Chen-Ricci Inequality for Contact CR-Warped Product Submanifolds of Cosymplectic Space Forms with Applications in Geometry and Physics

In this paper, we prove the Chen-Ricci inequality for contact $CR$-warped products in the cosymplectic space forms, Theorem 5.1, which involves an intrinsic invariant (Ricci curvature) controlled by an extrinsic one (the mean curvature vector). This inequality is useful in both differential geometry and physics. In geometry, we apply it to get necessary conditions for the immersed submanifold to be minimal in a cosymplectic space form, which presents new answers for the well-known problem proposed by S.S. Chern, Problem 3. In physics, it enables us to derive some relations for the Dirichlet energy of the warping function controlled by some geometric invariants. In further research directions, we address a couple of open problems, namely Problem 4 and Problem 5, and a potential extension in the generalized contact metric manifolds.