Tight Polyhedral Submanifolds and Tight Triangulations
This volume is an introduction and a monograph about tight polyhedra. The treatment of the 2-dimensional case is self- contained and fairly elementary. It would be suitable also for undergraduate seminars. Particular emphasis is given to the interplay of
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Wolfgang Kuhnel
Tight Polyhedral Submanifolds and Tight Triangulations
Springer
Author Wolfgang Kuhnel Mathematisches Institut B Universitat Stuttgart Pfaffenwaldring 57 D-70550 Stuttgart, Germany E-mail: [email protected]
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Die Deutsche Bibliothek - CIP-Einheitsaufnahme KUhneI, Wolfgang: Tight polyhedral submanifolds and tight triangulations I Wolfgang Ktihnel. - Berlin; Heidelberg; New York; Barcelona; Budapest; Hong Kong; London; Milan ; Paris ; Tokyo: Springer, 1995 (Lecture notes in mathematics; 1612) ISBN 3-540-60l21-X NE:GT
Mathematics Subject Classification (1991): 53C42, 52B70, 57Q15, 57Q35 ISBN 3-540-60121-X Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part ofthe material is concerned, specifically the rights oftranslation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1995 Printed in Germany Typesetting: Camera-ready TEX output by the author SPIN: 10479510 46/3142-543210 - Printed on acid-free paper
Preface A first version of this monograph was written several years ago, while the author was a guest of the I.H.E.S. at Bures-sur-Yvette, He gratefully acknowledges numerous discussions about tight submanifolds with N.H. Kuiper, T.F. Banchoff and others. Later this first draft was distributed as preprint Nr. 108 of the Math. Dept., University of Duisburg. The author also acknowledges the warm hospitality of the Landau-Center at the Hebrew University of Jerusalem in 1990. At that time Gil Kalai introduced the author to some of the mysteries of the Upper and Lower Bound Conjecture for polytopes, and certain parts of Chapter 4 were being developed. Fortunately or unfortunately, there are still a number of conjectures left open. Tightness is a concept from differential geometry that has many connections to other branches of mathematics. This monograph is a presentation of a part of mathematics sitting between various special disciplines such as differential geometry, topology, theory of convex polytopes, combinatorics. The main intention is to stimulate further fruitful interaction in this direction. The treatment of the 2-dimensional case in the - essentially self-contained - Chapter 2 is an example of an interplay between the theory of convex polytopes, graph theory, and elementary polyhedral topology. Finally, the author should like to thank S. Lukas for typing the first preprint version, B. Dunkel for typing the main part of the present version
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