Geometric inequalities for CR-submanifolds

TL;DR

This paper investigates new curvature invariants for CR-submanifolds in almost Hermitian manifolds, establishing geometric inequalities and exploring their implications for the existence of certain minimal submanifolds.

Contribution

It introduces and compares new mutual curvature invariants with Chen-type invariants, deriving inequalities and studying their geometric and topological consequences.

Findings

Established inequalities relating mutual curvature and mean curvature squared.

Introduced curvature invariants based on holomorphic bisectional curvature.

Derived conditions for the non-existence of certain D-minimal 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.