Differential geometry of varieties with degenerate Gauss maps
Complex varieties are the classical and main objects of algebraic geometry. Their singularities, considered as point sets, can be very complicated. The theory of polynomial ideals yields the adequate tools to investigate them on the algebraic side. Additional viewpoints have been brought into the game by applying infinitesimal methods from differential geometry; see the fundamental paper by \textit{P. Griffiths} and \textit{J. Harris} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 12, 355--452 (1979; Zbl 0426.14019)]. In fact this work, dealing mainly with complex analytic submanifolds in complex projective spaces, may be considered a starting point for the analogue real situation. The present authors first aim at an extension of the notion of a variety to the real \(C^{\infty}\)-category and, secondly, embark on a theory for varieties with degenerate Gauss map, including several structure theorems and a lot of concrete situations. Unfortunately, the foundations on varieties in the first part of the book are weak and ambiguous. The remainder of the book contains informative results, otherwise hardly available, which are mainly due to Russian authors, in particular to the book authors themselves. Geometrically, the authors want to study a subset \(X\) of the real projective space \( \mathbb{P}^{N} \) whose set of tangent spaces develops singularities, generalizing examples like cones or torses in the three-dimensional case. In the reviewer's opinion, in the book there are serious inaccuracies on the definition of a variety, the differentiability of it, the notion of its tangent spaces, and the nature of singular points. While the authors emphasize the importance of varieties in the differential geometry of \(\mathbb P^N\) it is strange enough that the book does not contain any satisfactory definition of this fundamental notion in a clean analytical language. To give an idea of the main object consider the following (simplified) situation: Let \(M\) be an \(n\)-dimensional connected differentiable manifold and \( f: M \to \mathbb P^N\) an immersion into the real projective space \(\mathbb P^N\) of dimension \(N > n\), everything being of class \(C^{\infty}\). By its tangential, \(f\) induces a map \( \gamma \), the Gauss map, of \(M\) into the Grassmannian of \(n\)-dimensional projective subspaces of \(\mathbb P^N\). The degeneracies of \( \gamma \) are those appealed to in the book title. It is assumed that \( \gamma \) is of constant rank \(r<n\), and then the image set \(X := f(M)\) is a variety with a degenerate Gauss map of rank \(r\). Observe that the tangent space \( T_{x}X \) for \(x \in X\) is only defined uniquely if \(f\) is injective, and then \(\gamma\) may be considered as defined on \(X\). In most cases the treatment is local. Then a variety foliates into leaves along which the tangent subspace \( T_{x}X \) is fixed (as for cones and torses in \(\mathbb P^3\)). The authors speak of a Monge-Ampère foliation. In fact, it is proved that the leaves of the Monge-Ampère foliation are (contained in) projective subspaces, the generators. The next decisive step is to introduce the focal points of the generators (like the vertex of a cone or the edge of regression of a torse), as initiated by the first author. The corresponding focal sets are then applied in order to a obtain a fine classification of varieties with degenerate Gauss map showing that their building blocks are three types of ``basic varieties'', namely cones, torsal varieties and hypersurfaces with degenerate Gauss map. Beyond structural results of this type, the topics include Grassmannians, Segre- and Veronese-manifolds, an affine analogue of the Hartman-Nirenberg cylinder theorem, and applications to de Sitter spaces and to smooth lines in projective planes over two-dimensional algebras. The method is almost thoroughly that of É. Cartan, i.e. adapted moving frames in connection with differential forms. Existence results are mostly proved with the aid of É. Cartan's test on Pfaffian systems. So typical existence theorems have a local form saying that the desired objects depend on a finite number of functions of a finite number of variables. An important tool is the second fundamental form which can be defined despite the absence of normal spaces. The book 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 the necessary assumptions and foundations on their own. Needless to say that the book is not suitable for beginners. It is noteworthy to see the Cartan method worked out by two masters of this field. The book has an ample bibliography of about \( 180 \) items. Every chapter is followed by copious historical notes.
scientific article; zbMATH DE number 2043374