Ruled Varieties An Introduction to Algebraic Differential Geometry
Ruled varieties are unions of a family of linear spaces. They are objects of algebraic geometry as well as differential geometry, especially if the ruling is developable. This book is an introduction to both aspects, the algebraic and differential one. St
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Advanced Lectures in Mathematics
Editorial board:
Prof. Dr. Martin Aigner, Freie Universitat Berlin, Germany Prof. Dr. Michael Griiter, Universitat des Saarlandes, Saarbriicken, Germany Prof. Dr. Rudolf Scharlau, Universitat Dortmund, Germany Prof. Dr. Gisbert Wiistholz, ETR Ziirich, Switzerland
Introduction to Markov Chains
Erhard Behrends Einfiihrung in die Symplektische Geometrie
Rolf Berndt Wavelets - Eine Einfiihrung
Christian Blatter Local Analytic Geometry
Theo de Jong, Gerhard Pfister Ruled Varieties
Gerd Fischer, Jens Piontkowski Dirac-Operatoren in der Riemannschen Geometrie
Thomas Friedrich Hypergeometric Summation
Wolfram Koepf The Steiner Tree Problem
Rans-Jiirgen Promel, Angelika Steger The Basic Theory of Power Series
Jesus M. Ruiz
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Gerd Fischer Jens Piontkowski
Ruled Varieties An Introduction to Algebraic Differential Geometry
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Prof. Dr. Gerd Fischer Dr. Jens Piontkowski Heinrich-Heine-Universitat Dusseldorf Mathematisches lnstitut UniversitatsstraBe 1 40255 Dusseldorf, Germany [email protected] [email protected]
Die Deutsche Bibliothek - CIP-Cataloguing-in-Publication-Data A catalogue record for this publication is available from Die Deutsche Bibliothek.
First edition, May 2001
All rights reserved © Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig/Wiesbaden, 2001 Vieweg is a company in the specialist publishing group BertelsmannSpringer.
No part of this publication may be reproduced, stored in a retrieval system or transmitted, mechanical, photocopying or otherwise without prior permission of the copyright holder.
www.vieweg.de Cover design: Ulrike Weigel, www.CorporateDesignGroup.de Printed on acid-free paper
ISBN-13: 978-3-528-03138-1 DOl: 10.1007/978-3-322-80217-0
e-ISBN-13: 978-3-322-80217-0
Er redet nur von Regelflachen, ich werd'mich an dem Flegel rachen! Dedicated to the memory of Karl Stein (1913 - 2000)
Preface The simplest surfaces, aside from planes, are the traces of a line moving in ambient space or, more precisely, the unions of one-parameter families of lines. The fact that these lines can be produced using a ruler explains their name, "ruled surfaces." The mechanical production of ruled surfaces is relatively easy, and they can be visualized by means of wire models. These surfaces are not only of practical use, but also provide artistic inspiration. Mathematically, ruled surfaces are the subject of several branches of geometry, especially differential geometry and algebraic geometry. In classical geometry, we know that surfaces of vanishing Gaussian curvature have a ruling that is even developable. Analytically, developable means that the tangent plane is the same for all points of the ruling line, which is equivalent to saying that the surface can be covered by pieces of paper. A classical result from algebraic geometry states that rulings are very rare for complex algebraic surfaces in three-space: Quadrics have two rulings, smooth cubics contain precisely twenty-seven lines,
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