Geometric Inequalities for a Submanifold Equipped with Distributions

The article introduces invariants of a Riemannian manifold related to the mutual curvature of several pairwise orthogonal subspaces of a tangent bundle. In the case of one-dimensional subspaces, this curvature is equal to half the scalar curvature of the subspace spanned by them, and in the case of complementary subspaces, this is the mixed scalar curvature. We compared our invariants with Chen invariants and proved geometric inequalities with intermediate mean curvature squared for a Riemannian submanifold. This gives sufficient conditions for the absence of minimal isometric immersions of Riemannian manifolds in a Euclidean space. As applications, geometric inequalities were obtained for isometric immersions of sub-Riemannian manifolds and Riemannian manifolds equipped with mutually orthogonal distributions.