Isometric immersions of space forms in space forms

Let M be a connected ^-dimensional space form isometrically immersed in a simply connected (2w-l)-dimensional space form of strictly larger curvature.If M is minimal, it is proven that it must be a piece of the flat Clifford torus in the (2w-l)-sphere.If M is complete and simply connected, it is proven that M possesses a global coordinate system whose coordinate vectors are unit-length asymptotic vectors.