It is the intention of the two authors to provide in their monograph a systematic introduction to the subject covering as much of the recent progress as possible. After a short historical summary they develop the theory (affine and projective immersions) from a very structural point of view using strictly index-free notations. In contrast to the book “Global affine differential geometry of hypersurfaces” (1993; Zbl 0808.53002) by A.-M. Li, U. Simon and G. Zhao local aspects have priority.
Some of the main topics should be emphasized: A better formulation of Radon’s fundamental theorem involving an affine connection \(\nabla\) and a nondegenerate metric \(h\) (Theorem 8.2 of Chapter II), the major results of E. Calabi and others about complete improper and proper affine hyperspheres (Theorem 11.5 and 7.6 of Chapter III, Note 7), the so-called affine Bernstein problem for affine minimal surfaces (Theorem 11.3 of Chapter III), the classification of all nondegenerate surfaces in \(\mathbb{R}^3\) which are homogeneous under equiaffine transformations resp. of all affine spheres with affine metric of constant curvature (Theorem 3.1 resp. Theorem 5.1 and 5.4 of Chapter III) and last not least a theory of hypersurfaces in the projective space \(\mathbb{P}^{n + 1}\) in connection with codimension 2 centro-affine immersions (Note 9). Finally we want to call attention to the extension of affine differential geometry to the complex case in Section 9 of Chapter IV.
Certainly, the (not very systematically arranged) monograph under review provides a helpful instrument for everybody interested in affine differential geometry.