On totally real 3-dimensional submanifolds of the nearly Kaehler 6-sphere

Dillen, F.; Opozda, B.; Verstraelen, L.; Vrancken, L.

doi:10.1090/S0002-9939-1987-0877050-8

On totally real $3$-dimensional submanifolds of the nearly Kaehler $6$-sphere
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by F. Dillen, B. Opozda, L. Verstraelen and L. Vrancken

Proc. Amer. Math. Soc. 99 (1987), 741-749

DOI: https://doi.org/10.1090/S0002-9939-1987-0877050-8

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Abstract:

Let $M$ be a compact $3$-dimensional totally real submanifold of the nearly Kaehler $6$-dimensional unit sphere. Let $K$ be the sectional curvature function of $M$. Then, if $K > 1/16$, $M$ is a totally geodesic submanifold (and $K \equiv 1$).

References

Norio Ejiri, Totally real submanifolds in a $6$-sphere, Proc. Amer. Math. Soc. 83 (1981), no. 4, 759–763. MR 630028, DOI 10.1090/S0002-9939-1981-0630028-6
Alfred Frölicher, Zur Differentialgeometrie der komplexen Strukturen, Math. Ann. 129 (1955), 50–95 (German). MR 68282, DOI 10.1007/BF01362360
Tetsuzo Fukami and Shigeru Ishihara, Almost Hermitian structure on $S^6$, Tohoku Math. J. (2) 7 (1955), 151–156. MR 77988, DOI 10.2748/tmj/1178245052
Alfred Gray, Minimal varieties and almost Hermitian submanifolds, Michigan Math. J. 12 (1965), 273–287. MR 184185
Antonio Ros, A characterization of seven compact Kaehler submanifolds by holomorphic pinching, Ann. of Math. (2) 121 (1985), no. 2, 377–382. MR 786353, DOI 10.2307/1971178
Kouei Sekigawa, Almost complex submanifolds of a $6$-dimensional sphere, Kodai Math. J. 6 (1983), no. 2, 174–185. MR 702939