#32: Tight and Taut Submanifolds

MSRI Publications -- Volume 32

Tight and Taut Submanifolds

Edited by Thomas E. Cecil and Shiing-shen Chern

Tight and taut submanifolds form an important class of manifolds with special curvature properties, one that has been studied intensively by differential geometers since the 1950's. They are in some ways the simplest figures after convex bodies: for example, tight manifolds in R^n are characterized by the fact that their intersection with any half-space is connected. Examples include many well-known manifolds such as spheres, Veronese surfaces, isoparametric hypersurfaces and the cyclides of Dupin.

This book contains six in-depth articles by leading experts in the field and an extensive bibliography. It is dedicated to the memory of Nicolaas H. Kuiper, and the first paper is an unfinished but insightful exposition of the subject of tight immersions and maps, written by Kuiper. Other papers survey topics such as the smooth and polyhedral portions of the theory of tight immersions; taut, Dupin, and isoparametric submanifolds of Euclidean space; taut submanifolds of arbitrary complete Riemannian manifolds; and real hypersurfaces in complex space forms with constant principal curvatures. Taken together these articles provide a comprehensive survey of the field and point toward several directions for future research.