Tight polyhedral submanifolds and tight triangulations
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.
scientific article; zbMATH DE number 803283