Transactions of the American Mathematical Society

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ISSN 1088-6850(e) ISSN 0002-9947(p)

Total absolute curvature and tightness of noncompact manifolds

Author(s): Martin van Gemmeren
Journal: Trans. Amer. Math. Soc. 348 (1996), 2413-2426.
MSC (1991): Primary 53C42; Secondary 57R45
MathSciNet review: 1355077
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Abstract: In the first part we prove an extension of the Chern-Lashof inequality for noncompact immersed manifolds with finitely many ends. For this we give a lower bound of the total absolute curvature in terms of topological invariants of the manifold. In the second part we discuss tightness properties for such immersions. Finally, we give an upper bound for the substantial codimension.

References:

1.: T.F. Banchoff, Tightly embedded 2-dimensional polyhedral manifold, Amer. J. Math. 87 (1965), 462-472. MR 31:2729
2.: ------, High codimensional $\begin{math}0\end{math}$ -tight maps on spheres, Proc. Amer. Math. Soc. 161 (1971), 259-267. MR 43:5541
3.: D. Braess, Morse-Theorie für berandete Mannigfaltigkeiten, Math. Annalen 208 (1974), 133-148. MR 50:1285
4.: P. Breuer, Straffe Immersionen von kompakten differenzierbaren Mannigfaltigkeiten in euklidische Räume und die homologische Kennzeichnung dieser Immersionen (insbesondere Zwei-Stück-Eigenschaft), Diploma Thesis, Köln (1990).
5.: T.E. Cecil, Taut immersions of non-compact surfaces into a Euclidean 3-space, J. Differential Geometry 11 (1976), 451-459. MR 55:11275
6.: T.E. Cecil and P.J. Ryan, Tight and Taut Immersions of Manifolds, Pitman, London, 1985. MR 87b:53089
7.: S.-s. Chern and R.K. Lashof, On the total curvature of immersed manifolds. I, Amer. J. Math. 79 (1957), 396-398. MR 18:927a
8.: ------, On the total curvature of immersed manifolds. II, Mich. J. Math. 5 (1958), 5-12. MR 20:4301
9.: A. Dold, Lectures on Algebraic Topology, Springer, Berlin, 1980. MR 82c:55001
10.: S. Eilenberg and N. Steenrod, Foundations of algebraic topology, Princeton University Press, Princeton, 1952. MR 14:398b
11.: Ky Fan and N. Gottesman, On compactifications of Freudenthal and Wallman, Indag. Math. 14 (1952), 504-510. MR 14:669c
12.: D. Ferus, Totale Absolutkrümmung in Differentialgeometrie und -topologie, Lecture Notes in Mathematics 66, Springer, 1968. MR 40:3468
13.: H. Freudenthal, Kompaktifizierungen und Bikompaktifizierungen, Indag. Math. 13 (1951), 184-192. MR 12:728b
14.: C.H. Houghton, Ends of locally compact groups and their coset spaces, J. Australian Math. Soc. 17 (1974), 274-284. MR 50:10147
15.: N.H. Kuiper, On surfaces in euclidean three spaces, Bull. Soc. Math. Belg. 12 (1960), 5-22. MR 23:A609
16.: ------, On convex maps, Nieuv Archief voor Wisk. 10 (1962), 147-164. MR 26:3076
17.: ------, Minimal total absolute curvature for immersions, Invent. Math. 10 (1970), 209-238. MR 42:2499
18.: J. Milnor, Morse Theory, Annals of Mathematical Studies 51, Princeton, 1963. MR 29:634
19.: M. Morse and S.S. Cairns, Critical point theory in global analysis and differential topology, Academic Press, New York, 1969. MR 39:6358
20.: R. Sulanke und P. Wintgen, Differentialgeometrie und Faserbündel, Birkhäuser, Berlin-Basel-Stuttgart, 1972. MR 54:1274
21.: P. Wintgen, On total absolute curvature of nonclosed submanifolds, Annals of Global Analysis and Geometry 2 (1984), 55-87. MR 85m:53071

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