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Some remarks on tight hypersurfaces

Published online by Cambridge University Press:  14 November 2011

Leslie Coghlan
Affiliation:
Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294, U.S.A.

Synopsis

We study the outer part of tight hypersurfaces. We explore in detail how the outer part of such hypersurfaces for n ≧ 3 is more complicated than in the case of tight surfaces in R3. We give a theorem describing tight hypersurfaces of arbitrary dimension.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1991

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References

1Cecil, T. E. and Ryan, P. J.. Tight and Taut Immersions of Manifolds, Research Notes in Mathematics 107 (New York: Pitman, 1985).Google Scholar
2Chern, S. S. and Lashof, R. K.. On the total curvature of immersed manifolds I. Amer. J. Math. 79 (1957), 306318.CrossRefGoogle Scholar
3Coghlan, L.. Some general constructions of tight mappings. Proc. Roy. Soc. Edinburgh Sect. A 111 (1989), 231247.CrossRefGoogle Scholar
4Coghlan, L.. Topsets of tight hypersurfaces. Geometriae Dedicata (submitted).Google Scholar
5Eells, J. and Kuiper, N. H.. Manifolds which are like projective planes. Publ. Math. IHES 14 (1962), 546.CrossRefGoogle Scholar
6Hirsch, M.. Immersions of manifolds. Trans. Amer. Math. Soc. 93 (1959), 242276.CrossRefGoogle Scholar
7Kuiper, N. H.. On surfaces in Euclidian three space. Bull. Soc. Math. Belg. 12 (1960), 512.Google Scholar
8Kuiper, N. H.. Convex immersions of closed surfaces in E 3. Comment. Math. Helv. 35 (1961), 8592.CrossRefGoogle Scholar
9Kuiper, N. H.. Minimal total absolute curvature for immersions. Invent. Math. 10 (1970), 209238.CrossRefGoogle Scholar
10Kuiper, N. H.. Tight Embeddings and Maps. Submanifolds of Geometrical Class Three in E n. The Chern Symposium 1979, Berkeley, 97145 (Berlin: Springer, 1980).CrossRefGoogle Scholar
11Kuiper, N. H.. There is no tight continuous immersion of the Klein bottle into R3 (IHES, Preprint 1983).Google Scholar
12Lickorish, W. B. R.. The unknotting number of a classical knot. Combinatorial Methods in Topology and Algebraic Geometry. Contemp. Math. 44 (1985), 117121.CrossRefGoogle Scholar
13Thorbergsson, G.. Tight immersions of highly connected manifolds. Comment. Math. Helv. 61 (1986), 102121.CrossRefGoogle Scholar