Codimension One Spheres Which Are Null Homotopic

Grove and Halperin [3] introduced a notion of taut immersions. Terng and Thorbergsson [5] give a slightly different definition and showed that taut immersions are a simultaneous generalization of taut immersions of manifolds into Euclidean spaces or spheres, and some interesting embeddings constructed by Bott and Samelson [1]. They go on to prove many theorems about such immersions. One particularly intriguing result, Theorem 6.25, concerned codimension one, null homotopic, tautly embedded spheres. Using a result of Ruberman, [4], they proved in many cases that this sphere had to be a distance sphere, that is, the image of a standard sphere under the exponential map, a generalization of a theorem of Chern and Lashof [2]. Here we observe that the methods of Terng–Thorbergsson and Ruberman suffice to classify tautly immersed, null homotopic, codimension one spheres. Informally one may say that the examples produced by Terng and Thorbergsson in [5] are all that there are. Precisely, we have