Tight polyhedral submanifolds and tight triangulations
type: Item
instance of: scholarly article
title: Tight polyhedral submanifolds and tight triangulations
zbMATH Open document ID: 0834.53004
published in: Lecture Notes in Mathematics
publication date: 1995-10-09T00:00:00Z
review text: The notion of tight submanifolds originated with the study of submanifolds of Euclidean space with minimal total absolute curvature. Tightness generalizes the idea of convexity and means that the submanifolds is embedded as convexly as possible in Euclidean space. A good introduction to this subject, for smooth manifolds, is the book ``Tight and taut immersions of manifolds'', Pitman, Boston (1985; Zbl 0596.53002) by \textit{T. E. Cecil} and \textit{P. J. Ryan}. The present monograph treats tight polyhedral submanifolds in Euclidean space and the related idea of tight triangulations. These are triangulations such that every simplexwise linear mapping into any Euclidean space is tight. The subject has a strong combinatorial flavor and contains surprises for someone only familiar with the smooth theory. For example, there is a tight polyhedral embedding of \(\mathbb{C} P^2 \# (- \mathbb{C} P^2)\) in \(E^8\), but no smooth right immersion into any Euclidean space. This book provides an up-to-date introduction to the area of tightness for polyhedral manifolds. One can obtain a sense of its contents from the list of chapter titles: 1. Introduction and basic notions, 2. Tight polyhedral surfaces, 3. Tightness and \(k\)-tightness, 4. \((k - 1)\)- connected \(2k\)-manifolds, 5. 3-manifolds and twisted sphere bundles, 6. Connected sums and manifolds with boundary, 7. Miscellaneous cases and pseudomanifolds.
Mathematics Subject Classification ID: 53-02
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 52B70
Mathematics Subject Classification ID: 57Q15
Mathematics Subject Classification ID: 57Q35
zbMATH DE Number: 803283
zbMATH Keywords: Heawood inequalities
zbMATH Keywords: polytopes
zbMATH Keywords: combinatorial manifolds
zbMATH Keywords: tight polyhedral submanifolds
zbMATH Keywords: tight triangulations
author: Wolfgang Kühnel
reviewed by: James Hebda
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bfb0096341
OpenAlex ID: W567226596
DOI: 10.1007/BFB0096341
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description: scientific article; zbMATH DE number 803283