Vol. 108, No. 2, 1983

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Appropriate cross-sectionally simple four-cells are flat

Steven Alan Pax

Vol. 108 (1983), No. 2, 379–384
Abstract

When X is a set in En, we let Xt = X Ht—where Ht is the horizontal hyperplane in En of height t. In this note, we prove that a 4-cell B in E4, such that each nonempty slice Bt is either a point or a 3-cell, is flat whenever, for all t, Bt is flat in Ht and Bd Bt is flat in Bd B.

Mathematical Subject Classification 2000
Primary: 57N13
Secondary: 57N45, 57N50
Milestones
Received: 19 August 1981
Revised: 14 January 1983
Published: 1 October 1983
Authors
Steven Alan Pax