On the curvature of minimal submanifolds in a sphere

S. T. Yau proved inAmer. J. Math.97 (1975), p. 95, Theorem 15 that if the sectional curvature of ann-dimensional compact minimal submanifold in the (n + p)-dimensional unit sphere is everywhere greater than (p – 1)/(2p – 1), then this minimal submanifold is totally geodesic. In this note we improve this bound for the casep 2 to (3p – 2)/(6p).