Let \(M^m\) be a compact Riemannian manifold immersed in Euclidean space \(\mathbb R^{m+k}\). The main result of the paper is that there are no stable currents in \(M^m\) if one of the following pinching conditions holds: 1) The sectional curvature \(K\) satisfies \(K > (k/4)(\lambda_0 - \mu_0)^2\) where \(\lambda_0\) and \(\mu_0\) are the maximum and the minimum of the principal curvatures. 2) \(\lambda\mu > (1/4) (\lambda-\mu)^2\) where \(\lambda\) and \(\mu\) are any two principal curvatures. The author also recognizes \(M^m\) as sphere if a pinching condition in terms of the principal curvatures is satisfied and shows that the nonexistence of stable currents can be extended to suitable compact submanifolds of \(M^m\).