Remarks on sphere-type theorems

Choi, Hyeong; Kim, Sang; Park, Sung

doi:10.1090/S0002-9939-97-03480-1

Remarks on sphere-type theorems
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by Hyeong In Choi, Sang Moon Kim and Sung Ho Park

Proc. Amer. Math. Soc. 125 (1997), 569-572

DOI: https://doi.org/10.1090/S0002-9939-97-03480-1

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Abstract:

We prove if $M$ is a complete Riemannian manifold with an embedded totally geodesic compact hypersurface $N$ such that $M$ has nonnegative sectional curvature, and the sectional curvature of $M$ is strictly positive in a neighborhood of $N$, then the pair $(M,N)$ is diffeomorphic to the pair $(S^n,S^{n-1})/\pi _1(M)$. This result gives an affirmative answer to a question of H. Wu in the case when $M$ is compact and simply connected.

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