Total absolute curvature and tightness of noncompact manifolds

van Gemmeren, Martin

doi:10.1090/S0002-9947-96-01632-7

Total absolute curvature and tightness of noncompact manifolds
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by Martin van Gemmeren PDF

Trans. Amer. Math. Soc. 348 (1996), 2413-2426 Request permission

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

Thomas F. Banchoff, Tightly embedded $2$-dimensional polyhedral manifolds, Amer. J. Math. 87 (1965), 462–472. MR 178472, DOI 10.2307/2373013
T. F. Banchoff, High codimensional $0$-tight maps on spheres, Proc. Amer. Math. Soc. 29 (1971), 133–137. MR 279820, DOI 10.1090/S0002-9939-1971-0279820-4
Dietrich Braess, Morse-Theorie für berandete Mannigfaltigkeiten, Math. Ann. 208 (1974), 133–148 (German). MR 348790, DOI 10.1007/BF01432381
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).
Thomas E. Cecil, Taut immersions of noncompact surfaces into a Euclidean $3$-space, J. Differential Geometry 11 (1976), no. 3, 451–459. MR 438360
T. E. Cecil and P. J. Ryan, Tight and taut immersions of manifolds, Research Notes in Mathematics, vol. 107, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 781126
Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
Shiing-shen Chern and Richard K. Lashof, On the total curvature of immersed manifolds. II, Michigan Math. J. 5 (1958), 5–12. MR 97834
Albrecht Dold, Lectures on algebraic topology, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 200, Springer-Verlag, Berlin-New York, 1980. MR 606196
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
Dirk Ferus, Totale Absolutkrümmung in Differentialgeometrie und -topologie, Lecture Notes in Mathematics, No. 66, Springer-Verlag, Berlin-New York, 1968 (German). MR 0250229, DOI 10.1007/BFb0076980
Charles Hopkins, Rings with minimal condition for left ideals, Ann. of Math. (2) 40 (1939), 712–730. MR 12, DOI 10.2307/1968951
C. H. Houghton, Ends of locally compact groups and their coset spaces, J. Austral. Math. Soc. 17 (1974), 274–284. Collection of articles dedicated to the memory of Hanna Neumann, VII. MR 0357679, DOI 10.1017/S1446788700017055
Nicolaas H. Kuiper, On surfaces in euclidean three-space, Bull. Soc. Math. Belg. 12 (1960), 5–22. MR 123281
Nicolaas H. Kuiper, On convex maps, Nieuw Arch. Wisk. (3) 10 (1962), 147–164. MR 145545
Nicolaas H. Kuiper, Minimal total absolute curvature for immersions, Invent. Math. 10 (1970), 209–238. MR 267597, DOI 10.1007/BF01403250
J. Milnor, Morse theory, Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. Based on lecture notes by M. Spivak and R. Wells. MR 0163331, DOI 10.1515/9781400881802
Marston Morse and Stewart S. Cairns, Critical point theory in global analysis and differential topology: An introduction, Pure and Applied Mathematics, Vol. 33, Academic Press, New York-London, 1969. MR 0245046
R. Sulanke and P. Wintgen, Differentialgeometrie und Faserbündel, Lehrbücher und Monographien aus dem Gebiete der Exakten Wissenschaften, Mathematische Reihe, Band 48, Birkhäuser Verlag, Basel-Stuttgart, 1972 (German). MR 0413153, DOI 10.1007/978-3-0348-5949-3
Peter Wintgen, On total absolute curvature of nonclosed submanifolds, Ann. Global Anal. Geom. 2 (1984), no. 1, 55–87. MR 755209, DOI 10.1007/BF01871945