Some pinching and classification theorems for minimal submanifolds. (English) Zbl 0811.53060

The author proves that the sectional curvature \(K\) and the scalar curvature \(\tau\) of a minimal submanifold of a Euclidean space satisfy the inequality \(K(\pi) \geq {1\over 2} \tau (p)\) for any plane section \(\pi \subset T_ p(M)\), \(p \in M\). Then, he constructs examples of minimal submanifolds of a Euclidean space which satisfy \(\inf K = {1\over 2} \tau\) and classifies minimal submanifolds in Euclidean space which satisfy \(\inf K = {1\over 2} \tau\).
Reviewer: M.Okumura (Urawa)


53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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[1] B. Y.Chen, Geometry of submanifolds. New York-Basel 1973. · Zbl 0262.53036
[2] B. Y.Chen, Total mean curvature and submanifolds of finite type. Singapore-New Jersey-London-Hong Kong 1984. · Zbl 0537.53049
[3] B. Y. Chen andM. Okumura, Scalar curvature, inequality and submanifold. Proc. Amer. Math. Soc.38, 605-608 (1973). · Zbl 0256.53041
[4] S. S. Chern, Minimal submanifolds in a Riemannian manifold. Lawrence, Kansas 1968.
[5] M. do Carmo andM. Dajczer, Rotation hypersurfaces in spaces of constant curvature. Trans. Amer. Math. Soc.277, 685-709 (1983). · Zbl 0518.53059
[6] F. Dillen, Semi-parallel hypersurfaces of a real space form. Israel J. Math.75, 193-202 (1991). · Zbl 0765.53012
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