Abstract
Immersions or maps of closed manifolds in Euclidean space, of minimal absolute total curvature are called tight in this paper. (They were called convex in [25].) After the definition in Chapter 1, many examples in Chapter 2, and some special topics in Chapter 3, we prove in Chapter 4 that topological tight immersions ofn-spheres are only of the expected type, namely embeddings onto the boundary of a convexn+1-dimensional body. This generalises a theorem of Chern and Lashof in the smooth case. In Chapter 5 we show that many manifolds exist that have no tight smooth immersion in any Euclidean space.
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This research was partially supported by National Science Foundation grant GP-7952X1.
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Kuiper, N.H. Minimal total absolute curvature for immersions. Invent Math 10, 209–238 (1970). https://doi.org/10.1007/BF01403250
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DOI: https://doi.org/10.1007/BF01403250