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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Inequalities relating sectional curvatures of a submanifold to the size of its second fundamental form and applications to pinching theorems for submanifolds


Authors: Ralph Howard and S. Walter Wei
Journal: Proc. Amer. Math. Soc. 94 (1985), 699-702
MSC: Primary 53C20
DOI: https://doi.org/10.1090/S0002-9939-1985-0792286-0
MathSciNet review: 792286
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Abstract: The Gauss curvature equation is used to prove inequalities relating the sectional curvatures of a submanifold with the corresponding sectional curvature of the ambient manifold and the size of the second fundamental form. These inequalities are then used to show that if a manifold $ \overline M $ is $ \delta $-pinched for some $ \delta > \tfrac{1}{4}$, then any submanifold $ M$ of $ \overline M $ that has small enough second fundamental form is $ {\delta _M}$-pinched for some $ {\delta _M} > \tfrac{1}{4}$. It then follows from the sphere theorem that the universal covering manifold of $ M$ is a sphere. Some related results are also given.


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DOI: https://doi.org/10.1090/S0002-9939-1985-0792286-0
Article copyright: © Copyright 1985 American Mathematical Society