We give a formula for the Laplacian of the second fundamental form of an n-dimensional compact minimal submanifold M in a complex projective space CPm. As an application of this formula, we prove that M is a geodesic minimal hypersphere in CPm if the sectional curvature satisfies K≥1/n, if the normal connection is flat, and if M satisfies an additional condition which is automatically satisfied when M is a CR submanifold. We also prove that M is the complex projective space CPn/2 if K≥3/n, and if the normal connection of M is semi-flat.
