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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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Article copyright: © Copyright 1990 American Mathematical Society