Tight and Taut Mappings

Banchoff's thesis [1], written under the direction of Prof. Shiing-Shen Chern, introduced the term "tight" into the subject of minimal total absolute curvature, and produced the first examples of tight embeddings of polyhedral surfaces in dimensions six and higher. Also from his thesis are the first examples in [7] of globally non-rigid tight polyhedral tori.

The two-piece property was first developed in [8] and the spherical version in [5] was the first paper in what has become the study of "taut" mappings. Higher-codimension examples of tight polyhedral embeddings of spheres first appeared in [10]. The first examples of tight polyhedral Klein bottles were constructed in [7], which also proved uniqueness for the six-vertex embedding of the real projective plane into 6-space, the counterpart of the uniqueness theorem for smooth surfaces of Banchoff's mentor, Nicolaas Kuiper, and his colleague, William Pohl.

Tightness for surfaces with boundary was the thesis subject of Lucio Rodriguez, Banchoff's first Ph.D. student, and intermediate tautness was first introduced in the thesis of another of his doctoral students, Eugene Curtin. A third Ph.D. student, Leslie Coghlan, classified tight general position mappings of the real projective plane. Wolfgang Kühnel and Banchoff published two extensive investigations of Kühnel's remarkable nine-vertex tight triangulation of the complex projective plane in [37] and [62].

The story of the first decade in the theories of tight and taut immersions appears in [47]. For a treatment of the entire subject over the past thirty years, see the survey article [73], written with Wolfgang Kühnel, in the MSRI volume "Tight and Taut Submanifolds", edited by Chern and Thomas Cecil, Banchoff's former student and long-time colleague in the Clavius Research Group. That volume was dedicated to the memory of Kuiper and it included a posthumous article which Cecil and Banchoff helped to prepare for publication [75], [76].