Codimension Two Submanifolds of Positive Curvature

In this note it is proven that a compact connected $n$-dimensional Riemannian manifold of positive curvature, isometrically immersed in $(n + 2)$-dimensional Euclidean space, is a homotopy sphere if $n \geqslant 3$; hence it is homeomorphic to a sphere if $n \geqslant 5$.

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 1978-06, Vol.70 (1), p.72-74
Main Author: Moore, John Douglas
Format: Article
Language:English
Subjects:
Critical points
Curvature
Euclidean space
Mathematical duality
Mathematical functions
Mathematical theorems
Morse theory
Riemann manifold
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Description
Summary:In this note it is proven that a compact connected $n$-dimensional Riemannian manifold of positive curvature, isometrically immersed in $(n + 2)$-dimensional Euclidean space, is a homotopy sphere if $n \geqslant 3$; hence it is homeomorphic to a sphere if $n \geqslant 5$.
ISSN:0002-9939
1088-6826
DOI:10.1090/S0002-9939-1978-0487560-8