In 1973, Lawson and Simons conjectured that there are no stable currents in any compact, simply connectedRiemannian manifold M^m which is 1/4-pinched. In this paper, we regard M^m as a submanifold immersed in a Euclidean space and prove the conjecture under some pinched conditions about the sectional curvatures and the principal curvatures of M^m. We also show that there is no stable p-current in a submanifold of M^m and the p-th homology group vanishes when the shape operator of the submanifold satisfies certain conditions.
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