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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Isometric homotopy in codimension two


Author: John Douglas Moore
Journal: Trans. Amer. Math. Soc. 292 (1985), 653-663
MSC: Primary 53C42
MathSciNet review: 808744
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Abstract: This article investigates the structure of the space of isometric immersions from a simply connected $ n$-dimensional Riemannian manifold with positive sectional curvatures into $ (n + 2)$-dimensional Euclidean space $ {E^{n + 2}}$. It is proven that if $ n \geqslant 4$ and $ {M^n}$ is such a manifold which admits a $ {C^\infty }$ isometric immersion as a hypersurface in $ {E^{n + 1}}$, then any $ {C^\infty }$ isometric immersion from $ {M^n}$ into $ {E^{n + 2}}$ is $ {C^{2n - 4}}$ homotopic through isometric immersions to an immersion whose image lies in some hyperplane.


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  • [A] Shmuel Agmon, The 𝐿_{𝑝} approach to the Dirichlet problem. I. Regularity theorems, Ann. Scuola Norm. Sup. Pisa (3) 13 (1959), 405–448. MR 0125306
  • [BC] Richard L. Bishop and Richard J. Crittenden, Geometry of manifolds, Pure and Applied Mathematics, Vol. XV, Academic Press, New York-London, 1964. MR 0169148
  • [H1] Wolfgang Henke, Riemannsche Mannigfaltigkeiten konstanter positiver Krümmung in euklidischen Räumen der Kodimension 2, Math. Ann. 193 (1971), 265–278 (German). MR 0288705
  • [H2] Wolfgang Henke, Über die isometrische Fortsetzbarkeit isometrischer Immersionen der Standard-𝑚-Sphäre 𝑆^{𝑚}(⊂𝑅^{𝑚+1}) in 𝑅^{𝑚+2}, Math. Ann. 219 (1976), no. 3, 261–276. MR 0407770
  • [M] John Douglas Moore, Conformally flat submanifolds of Euclidean space, Math. Ann. 225 (1977), no. 1, 89–97. MR 0431046
  • [R] Walter Rudin, Functional analysis, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. McGraw-Hill Series in Higher Mathematics. MR 0365062
  • [W] Lee Whitt, Isometric homotopy and codimension-two isometric immersions of the 𝑛-sphere into Euclidean space, J. Differential Geom. 14 (1979), no. 2, 295–302. MR 587554

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DOI: https://doi.org/10.1090/S0002-9947-1985-0808744-0
Article copyright: © Copyright 1985 American Mathematical Society