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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Immersions of positively curved manifolds into manifolds with curvature bounded above


Author: Nadine L. Menninga
Journal: Trans. Amer. Math. Soc. 318 (1990), 809-821
MSC: Primary 53C42; Secondary 53C40
MathSciNet review: 962285
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Abstract: Let $ M$ be a compact, connected, orientable Riemannian manifold of dimension $ n - 1 \geqslant 2$, and let $ x$ be an isometric immersion of $ M$ into an $ n$-dimensional Riemannian manifold $ N$. Let $ K$ denote sectional curvature and $ i$ denote the injectivity radius. Assume, for some constant positive constant $ c$, that $ K(N) \leqslant 1/(4{c^2}),\quad 1/{c^2} \leqslant K(M)$, and $ \pi c \leqslant i(N)$. Then the radius of the smallest $ N$-ball containing $ x(M)$ is less than $ \tfrac{1} {2}\pi c$ and $ x$ is in fact an imbedding of $ M$ into $ N$, whose image bounds a convex body.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1990-0962285-0
PII: S 0002-9947(1990)0962285-0
Article copyright: © Copyright 1990 American Mathematical Society