53C21, 53C24, 49K35.

In the theory of minimal submanifolds, the following problem is fundamental: when does a given Riemannian manifold admit (or does not admit) a minimal isometric immersion into an Euclidean space of arbitrary dimension? S.S. Chern, in his monograph [6] Minimal submanifolds in a Riemannian manifold, remarked that the result of Takahashi (the Ricci tensor of a minimal submanifold into a Euclidean space is negative semidefinite) was the only known Riemannian obstruction to minimal isometric immersions in Euclidean spaces. A second obstruction was obtained by B.Y. Chen as an immediate application of his fundamental inequality [1]: the scalar curvature and the sectional curvature of a minimal submanifold into a Euclidean space satisfies the inequality We find a new relation between the Chen invariant, the dimension of the submanifold, the length of the mean curvature vector field and a deviation parameter. This result implies a new obstruction: the sectional curvature of a minimal submanifold into a Euclidean space also satisfies the inequality