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On the topology of a compact submanifold of a sphere with bounded second fundamental form. (English) Zbl 0788.53050

Summary: Let \(h\) be the second fundamental form of a compact submanifold of a unit sphere. We show that if \(\| h(u,u)\|^ 2 < {1\over 3}\) holds for any unit tangent vector \(u\) at any point on the submanifold then it is a homotopy sphere.

MSC:

53C40 Global submanifolds
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References:

[1] H. Gauchman, Minimal submanifolds of a sphere with bounded second fundamental form. Trans. Amer. Math. Soc. 298 (1986) 779-791 · Zbl 0608.53056 · doi:10.1090/S0002-9947-1986-0860393-5
[2] H.B. Lawson and J. Simons, On stable currents and their application to global problems in real and complex geometry. Ann. of Math. 98 (1973) 427-450 · Zbl 0283.53049 · doi:10.2307/1970913
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