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A pinching theorem for the normal scalar curvature of invariant submanifolds. (English) Zbl 1108.53020
Summary: We prove some inequalities relating intrinsic and extrinsic curvature invariants for invariant submanifolds of Kählerian and Sasakian space forms. When we restrict to invariant submanifolds of odd-dimensional unit spheres or invariant submanifolds of complex Euclidean space, one of the inequalities gives a positive answer to a conjecture, proposed by P. J. De Smet, F. Dillen, L. Verstraelen and L. Vrancken [Arch. Math., Brno 35, No. 2, 115–128 (1999; Zbl 1054.53075)].

MSC:
53C20 Global Riemannian geometry, including pinching
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