Geometric inequalities for CR-submanifolds

Abstract: We study two kinds of curvature invariants of Riemannian manifold equip-ped with a complex distribution $D$ (for example, a CR-submanifold of an almost Hermitian manifold) related to sets of pairwise orthogonal subspaces of the distribution. One kind of invariant is based on the mutual curvature of the subspaces and another is similar to Chen's $\delta$-invariants. We compare the mutual curvature invariants with Chen-type 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 introduce and 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.