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Extrinsic radius pinching in space forms of nonnegative sectional curvature. (English) Zbl 1131.53027

The author considers compact connected and oriented \(n\)-dimensional Riemannian manifolds \((M,g)\) without boundary isometrically immersed into an \((n+1)\)-dimensional simply connected space-form. He gives estimates for the corresponding extrinsic radius of this immersion in terms of higher order mean curvatures. These estimates are extended to so-called pinching cases. The author is able to prove that under a suitable pinching condition \(M\) is diffeomorphic and almost isometric to a sphere.

MSC:

53C40 Global submanifolds
53C20 Global Riemannian geometry, including pinching
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