This second edition extends the first book of these authors published in 1993, by including significant developments since then. As the authors write in the introduction, this has been the main motivation for the second edition.
Chapter 1, very well structured, contains preliminaries and elementary structural aspects about the affine theory of hypersurfaces. It contains a review of the Euclidean hypersurface theory and preparatory material about transversal and conormal fields of hypersurfaces. It closes by the introduction of the three most important relative normalizations: the Euclidean, the equiaffine and the centroaffine ones.
Chapter 2, Local equiaffine hypersurface theory, is devoted to the local theory and uses the invariant and the local calculus, together with Cartan’s moving frames.
Chapter 3, Affine hyperspheres, recalls problems solved during the last decade. The authors present a local classification of all locally strongly convex affine spheres with equiaffine metric of constant sectional curvature. All equiaffine hypersurfaces with a parallel Fubini-Pick form are affine hyperspheres and if the hypersurfaces are locally strongly convex, they are either hyperquadrics or hyperbolic hyperspheres, completely and explicitely classified by Z. Hu, H. Li and L. Vrancken. Also, affine spheres with complete Blaschke metric are investigated. The first global result is due to Blaschke and later contributions of Calabi, Schneider, Pogorelov, Cheng, Yau, Sasaki, Gigena, A.-M. Li are mentioned. As the authors point out, this classification requires subtle estimates and this chapter, especially Section 3.4, is an excellent demonstration of strong PDE methods.
Chapter 4, Rigidity and uniqueness theorems, presents two different methods: the application of an integral formula and the index method, several integral formulas of Minkowski and Lichnerowicz type are derived. Another important result is about the global solution of an important Schrödinger-type equation.
In Chapter 5, Variational problems and affine maximal surfaces, two variational problems are studied: variational formulas for the affine curvature integrals are proved and recent results about affine minimal surfaces (nowadays called affine maximal surfaces, following Calabi) are presented.
Chapter 6 studies hypersurfaces with constant affine Gauss-Kronecker curvature. The contents of this chapter in this second edition are completely new.
Chapter 7, Geometric inequalities, gives an introduction to affine problems related to the theory of convex bodies and proves the affine isoperimetric inequality and some integral inequalities for affine curvature integrals.
There are two appendices. Appendix A recalls the basic concepts from differential geometry. In Appendix B, the Laplacian comparison theorem is stated and proved.
The bibliography contains 413 references and is followed by an index.
It follows that the second edition is substantially improved. Some important new contributions are listed below (see also the introduction, page 3):
i) Proofs of both versions of an affine Bernstein conjecture in dimension two, posed by Calabi and Chern, respectively.
ii) New results on the class of locally strongly convex Blaschke hypersurfaces with constant Gauss-Kronecker curvature which includes the class of affine hyperspheres.
iii) An explicit classification of the locally strongly convex Blaschke hypersurfaces with Fubini-Pick form being parallel with respect to the Levi-Civita connection of the Blaschke metric.
Besides the valuable contents of this monograph, a few things should be certainly mentioned.
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.