Differential geometry of varieties with degenerate Gauss maps
type: Item
instance of: scholarly article
title: Differential geometry of varieties with degenerate Gauss maps
zbMATH Open document ID: 1114.53009
author: Maks A. Akivis
author: Vladislav V. Goldberg
published in: CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC
publication date: 2004-02-16T00:00:00Z
review text: 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 con…
Mathematics Subject Classification ID: 53A20
Mathematics Subject Classification ID: 53-02
Mathematics Subject Classification ID: 53A15
zbMATH DE Number: 2043374
zbMATH Keywords: projective differential geometry
zbMATH Keywords: degenerate Gauss map
zbMATH Keywords: variety
zbMATH Keywords: torsal
reviewed by: Rolf Walter
MaRDI profile type: Publication
full work available at URL: b97221
OpenAlex ID: W628476126
DOI: 10.1007/B97221
description: scientific article; zbMATH DE number 2043374