  We study two kinds of curvature invariants of Riemannian manifold equipped
with a distribution (for example, a CR-submanifold of an almost Hermitian
manifold) related to sets of pairwise orthogonal subspaces of the distribution:
one is similar to Chen's $\delta$-invariants and another kind of invariants is
based on the mutual curvature of the subspaces. We compare Chen-type invariants
with the mutual curvature invariants and prove geometric inequalities with
intermediate mean curvature squared for CR-submanifolds in almost Hermitian
spaces. In the case of a set of complex planes, we study curvature invariants
based on the concept of holomorphic bisectional curvature. As applications, we
give consequences of the absence of some $D$-minimal CR-submanifolds in almost
Hermitian manifolds.
