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We establish a nice orthonormal frame field on a closed surface minimally immersed in a unit sphere S n , under which the shape operators take very simple forms. Using this frame field, we obtain an interesting property K + K N  = 1 for the Gauss curvature K and the normal curvature K N  if the Gauss curvature is positive. Moreover, using this property we obtain the pinching on the intrinsic curvature and normal curvature, the pinching on the normal curvature, respectively.

Minimal surfaces in a unit sphere pinched by intrinsic curvature and normal curvature