1970 (Publication #7)
The spherical two piece property and tight surfaces in spheres
J. Differential Geometry, 4 (1970), 193-205.
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A set, A, in E^3, is said to have the spherical two-piece property (STPP) if no plane or sphere in E^3 separtes A into more than two pieces. Spheres, planes, circles, and tori all have the STPP. The first part of this paper gives a characterization of all closed sets in the plane that have the STPP.
A surface, M^2, in E^n is said to be tight if no hyperplane separtes M^2 into more than two pieces. This paper identifies all tight smooth surfaces that are subsets of spheres in E^n. The main result in this paper in that the only tight surfaces in S^3 besides S^2 are the images under inverse stereographic projection of cyclides of Dupin, a class of alegraic tori in E^3.