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On some isotropic submanifolds in spheres. (English) Zbl 1007.53045

In a joint paper with P. Verheyen in Math. Ann. 269, 417-429 (1984; Zbl 0536.53053), B.-Y. Chen proved the following result: Let \(M^n\) be a connected compact isotropic submanifold of a sphere \(S^{n+p}\) of constant curvature \(\widetilde c\). If \(M^n\) has constant mean curvature \(H\), and if the sectional curvature function \(K\) of \(M^n\) satisfies \(K \geq (H^2+\widetilde c)/2\), then \(M^n\) is totally umbilical in \(S^{n+p}\). In the present paper the author constructs counterexamples to the above result.

MSC:

53C40 Global submanifolds

Citations:

Zbl 0536.53053
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References:

[1] O’Neill, B.: Isotropic and Kaehler immesions. Canadian J. Math., 17 , 905-915 (1965). · Zbl 0171.20503 · doi:10.4153/CJM-1965-086-7
[2] Shen, Y. B.: On isotropic submanifolds with constant mean curvature. Chinese Ann. Math. Ser. A, 5 , 117-122 (1984) (in Chinese). · Zbl 0518.53056
[3] Takahashi, T.: Minimal immersions of Riemannian manifolds. J. Math. Soc. Japan, 18 , 380-385 (1966). · Zbl 0145.18601 · doi:10.2969/jmsj/01840380
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