Geodesic
Normal curvature of minimal submanifolds in a sphere
Glasgow mathematical journal, Tome 39 (1997) no. 1, pp. 29-33

Voir la notice de l'article provenant de la source Cambridge University Press

Simons [5] has proved a pinching theorem for compact minimal submanifolds in a unit sphere, which led to an intrinsic rigidity result. Sakaki [4] improved this result of Simons for arbitrary codimension and has proved that if the scalar curvature S of the minimal submanifold Mn of Sn+P satisfiesthen either Mn is totally geodesic or S= 2/3 in which case n = 2 and M2 is the Veronese surface in a totally geodesic 4-sphere. This result of Sakaki was further improved by Shen [6] but only for dimension n=3, where it is shown that if S>4, then M3 is totally geodesic (cf. Theorem 3, p. 791). 
Deshmukh, Sharief. Normal curvature of minimal submanifolds in a sphere. Glasgow mathematical journal, Tome 39 (1997) no. 1, pp. 29-33. doi: 10.1017/S0017089500031876
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