the complex- analytic sense).
The core of the text is devoted to four important subfields [. . .]: Affine hyperspheres; Rigidity and uniqueness theorems; Variational problems and affine maximal surfaces; Geometric inequalities. There is a comprehensive introduction [. . .], starting at the level of students with a general background in Euclidean differential geometry and basic Riemannian geometry. [. . .] The bibliography contains about 625 items [. . .], such that researchers and newcomers are provided with an almost complete list, starting right with the beginnings.
The book is written in a clear style, and almost all proofs are carried out in detail. Even auxiliary parts from other fields are explained and sometimes proved. This underlines in addition, how close this field is to the broad flow of modern mathematics. [. . .] Summary: This is a fine book, inviting to an active and interesting field of research."
"Indeed, this book is written very carefully; starting from almost nothing (with nice appendices giving a brief introduction to manifolds, tensor calculus, affine connections and Riemannian geometry), it takes the reader through the basic theory and the classical theorems, and finishes with the latest developments in global affine differential geometry, using the language of moving frames. It is written in such a way that it can be used as an introduction to affine differential geometry, but also as a handbook for experts. [. . .] I really enjoyed reading this marvelous book, which, I believe, will be of great value to everyone interested in affine differential geometry." Mathematical Reviews (review of the first edition)
"The first edition of this revised and extended monograph appeared in 1993 [. . .]. Since then everyone interested in affine differential geometry has been indebted to the authors. They wrote an excellent book from elementary aspects up to (then-)recent research. In the new edition, they include some significant developments of the last two decades." Mathematical Reviews
"The book is written in a very clear, rigorous manner, many proofs are given and then a reader (in particular a student) should be able to follow it from the beginning up to recent research. This monograph also contains some historical aspects of the development of the affine differential geometry (in the introduction and also in each chapter), interesting and useful at the same time. It is obvious that the authors succeeded to present in this book new results, but also important global methods in affine differential geometry. Their expertise in this area of research has a major contribution in making this monograph, in the reviewer's opinion, one of the best actual and modern books in affine differential geometry." Zentralblatt für Mathematik
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