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Let h be the second fundamental form of an n-dimensional minimal submanifold M of a unit sphere Sn+P (p > 2), S be the square of the length of h, and a(u) = ||/i(u,u)||2 for any unit vector u E TM. Simons proved that if S < n/(2 -1/p) on M, then either S = 0, or S = n/(2 -1/p). Chern, do Carmo, and Kobayashi determined all minimal submanifolds satisfying S = n/(2 -1/p). In this paper the analogous results for a(u) are obtained. It is proved that if a(u) < ¿, then either <r(u) = 0, or c(u) = J.doi:10.2307/2000649 fatcat:cx63qpvzl5f4pdof3jeehrpzpa